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Theorem isnumbasgrplem2 27372
Description: If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.)
Assertion
Ref Expression
isnumbasgrplem2  |-  ( ( S  u.  (har `  S ) )  e.  ( Base " Grp )  ->  S  e.  dom  card )

Proof of Theorem isnumbasgrplem2
Dummy variables  a 
b  c  d  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 basfn 27368 . . 3  |-  Base  Fn  _V
2 ssv 3211 . . 3  |-  Grp  C_  _V
3 fvelimab 5594 . . 3  |-  ( (
Base  Fn  _V  /\  Grp  C_ 
_V )  ->  (
( S  u.  (har `  S ) )  e.  ( Base " Grp ) 
<->  E. x  e.  Grp  ( Base `  x )  =  ( S  u.  (har `  S ) ) ) )
41, 2, 3mp2an 653 . 2  |-  ( ( S  u.  (har `  S ) )  e.  ( Base " Grp ) 
<->  E. x  e.  Grp  ( Base `  x )  =  ( S  u.  (har `  S ) ) )
5 harcl 7291 . . . . . 6  |-  (har `  S )  e.  On
6 onenon 7598 . . . . . 6  |-  ( (har
`  S )  e.  On  ->  (har `  S
)  e.  dom  card )
75, 6ax-mp 8 . . . . 5  |-  (har `  S )  e.  dom  card
8 xpnum 7600 . . . . 5  |-  ( ( (har `  S )  e.  dom  card  /\  (har `  S )  e.  dom  card )  ->  ( (har `  S )  X.  (har `  S ) )  e. 
dom  card )
97, 7, 8mp2an 653 . . . 4  |-  ( (har
`  S )  X.  (har `  S )
)  e.  dom  card
10 ssun1 3351 . . . . . . . 8  |-  S  C_  ( S  u.  (har `  S ) )
11 simpr 447 . . . . . . . 8  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  ( Base `  x )  =  ( S  u.  (har `  S ) ) )
1210, 11syl5sseqr 3240 . . . . . . 7  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  C_  ( Base `  x ) )
13 fvex 5555 . . . . . . . 8  |-  ( Base `  x )  e.  _V
1413ssex 4174 . . . . . . 7  |-  ( S 
C_  ( Base `  x
)  ->  S  e.  _V )
1512, 14syl 15 . . . . . 6  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  e.  _V )
167a1i 10 . . . . . 6  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  (har `  S
)  e.  dom  card )
17 simp1l 979 . . . . . . . 8  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  x  e.  Grp )
18123ad2ant1 976 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  S  C_  ( Base `  x
) )
19 simp2 956 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  a  e.  S )
2018, 19sseldd 3194 . . . . . . . 8  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  a  e.  ( Base `  x
) )
21 ssun2 3352 . . . . . . . . . . 11  |-  (har `  S )  C_  ( S  u.  (har `  S
) )
2221, 11syl5sseqr 3240 . . . . . . . . . 10  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  (har `  S
)  C_  ( Base `  x ) )
23223ad2ant1 976 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  (har `  S )  C_  ( Base `  x ) )
24 simp3 957 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  c  e.  (har `  S )
)
2523, 24sseldd 3194 . . . . . . . 8  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  c  e.  ( Base `  x
) )
26 eqid 2296 . . . . . . . . 9  |-  ( Base `  x )  =  (
Base `  x )
27 eqid 2296 . . . . . . . . 9  |-  ( +g  `  x )  =  ( +g  `  x )
2826, 27grpcl 14511 . . . . . . . 8  |-  ( ( x  e.  Grp  /\  a  e.  ( Base `  x )  /\  c  e.  ( Base `  x
) )  ->  (
a ( +g  `  x
) c )  e.  ( Base `  x
) )
2917, 20, 25, 28syl3anc 1182 . . . . . . 7  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  (
a ( +g  `  x
) c )  e.  ( Base `  x
) )
30 simp1r 980 . . . . . . 7  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  ( Base `  x )  =  ( S  u.  (har `  S ) ) )
3129, 30eleqtrd 2372 . . . . . 6  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  (
a ( +g  `  x
) c )  e.  ( S  u.  (har `  S ) ) )
32 simplll 734 . . . . . . 7  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  x  e.  Grp )
3322ad2antrr 706 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  (har `  S
)  C_  ( Base `  x ) )
34 simprl 732 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  c  e.  (har
`  S ) )
3533, 34sseldd 3194 . . . . . . 7  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  c  e.  (
Base `  x )
)
36 simprr 733 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  d  e.  (har
`  S ) )
3733, 36sseldd 3194 . . . . . . 7  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  d  e.  (
