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Theorem chscllem3 23098
Description: Lemma for chscl 23100. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
chscl.1  |-  ( ph  ->  A  e.  CH )
chscl.2  |-  ( ph  ->  B  e.  CH )
chscl.3  |-  ( ph  ->  B  C_  ( _|_ `  A ) )
chscl.4  |-  ( ph  ->  H : NN --> ( A  +H  B ) )
chscl.5  |-  ( ph  ->  H  ~~>v  u )
chscl.6  |-  F  =  ( n  e.  NN  |->  ( ( proj  h `  A ) `  ( H `  n )
) )
chscllem3.7  |-  ( ph  ->  N  e.  NN )
chscllem3.8  |-  ( ph  ->  C  e.  A )
chscllem3.9  |-  ( ph  ->  D  e.  B )
chscllem3.10  |-  ( ph  ->  ( H `  N
)  =  ( C  +h  D ) )
Assertion
Ref Expression
chscllem3  |-  ( ph  ->  C  =  ( F `
 N ) )
Distinct variable groups:    u, n, A    ph, n    B, n, u    n, H, u    n, N
Allowed substitution hints:    ph( u)    C( u, n)    D( u, n)    F( u, n)    N( u)

Proof of Theorem chscllem3
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 chscllem3.7 . . . . . 6  |-  ( ph  ->  N  e.  NN )
2 fveq2 5691 . . . . . . . 8  |-  ( n  =  N  ->  ( H `  n )  =  ( H `  N ) )
32fveq2d 5695 . . . . . . 7  |-  ( n  =  N  ->  (
( proj  h `  A
) `  ( H `  n ) )  =  ( ( proj  h `  A ) `  ( H `  N )
) )
4 chscl.6 . . . . . . 7  |-  F  =  ( n  e.  NN  |->  ( ( proj  h `  A ) `  ( H `  n )
) )
5 fvex 5705 . . . . . . 7  |-  ( (
proj  h `  A ) `
 ( H `  N ) )  e. 
_V
63, 4, 5fvmpt 5769 . . . . . 6  |-  ( N  e.  NN  ->  ( F `  N )  =  ( ( proj 
h `  A ) `  ( H `  N
) ) )
71, 6syl 16 . . . . 5  |-  ( ph  ->  ( F `  N
)  =  ( (
proj  h `  A ) `
 ( H `  N ) ) )
87eqcomd 2413 . . . 4  |-  ( ph  ->  ( ( proj  h `  A ) `  ( H `  N )
)  =  ( F `
 N ) )
9 chscl.1 . . . . 5  |-  ( ph  ->  A  e.  CH )
10 chscl.2 . . . . . . . . 9  |-  ( ph  ->  B  e.  CH )
11 chsh 22684 . . . . . . . . 9  |-  ( B  e.  CH  ->  B  e.  SH )
1210, 11syl 16 . . . . . . . 8  |-  ( ph  ->  B  e.  SH )
13 chsh 22684 . . . . . . . . . 10  |-  ( A  e.  CH  ->  A  e.  SH )
149, 13syl 16 . . . . . . . . 9  |-  ( ph  ->  A  e.  SH )
15 shocsh 22743 . . . . . . . . 9  |-  ( A  e.  SH  ->  ( _|_ `  A )  e.  SH )
1614, 15syl 16 . . . . . . . 8  |-  ( ph  ->  ( _|_ `  A
)  e.  SH )
17 chscl.3 . . . . . . . 8  |-  ( ph  ->  B  C_  ( _|_ `  A ) )
18 shless 22818 . . . . . . . 8  |-  ( ( ( B  e.  SH  /\  ( _|_ `  A
)  e.  SH  /\  A  e.  SH )  /\  B  C_  ( _|_ `  A ) )  -> 
( B  +H  A
)  C_  ( ( _|_ `  A )  +H  A ) )
1912, 16, 14, 17, 18syl31anc 1187 . . . . . . 7  |-  ( ph  ->  ( B  +H  A
)  C_  ( ( _|_ `  A )  +H  A ) )
20 shscom 22778 . . . . . . . 8  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B
)  =  ( B  +H  A ) )
2114, 12, 20syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( A  +H  B
)  =  ( B  +H  A ) )
22 shscom 22778 . . . . . . . 8  |-  ( ( A  e.  SH  /\  ( _|_ `  A )  e.  SH )  -> 
( A  +H  ( _|_ `  A ) )  =  ( ( _|_ `  A )  +H  A
) )
2314, 16, 22syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( A  +H  ( _|_ `  A ) )  =  ( ( _|_ `  A )  +H  A
) )
2419, 21, 233sstr4d 3355 . . . . . 6  |-  ( ph  ->  ( A  +H  B
)  C_  ( A  +H  ( _|_ `  A
) ) )
25 chscl.4 . . . . . . 7  |-  ( ph  ->  H : NN --> ( A  +H  B ) )
2625, 1ffvelrnd 5834 . . . . . 6  |-  ( ph  ->  ( H `  N
)  e.  ( A  +H  B ) )
2724, 26sseldd 3313 . . . . 5  |-  ( ph  ->  ( H `  N
)  e.  ( A  +H  ( _|_ `  A
) ) )
28 pjpreeq 22857 . . . . 5  |-  ( ( A  e.  CH  /\  ( H `  N )  e.  ( A  +H  ( _|_ `  A ) ) )  ->  (
( ( proj  h `  A ) `  ( H `  N )
)  =  ( F `
 N )  <->  ( ( F `  N )  e.  A  /\  E. z  e.  ( _|_ `  A
) ( H `  N )  =  ( ( F `  N
)  +h  z ) ) ) )
