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Theorem chscllem1 22208
Description: Lemma for chscl 22212. (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
Dummy variable  x is distinct from all other variables.
Allowed substitution hints:    ph( u)    F( u, n)

Proof of Theorem chscllem1
StepHypRef Expression
1 eqid 2284 . . . 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 5624 . . . . . . 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 21796 . . . . . . . . . 10  |-  ( B  e.  CH  ->  B  e.  SH )
97, 8syl 17 . . . . . . . . 9  |-  ( ph  ->  B  e.  SH )
10 chsh 21796 . . . . . . . . . . 11  |-  ( A  e.  CH  ->  A  e.  SH )
112, 10syl 17 . . . . . . . . . 10  |-  ( ph  ->  A  e.  SH )
12 shocsh 21855 . . . . . . . . . 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 21930 . . . . . . . . 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 1187 . . . . . . . 8  |-  ( ph  ->  ( B  +H  A
)  C_  ( ( _|_ `  A )  +H  A ) )
17 shscom 21890 . . . . . . . . 9  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B
)  =  ( B  +H  A ) )
1811, 9, 17syl2anc 644 . . . . . . . 8  |-  ( ph  ->  ( A  +H  B
)  =  ( B  +H  A ) )
19 shscom 21890 . . . . . . . . 9  |-  ( ( A  e.  SH  /\  ( _|_ `  A )  e.  SH )  -> 
( A  +H  ( _|_ `  A ) )  =  ( ( _|_ `  A )  +H  A
) )
2011, 13, 19syl2anc 644 . . . . . . . 8  |-  ( ph  ->  ( A  +H  ( _|_ `  A ) )  =  ( ( _|_ `  A )  +H  A
) )
2116, 18, 203sstr4d 3222 . . . . . . 7  |-  ( ph  ->  ( A  +H  B
)  C_  ( A  +H  ( _|_ `  A
) ) )
2221sselda 3181 . . . . . 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 21969 . . . . 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 644 . . . 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 5645 1  |-  ( ph  ->  F : NN --> A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685   E.wrex 2545    C_ wss 3153   class class class wbr 4024    e. cmpt 4078   -->wf 5217   ` cfv 5221  (class class class)co 5819   NNcn 9741    +h cva 21492    ~~>v chli 21499   SHcsh 21500   CHcch 21501   _|_cort 21502    +H cph 21503   proj 
hcpjh 21509
This theorem is referenced by:  chscllem2  22209  chscllem3  22210  chscllem4  22211
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-hilex 21571  ax-hfvadd 21572  ax-hvcom 21573  ax-hvass 21574  ax-hv0cl 21575  ax-hvaddid 21576  ax-hfvmul 21577  ax-hvmulid 21578  ax-hvmulass 21579  ax-hvdistr1 21580  ax-hvdistr2 21581  ax-hvmul0 21582  ax-hfi 21650  ax-his2 21654  ax-his3 21655  ax-his4 21656
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-iota 6252  df-riota 6299  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-grpo 20850  df-ablo 20941  df-hvsub 21543  df-sh 21778  df-ch 21793  df-oc 21823  df-ch0 21824  df-shs 21879  df-pjh 21966
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