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Theorem isnumbasgrplem2 27237
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 27233 . . 3  |-  Base  Fn  _V
2 ssv 3360 . . 3  |-  Grp  C_  _V
3 fvelimab 5774 . . 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 654 . 2  |-  ( ( S  u.  (har `  S ) )  e.  ( Base " Grp ) 
<->  E. x  e.  Grp  ( Base `  x )  =  ( S  u.  (har `  S ) ) )
5 harcl 7521 . . . . . 6  |-  (har `  S )  e.  On
6 onenon 7828 . . . . . 6  |-  ( (har
`  S )  e.  On  ->  (har `  S
)  e.  dom  card )
75, 6ax-mp 8 . . . . 5  |-  (har `  S )  e.  dom  card
8 xpnum 7830 . . . . 5  |-  ( ( (har `  S )  e.  dom  card  /\  (har `  S )  e.  dom  card )  ->  ( (har `  S )  X.  (har `  S ) )  e. 
dom  card )
97, 7, 8mp2an 654 . . . 4  |-  ( (har
`  S )  X.  (har `  S )
)  e.  dom  card
10 ssun1 3502 . . . . . . . 8  |-  S  C_  ( S  u.  (har `  S ) )
11 simpr 448 . . . . . . . 8  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  ( Base `  x )  =  ( S  u.  (har `  S ) ) )
1210, 11syl5sseqr 3389 . . . . . . 7  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  C_  ( Base `  x ) )
13 fvex 5734 . . . . . . . 8  |-  ( Base `  x )  e.  _V
1413ssex 4339 . . . . . . 7  |-  ( S 
C_  ( Base `  x
)  ->  S  e.  _V )
1512, 14syl 16 . . . . . 6  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  e.  _V )
167a1i 11 . . . . . 6  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  (har `  S
)  e.  dom  card )
17 simp1l 981 . . . . . . . 8  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  x  e.  Grp )
18123ad2ant1 978 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  S  C_  ( Base `  x
) )
19 simp2 958 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  a  e.  S )
2018, 19sseldd 3341 . . . . . . . 8  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  a  e.  ( Base `  x
) )
21 ssun2 3503 . . . . . . . . . . 11  |-  (har `  S )  C_  ( S  u.  (har `  S
) )
2221, 11syl5sseqr 3389 . . . . . . . . . 10  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  (har `  S
)  C_  ( Base `  x ) )
23223ad2ant1 978 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  (har `  S )  C_  ( Base `  x ) )
24 simp3 959 . . . . . . . . 9  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  c  e.  (har `  S )
)
2523, 24sseldd 3341 . . . . . . . 8  |-  ( ( ( x  e.  Grp  /\  ( Base `  x
)  =  ( S  u.  (har `  S
) ) )  /\  a  e.  S  /\  c  e.  (har `  S
) )  ->  c  e.  ( Base `  x
) )
26 eqid 2435 . . . . . . . . 9  |-  ( Base `  x )  =  (
Base `  x )
27 eqid 2435 . . . . . . . . 9  |-  ( +g  `  x )  =  ( +g  `  x )
2826, 27grpcl 14810 . . . . . . . 8  |-  ( ( x  e.  Grp  /\  a  e.  ( Base `  x )  /\  c  e.  ( Base `  x
) )  ->  (
a ( +g  `  x
) c )  e.  ( Base `  x
) )
2917, 20, 25, 28syl3anc 1184 . . . . . . 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 982 . . . . . . 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 2511 . . . . . 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 735 . . . . . . 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 707 . . . . . . . 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 733 . . . . . . . 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 3341 . . . . . . 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 734 . . . . . . . 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 3341 . . . . . . 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 707 . . . . . . . 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 732 . . . . . . . 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 3341 . . . . . . 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 14849 . . . . . . 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 1186 . . . . . 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 735 . . . . . . 7  |-  ( ( ( ( x  e. 
Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  /\  b  e.  (har
`  S ) )  /\  ( a  e.  S  /\  d  e.  S ) )  ->  x  e.  Grp )
4412ad2antrr 707 . . . . . . . 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 734 . . . . . . . 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 3341 . . . . . . 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 733 . . . . . . . 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 3341 . . . . . . 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 707 . . . . . . . 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 732 . . . . . . . 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 3341 . . . . . . 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 14830 . . . . . . 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 1186 . . . . . 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 7524 . . . . . . 7  |-  -.  (har `  S )  ~<_  S
5554a1i 11 . . . . . 6  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  -.  (har `  S )  ~<_  S )
5615, 16, 16, 31, 42, 53, 55unxpwdom3 27224 . . . . 5  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  ~<_*  ( (har `  S
)  X.  (har `  S ) ) )
57 wdomnumr 7937 . . . . . 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 189 . . . 4  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  ~<_  ( (har
`  S )  X.  (har `  S )
) )
60 numdom 7911 . . . 4  |-  ( ( ( (har `  S
)  X.  (har `  S ) )  e. 
dom  card  /\  S  ~<_  ( (har
`  S )  X.  (har `  S )
) )  ->  S  e.  dom  card )
619, 59, 60sylancr 645 . . 3  |-  ( ( x  e.  Grp  /\  ( Base `  x )  =  ( S  u.  (har `  S ) ) )  ->  S  e.  dom  card )
6261rexlimiva 2817 . 2  |-  ( E. x  e.  Grp  ( Base `  x )  =  ( S  u.  (har `  S ) )  ->  S  e.  dom  card )
634, 62sylbi 188 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2698   _Vcvv 2948    u. cun 3310    C_ wss 3312   class class class wbr 4204   Oncon0 4573    X. cxp 4868   dom cdm 4870   "cima 4873    Fn wfn 5441   ` cfv 5446  (class class class)co 6073    ~<_ cdom 7099  harchar 7516    ~<_* cwdom 7517   cardccrd 7814   Basecbs 13461   +g cplusg 13521   Grpcgrp 14677
This theorem is referenced by:  isnumbasabl  27239  isnumbasgrp  27240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-omul 6721  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-oi 7471  df-har 7518  df-wdom 7519  df-card 7818  df-acn 7821  df-slot 13465  df-base 13466  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805
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