299, 27, 28syl2anc 643 . . . 4  |-  ( ph  ->  ( ( ( proj 
h `  A ) `  ( H `  N
) )  =  ( F `  N )  <-> 
( ( F `  N )  e.  A  /\  E. z  e.  ( _|_ `  A ) ( H `  N
)  =  ( ( F `  N )  +h  z ) ) ) )
308, 29mpbid 202 . . 3  |-  ( ph  ->  ( ( F `  N )  e.  A  /\  E. z  e.  ( _|_ `  A ) ( H `  N
)  =  ( ( F `  N )  +h  z ) ) )
3130simprd 450 . 2  |-  ( ph  ->  E. z  e.  ( _|_ `  A ) ( H `  N
)  =  ( ( F `  N )  +h  z ) )
3214adantr 452 . . . 4  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  A  e.  SH )
3316adantr 452 . . . 4  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( _|_ `  A )  e.  SH )
34 ocin 22755 . . . . . 6  |-  ( A  e.  SH  ->  ( A  i^i  ( _|_ `  A
) )  =  0H )
3514, 34syl 16 . . . . 5  |-  ( ph  ->  ( A  i^i  ( _|_ `  A ) )  =  0H )
3635adantr 452 . . . 4  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( A  i^i  ( _|_ `  A
) )  =  0H )
37 chscllem3.8 . . . . 5  |-  ( ph  ->  C  e.  A )
3837adantr 452 . . . 4  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  C  e.  A )
3917adantr 452 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  B  C_  ( _|_ `  A
) )
40 chscllem3.9 . . . . . 6  |-  ( ph  ->  D  e.  B )
4140adantr 452 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  D  e.  B )
4239, 41sseldd 3313 . . . 4  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  D  e.  ( _|_ `  A
) )
43 chscl.5 . . . . . . 7  |-  ( ph  ->  H  ~~>v  u )
449, 10, 17, 25, 43, 4chscllem1 23096 . . . . . 6  |-  ( ph  ->  F : NN --> A )
4544, 1ffvelrnd 5834 . . . . 5  |-  ( ph  ->  ( F `  N
)  e.  A )
4645adantr 452 . . . 4  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( F `  N )  e.  A )
47 simprl 733 . . . 4  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  z  e.  ( _|_ `  A
) )
48 chscllem3.10 . . . . . 6  |-  ( ph  ->  ( H `  N
)  =  ( C  +h  D ) )
4948adantr 452 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( H `  N )  =  ( C  +h  D ) )
50 simprr 734 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( H `  N )  =  ( ( F `
 N )  +h  z ) )
5149, 50eqtr3d 2442 . . . 4  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( C  +h  D )  =  ( ( F `  N )  +h  z
) )
5232, 33, 36, 38, 42, 46, 47, 51shuni 22759 . . 3  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( C  =  ( F `  N )  /\  D  =  z ) )
5352simpld 446 . 2  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  C  =  ( F `  N ) )
5431, 53rexlimddv 2798 1  |-  ( ph  ->  C  =  ( F `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2671    i^i cin 3283    C_ wss 3284   class class class wbr 4176    e. cmpt 4230   -->wf 5413   ` cfv 5417  (class class class)co 6044   NNcn 9960    +h cva 22380    ~~>v chli 22387   SHcsh 22388   CHcch 22389   _|_cort 22390    +H cph 22391   0Hc0h 22395   proj 
hcpjh 22397
This theorem is referenced by:  chscllem4  23099
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-hilex 22459  ax-hfvadd 22460  ax-hvcom 22461  ax-hvass 22462  ax-hv0cl 22463  ax-hvaddid 22464  ax-hfvmul 22465  ax-hvmulid 22466  ax-hvmulass 22467  ax-hvdistr1 22468  ax-hvdistr2 22469  ax-hvmul0 22470  ax-hfi 22538  ax-his2 22542  ax-his3 22543  ax-his4 22544
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-po 4467  df-so 4468  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-riota 6512  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-grpo 21736  df-ablo 21827  df-hvsub 22431  df-sh 22666  df-ch 22681  df-oc 22711  df-ch0 22712  df-shs 22767  df-pjh 22854
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