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Theorem chscllem3 25047
Description: Lemma for chscl 25049. (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 5696 . . . . . . . 8  |-  ( n  =  N  ->  ( H `  n )  =  ( H `  N ) )
32fveq2d 5700 . . . . . . 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 5706 . . . . . . 7  |-  ( (
proj h `  A ) `
 ( H `  N ) )  e. 
_V
63, 4, 5fvmpt 5779 . . . . . 6  |-  ( N  e.  NN  ->  ( F `  N )  =  ( ( proj h `  A ) `  ( H `  N
) ) )
71, 6syl 16 . . . . 5  |-  ( ph  ->  ( F `  N
)  =  ( (
proj h `  A ) `
 ( H `  N ) ) )
87eqcomd 2448 . . . 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 24632 . . . . . . . . 9  |-  ( B  e.  CH  ->  B  e.  SH )
1210, 11syl 16 . . . . . . . 8  |-  ( ph  ->  B  e.  SH )
13 chsh 24632 . . . . . . . . . 10  |-  ( A  e.  CH  ->  A  e.  SH )
149, 13syl 16 . . . . . . . . 9  |-  ( ph  ->  A  e.  SH )
15 shocsh 24692 . . . . . . . . 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 24767 . . . . . . . 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 1221 . . . . . . 7  |-  ( ph  ->  ( B  +H  A
)  C_  ( ( _|_ `  A )  +H  A ) )
20 shscom 24727 . . . . . . . 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 24727 . . . . . . . 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 3404 . . . . . 6  |-  ( ph  ->  ( A  +H  B
)  C_  ( A  +H  ( _|_ `  A
) ) )
25 chscl.4 . . . . . . 7  |-  ( ph  ->  H : NN --> ( A  +H  B ) )
2625, 1ffvelrnd 5849 . . . . . 6  |-  ( ph  ->  ( H `  N
)  e.  ( A  +H  B ) )
2724, 26sseldd 3362 . . . . 5  |-  ( ph  ->  ( H `  N
)  e.  ( A  +H  ( _|_ `  A
) ) )
28 pjpreeq 24806 . . . . 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 24704 . . . . . 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 3362 . . . 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 25045 . . . . . 6  |-  ( ph  ->  F : NN --> A )
4544, 1ffvelrnd 5849 . . . . 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 755 . . . 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 756 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( _|_ `  A
)  /\  ( H `  N )  =  ( ( F `  N
)  +h  z ) ) )  ->  ( H `  N )  =  ( ( F `
 N )  +h  z ) )
5149, 50eqtr3d 2477 . . . 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 24708 . . 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 2850 1  |-  ( ph  ->  C  =  ( F `
 N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2721    i^i cin 3332    C_ wss 3333   class class class wbr 4297    e. cmpt 4355   -->wf 5419   ` cfv 5423  (class class class)co 6096   NNcn 10327    +h cva 24327    ~~>v chli 24334   SHcsh 24335   CHcch 24336   _|_cort 24337    +H cph 24338   0Hc0h 24342   proj hcpjh 24344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-hilex 24406  ax-hfvadd 24407  ax-hvcom 24408  ax-hvass 24409  ax-hv0cl 24410  ax-hvaddid 24411  ax-hfvmul 24412  ax-hvmulid 24413  ax-hvmulass 24414  ax-hvdistr1 24415  ax-hvdistr2 24416  ax-hvmul0 24417  ax-hfi 24486  ax-his2 24490  ax-his3 24491  ax-his4 24492
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-grpo 23683  df-ablo 23774  df-hvsub 24378  df-sh 24614  df-ch 24629  df-oc 24660  df-ch0 24661  df-shs 24716  df-pjh 24803
This theorem is referenced by:  chscllem4  25048
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