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Theorem chscllem3 26683
Description: Lemma for chscl 26685. (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 5872 . . . . . . . 8  |-  ( n  =  N  ->  ( H `  n )  =  ( H `  N ) )
32fveq2d 5876 . . . . . . 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 5882 . . . . . . 7  |-  ( (
proj h `  A ) `
 ( H `  N ) )  e. 
_V
63, 4, 5fvmpt 5956 . . . . . 6  |-  ( N  e.  NN  ->  ( F `  N )  =  ( ( proj h `  A ) `  ( H `  N
) ) )
71, 6syl 16 . . . . 5  |-  ( ph  ->  ( F `  N
)  =  ( (
proj h `  A ) `
 ( H `  N ) ) )
87eqcomd 2465 . . . 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 26268 . . . . . . . . 9  |-  ( B  e.  CH  ->  B  e.  SH )
1210, 11syl 16 . . . . . . . 8  |-  ( ph  ->  B  e.  SH )
13 chsh 26268 . . . . . . . . . 10  |-  ( A  e.  CH  ->  A  e.  SH )
149, 13syl 16 . . . . . . . . 9  |-  ( ph  ->  A  e.  SH )
15 shocsh 26328 . . . . . . . . 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 26403 . . . . . . . 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 1231 . . . . . . 7  |-  ( ph  ->  ( B  +H  A
)  C_  ( ( _|_ `  A )  +H  A ) )
20 shscom 26363 . . . . . . . 8  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B
)  =  ( B  +H  A ) )
2114, 12, 20syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( A  +H  B
)  =  ( B  +H  A ) )
22 shscom 26363 . . . . . . . 8  |-  ( ( A  e.  SH  /\  ( _|_ `  A )  e.  SH )  -> 
( A  +H  ( _|_ `  A ) )  =  ( ( _|_ `  A )  +H  A
) )
2314, 16, 22syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( A  +H  ( _|_ `  A ) )  =  ( ( _|_ `  A )  +H  A
) )
2419, 21, 233sstr4d 3542 . . . . . 6  |-  ( ph  ->  ( A  +H  B
)  C_  ( A  +H  ( _|_ `  A
) ) )
25 chscl.4 . . . . . . 7  |-  ( ph  ->  H : NN --> ( A  +H  B ) )
2625, 1ffvelrnd 6033 . . . . . 6  |-  ( ph  ->  ( H `  N
)  e.  ( A  +H  B ) )
2724, 26sseldd 3500 . . . . 5  |-  ( ph  ->  ( H `  N
)  e.  ( A  +H  ( _|_ `  A
) ) )
28 pjpreeq 26442 . . . . 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 661 . . . 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 210 . . 3  |-  ( ph  ->  ( ( F `  N )  e.  A  /\  E. z  e.  ( _|_ `  A ) ( H `  N
)  =  ( ( F `  N )  +h  z ) ) )
3130simprd 463 . 2  |-  ( ph  ->  E. z  e.  ( _|_ `  A ) ( H `  N
)  =  ( ( F `  N )  +h  z ) )
3214adantr 465 . . . 4  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  A  e.  SH )
3316adantr 465 . . . 4  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( _|_ `  A )  e.  SH )
34 ocin 26340 . . . . . 6  |-  ( A  e.  SH  ->  ( A  i^i  ( _|_ `  A
) )  =  0H )
3514, 34syl 16 . . . . 5  |-  ( ph  ->  ( A  i^i  ( _|_ `  A ) )  =  0H )
3635adantr 465 . . . 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 465 . . . 4  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  C  e.  A )
3917adantr 465 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  B  C_  ( _|_ `  A
) )
40 chscllem3.9 . . . . . 6  |-  ( ph  ->  D  e.  B )
4140adantr 465 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  D  e.  B )
4239, 41sseldd 3500 . . . 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 26681 . . . . . 6  |-  ( ph  ->  F : NN --> A )
4544, 1ffvelrnd 6033 . . . . 5  |-  ( ph  ->  ( F `  N
)  e.  A )
4645adantr 465 . . . 4  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( F `  N )  e.  A )
47 simprl 756 . . . 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 465 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( H `  N )  =  ( C  +h  D ) )
50 simprr 757 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( H `  N )  =  ( ( F `
 N )  +h  z ) )
5149, 50eqtr3d 2500 . . . 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 26344 . . 3  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( C  =  ( F `  N )  /\  D  =  z ) )
5352simpld 459 . 2  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  C  =  ( F `  N ) )
5431, 53rexlimddv 2953 1  |-  ( ph  ->  C  =  ( F `
 N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808    i^i cin 3470    C_ wss 3471   class class class wbr 4456    |-> cmpt 4515   -->wf 5590   ` cfv 5594  (class class class)co 6296   NNcn 10556    +h cva 25963    ~~>v chli 25970   SHcsh 25971   CHcch 25972   _|_cort 25973    +H cph 25974   0Hc0h 25978   proj hcpjh 25980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-hilex 26042  ax-hfvadd 26043  ax-hvcom 26044  ax-hvass 26045  ax-hv0cl 26046  ax-hvaddid 26047  ax-hfvmul 26048  ax-hvmulid 26049  ax-hvmulass 26050  ax-hvdistr1 26051  ax-hvdistr2 26052  ax-hvmul0 26053  ax-hfi 26122  ax-his2 26126  ax-his3 26127  ax-his4 26128
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-grpo 25319  df-ablo 25410  df-hvsub 26014  df-sh 26250  df-ch 26265  df-oc 26296  df-ch0 26297  df-shs 26352  df-pjh 26439
This theorem is referenced by:  chscllem4  26684
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