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Theorem chscllem1 22064
Description: Lemma for chscl 22068. (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 )
) )
Assertion
Ref Expression
chscllem1  |-  ( ph  ->  F : NN --> A )
Distinct variable groups:    u, n, A    ph, n    B, n, u    n, H, u
Allowed substitution hints:    ph( u)    F( u, n)

Proof of Theorem chscllem1
StepHypRef Expression
1 eqid 2253 . . . 4  |-  ( (
proj  h `  A ) `
 ( H `  n ) )  =  ( ( proj  h `  A ) `  ( H `  n )
)
2 chscl.1 . . . . . 6  |-  ( ph  ->  A  e.  CH )
32adantr 453 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  A  e. 
CH )
4 chscl.4 . . . . . . 7  |-  ( ph  ->  H : NN --> ( A  +H  B ) )
5 ffvelrn 5515 . . . . . . 7  |-  ( ( H : NN --> ( A  +H  B )  /\  n  e.  NN )  ->  ( H `  n
)  e.  ( A  +H  B ) )
64, 5sylan 459 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( H `
 n )  e.  ( A  +H  B
) )
7 chscl.2 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CH )
8 chsh 21634 . . . . . . . . . 10  |-  ( B  e.  CH  ->  B  e.  SH )
97, 8syl 17 . . . . . . . . 9  |-  ( ph  ->  B  e.  SH )
10 chsh 21634 . . . . . . . . . . 11  |-  ( A  e.  CH  ->  A  e.  SH )
112, 10syl 17 . . . . . . . . . 10  |-  ( ph  ->  A  e.  SH )
12 shocsh 21693 . . . . . . . . . 10  |-  ( A  e.  SH  ->  ( _|_ `  A )  e.  SH )
1311, 12syl 17 . . . . . . . . 9  |-  ( ph  ->  ( _|_ `  A
)  e.  SH )
14 chscl.3 . . . . . . . . 9  |-  ( ph  ->  B  C_  ( _|_ `  A ) )
15 shless 21768 . . . . . . . . 9  |-  ( ( ( B  e.  SH  /\  ( _|_ `  A
)  e.  SH  /\  A  e.  SH )  /\  B  C_  ( _|_ `  A ) )  -> 
( B  +H  A
)  C_  ( ( _|_ `  A )  +H  A ) )
169, 13, 11, 14, 15syl31anc 1190 . . . . . . . 8  |-  ( ph  ->  ( B  +H  A
)  C_  ( ( _|_ `  A )  +H  A ) )
17 shscom 21728 . . . . . . . . 9  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B
)  =  ( B  +H  A ) )
1811, 9, 17syl2anc 645 . . . . . . . 8  |-  ( ph  ->  ( A  +H  B
)  =  ( B  +H  A ) )
19 shscom 21728 . . . . . . . . 9  |-  ( ( A  e.  SH  /\  ( _|_ `  A )  e.  SH )  -> 
( A  +H  ( _|_ `  A ) )  =  ( ( _|_ `  A )  +H  A
) )
2011, 13, 19syl2anc 645 . . . . . . . 8  |-  ( ph  ->  ( A  +H  ( _|_ `  A ) )  =  ( ( _|_ `  A )  +H  A
) )
2116, 18, 203sstr4d 3142 . . . . . . 7  |-  ( ph  ->  ( A  +H  B
)  C_  ( A  +H  ( _|_ `  A
) ) )
2221sselda 3103 . . . . . 6  |-  ( (
ph  /\  ( H `  n )  e.  ( A  +H  B ) )  ->  ( H `  n )  e.  ( A  +H  ( _|_ `  A ) ) )
236, 22syldan 458 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( H `
 n )  e.  ( A  +H  ( _|_ `  A ) ) )
24 pjpreeq 21807 . . . . 5  |-  ( ( A  e.  CH  /\  ( H `  n )  e.  ( A  +H  ( _|_ `  A ) ) )  ->  (
( ( proj  h `  A ) `  ( H `  n )
)  =  ( (
proj  h `  A ) `
 ( H `  n ) )  <->  ( (
( proj  h `  A
) `  ( H `  n ) )  e.  A  /\  E. x  e.  ( _|_ `  A
) ( H `  n )  =  ( ( ( proj  h `  A ) `  ( H `  n )
)  +h  x ) ) ) )
253, 23, 24syl2anc 645 . . . 4  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( proj  h `  A
) `  ( H `  n ) )  =  ( ( proj  h `  A ) `  ( H `  n )
)  <->  ( ( (
proj  h `  A ) `
 ( H `  n ) )  e.  A  /\  E. x  e.  ( _|_ `  A
) ( H `  n )  =  ( ( ( proj  h `  A ) `  ( H `  n )
)  +h  x ) ) ) )
261, 25mpbii 204 . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  ( ( ( proj  h `  A
) `  ( H `  n ) )  e.  A  /\  E. x  e.  ( _|_ `  A
) ( H `  n )  =  ( ( ( proj  h `  A ) `  ( H `  n )
)  +h  x ) ) )
2726simpld 447 . 2  |-  ( (
ph  /\  n  e.  NN )  ->  ( (
proj  h `  A ) `
 ( H `  n ) )  e.  A )
28 chscl.6 . 2  |-  F  =  ( n  e.  NN  |->  ( ( proj  h `  A ) `  ( H `  n )
) )
2927, 28fmptd 5536 1  |-  ( ph  ->  F : NN --> A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   E.wrex 2510    C_ wss 3078   class class class wbr 3920    e. cmpt 3974   -->wf 4588   ` cfv 4592  (class class class)co 5710   NNcn 9626    +h cva 21330    ~~>v chli 21337   SHcsh 21338   CHcch 21339   _|_cort 21340    +H cph 21341   proj 
hcpjh 21347
This theorem is referenced by:  chscllem2  22065  chscllem3  22066  chscllem4  22067
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-hilex 21409  ax-hfvadd 21410  ax-hvcom 21411  ax-hvass 21412  ax-hv0cl 21413  ax-hvaddid 21414  ax-hfvmul 21415  ax-hvmulid 21416  ax-hvmulass 21417  ax-hvdistr1 21418  ax-hvdistr2 21419  ax-hvmul0 21420  ax-hfi 21488  ax-his2 21492  ax-his3 21493  ax-his4 21494
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-iota 6143  df-riota 6190  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-grpo 20688  df-ablo 20779  df-hvsub 21381  df-sh 21616  df-ch 21631  df-oc 21661  df-ch0 21662  df-shs 21717  df-pjh 21804
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