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Theorem chscllem1 22271
Description: Lemma for chscl 22275. (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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2316 . . . 4  |-  ( (
proj  h `  A ) `
 ( H `  n ) )  =  ( ( proj  h `  A ) `  ( H `  n )
)
2 chscl.1 . . . . . 6  |-  ( ph  ->  A  e.  CH )
32adantr 451 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  A  e. 
CH )
4 chscl.4 . . . . . . 7  |-  ( ph  ->  H : NN --> ( A  +H  B ) )
5 ffvelrn 5701 . . . . . . 7  |-  ( ( H : NN --> ( A  +H  B )  /\  n  e.  NN )  ->  ( H `  n
)  e.  ( A  +H  B ) )
64, 5sylan 457 . . . . . 6  |-  ( (
ph  /\  n  e.  NN )  ->  ( H `
 n )  e.  ( A  +H  B
) )
7 chscl.2 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CH )
8 chsh 21859 . . . . . . . . . 10  |-  ( B  e.  CH  ->  B  e.  SH )
97, 8syl 15 . . . . . . . . 9  |-  ( ph  ->  B  e.  SH )
10 chsh 21859 . . . . . . . . . . 11  |-  ( A  e.  CH  ->  A  e.  SH )
112, 10syl 15 . . . . . . . . . 10  |-  ( ph  ->  A  e.  SH )
12 shocsh 21918 . . . . . . . . . 10  |-  ( A  e.  SH  ->  ( _|_ `  A )  e.  SH )
1311, 12syl 15 . . . . . . . . 9  |-  ( ph  ->  ( _|_ `  A
)  e.  SH )
14 chscl.3 . . . . . . . . 9  |-  ( ph  ->  B  C_  ( _|_ `  A ) )
15 shless 21993 . . . . . . . . 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 1185 . . . . . . . 8  |-  ( ph  ->  ( B  +H  A
)  C_  ( ( _|_ `  A )  +H  A ) )
17 shscom 21953 . . . . . . . . 9  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B
)  =  ( B  +H  A ) )
1811, 9, 17syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( A  +H  B
)  =  ( B  +H  A ) )
19 shscom 21953 . . . . . . . . 9  |-  ( ( A  e.  SH  /\  ( _|_ `  A )  e.  SH )  -> 
( A  +H  ( _|_ `  A ) )  =  ( ( _|_ `  A )  +H  A
) )
2011, 13, 19syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( A  +H  ( _|_ `  A ) )  =  ( ( _|_ `  A )  +H  A
) )
2116, 18, 203sstr4d 3255 . . . . . . 7  |-  ( ph  ->  ( A  +H  B
)  C_  ( A  +H  ( _|_ `  A
) ) )
2221sselda 3214 . . . . . 6  |-  ( (
ph  /\  ( H `  n )  e.  ( A  +H  B ) )  ->  ( H `  n )  e.  ( A  +H  ( _|_ `  A ) ) )
236, 22syldan 456 . . . . 5  |-  ( (
ph  /\  n  e.  NN )  ->  ( H `
 n )  e.  ( A  +H  ( _|_ `  A ) ) )
24 pjpreeq 22032 . . . . 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 642 . . . 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 202 . . 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 445 . 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 5722 1  |-  ( ph  ->  F : NN --> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701   E.wrex 2578    C_ wss 3186   class class class wbr 4060    e. cmpt 4114   -->wf 5288   ` cfv 5292  (class class class)co 5900   NNcn 9791    +h cva 21555    ~~>v chli 21562   SHcsh 21563   CHcch 21564   _|_cort 21565    +H cph 21566   proj 
hcpjh 21572
This theorem is referenced by:  chscllem2  22272  chscllem3  22273  chscllem4  22274
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-hilex 21634  ax-hfvadd 21635  ax-hvcom 21636  ax-hvass 21637  ax-hv0cl 21638  ax-hvaddid 21639  ax-hfvmul 21640  ax-hvmulid 21641  ax-hvmulass 21642  ax-hvdistr1 21643  ax-hvdistr2 21644  ax-hvmul0 21645  ax-hfi 21713  ax-his2 21717  ax-his3 21718  ax-his4 21719
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-po 4351  df-so 4352  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-riota 6346  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-grpo 20911  df-ablo 21002  df-hvsub 21606  df-sh 21841  df-ch 21856  df-oc 21886  df-ch0 21887  df-shs 21942  df-pjh 22029
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