Base `  x )
)
3812ad2antrr 706 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  S  C_  ( Base `  x ) )
39 simplr 731 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  a  e.  S
)
4038, 39sseldd 3194 . . . . . . 7  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  a  e.  (
Base `  x )
)
4126, 27grplcan 14550 . . . . . . 7  |-  ( ( x  e.  Grp  /\  ( c  e.  (
Base `  x )  /\  d  e.  ( Base `  x )  /\  a  e.  ( Base `  x ) ) )  ->  ( ( a ( +g  `  x
) c )  =  ( a ( +g  `  x ) d )  <-> 
c  =  d ) )
4232, 35, 37, 40, 41syl13anc 1184 . . . . . 6  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  a  e.  S
)  /\  ( c  e.  (har `  S )  /\  d  e.  (har `  S ) ) )  ->  ( ( a ( +g  `  x
) c )  =  ( a ( +g  `  x ) d )  <-> 
c  =  d ) )
43 simplll 734 . . . . . . 7  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  ->  x  e.  Grp )
4412ad2antrr 706 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  ->  S  C_  ( Base `  x
) )
45 simprr 733 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  -> 
d  e.  S )
4644, 45sseldd 3194 . . . . . . 7  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  -> 
d  e.  ( Base `  x ) )
47 simprl 732 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  -> 
a  e.  S )
4844, 47sseldd 3194 . . . . . . 7  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  -> 
a  e.  ( Base `  x ) )
4922ad2antrr 706 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  -> 
(har `  S )  C_  ( Base `  x
) )
50 simplr 731 . . . . . . . 8  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  -> 
b  e.  (har `  S ) )
5149, 50sseldd 3194 . . . . . . 7  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  -> 
b  e.  ( Base `  x ) )
5226, 27grprcan 14531 . . . . . . 7  |-  ( ( x  e.  Grp  /\  ( d  e.  (
Base `  x )  /\  a  e.  ( Base `  x )  /\  b  e.  ( Base `  x ) ) )  ->  ( ( d ( +g  `  x
) b )  =  ( a ( +g  `  x ) b )  <-> 
d  =  a ) )
5343, 46, 48, 51, 52syl13anc 1184 . . . . . 6  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  -> 
( ( d ( +g  `  x ) b )  =  ( a ( +g  `  x
) b )  <->  d  =  a ) )
54 harndom 7294 . . . . . . 7  |-  -.  (har `  S )  ~<_  S
5554a1i 10 . . . . . 6  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  -.  (har `  S )  ~<_  S )
5615, 16, 16, 31, 42, 53, 55unxpwdom3 27359 . . . . 5  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  ~<_*  ( (har `  S
)  X.  (har `  S ) ) )
57 wdomnumr 7707 . . . . . 6  |-  ( ( (har `  S )  X.  (har `  S )
)  e.  dom  card  -> 
( S  ~<_*  ( (har `  S
)  X.  (har `  S ) )  <->  S  ~<_  ( (har
`  S )  X.  (har `  S )
) ) )
589, 57ax-mp 8 . . . . 5  |-  ( S  ~<_*  ( (har `  S )  X.  (har `  S )
)  <->  S  ~<_  ( (har `  S )  X.  (har `  S ) ) )
5956, 58sylib 188 . . . 4  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  ~<_  ( (har
`  S )  X.  (har `  S )
) )
60 numdom 7681 . . . 4  |-  ( ( ( (har `  S
)  X.  (har `  S ) )  e. 
dom  card  /\  S  ~<_  ( (har
`  S )  X.  (har `  S )
) )  ->  S  e.  dom  card )
619, 59, 60sylancr 644 . . 3  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  e.  dom  card )
6261rexlimiva 2675 . 2  |-  ( E. x  e.  Grp  ( Base `  x )  =  ( S  u.  (har `  S ) )  ->  S  e.  dom  card )
634, 62sylbi 187 1  |-  ( ( S  u.  (har `  S ) )  e.  ( Base " Grp )  ->  S  e.  dom  card )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    u. cun 3163    C_ wss 3165   class class class wbr 4039   Oncon0 4408    X. cxp 4703   dom cdm 4705   "cima 4708    Fn wfn 5266   ` cfv 5271  (class class class)co 5874    ~<_ cdom 6877  harchar 7286    ~<_* cwdom 7287   cardccrd 7584   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378
This theorem is referenced by:  isnumbasabl  27374  isnumbasgrp  27375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-oi 7241  df-har 7288  df-wdom 7289  df-card 7588  df-acn 7591  df-slot 13168  df-base 13169  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506
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