Theorem chscllem1 28496

Description: Lemma for chscl 28500. (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.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( ( proj h ` A ) ` ( H ` n ) ) = ( ( proj h ` A ) ` ( H ` n ) )
2		chscl.1	. . . . . 6 |- ( ph -> A e. CH )
3	2	adantr 481	. . . . 5 |- ( ( ph /\ n e. NN ) -> A e. CH )
4		chscl.4	. . . . . . 7 |- ( ph -> H : NN --> ( A +H B ) )
5	4	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ n e. NN ) -> ( H ` n ) e. ( A +H B ) )
6		chscl.2	. . . . . . . . . 10 |- ( ph -> B e. CH )
7		chsh 28081	. . . . . . . . . 10 |- ( B e. CH -> B e. SH )
8	6, 7	syl 17	. . . . . . . . 9 |- ( ph -> B e. SH )
9		chsh 28081	. . . . . . . . . . 11 |- ( A e. CH -> A e. SH )
10	2, 9	syl 17	. . . . . . . . . 10 |- ( ph -> A e. SH )
11		shocsh 28143	. . . . . . . . . 10 |- ( A e. SH -> ( _|_ ` A ) e. SH )
12	10, 11	syl 17	. . . . . . . . 9 |- ( ph -> ( _|_ ` A ) e. SH )
13		chscl.3	. . . . . . . . 9 |- ( ph -> B C_ ( _|_ ` A ) )
14		shless 28218	. . . . . . . . 9 |- ( ( ( B e. SH /\ ( _|_ ` A ) e. SH /\ A e. SH ) /\ B C_ ( _|_ ` A ) ) -> ( B +H A ) C_ ( ( _|_ ` A ) +H A ) )
15	8, 12, 10, 13, 14	syl31anc 1329	. . . . . . . 8 |- ( ph -> ( B +H A ) C_ ( ( _|_ ` A ) +H A ) )
16		shscom 28178	. . . . . . . . 9 |- ( ( A e. SH /\ B e. SH ) -> ( A +H B ) = ( B +H A ) )
17	10, 8, 16	syl2anc 693	. . . . . . . 8 |- ( ph -> ( A +H B ) = ( B +H A ) )
18		shscom 28178	. . . . . . . . 9 |- ( ( A e. SH /\ ( _|_ ` A ) e. SH ) -> ( A +H ( _|_ ` A ) ) = ( ( _|_ ` A ) +H A ) )
19	10, 12, 18	syl2anc 693	. . . . . . . 8 |- ( ph -> ( A +H ( _|_ ` A ) ) = ( ( _|_ ` A ) +H A ) )
20	15, 17, 19	3sstr4d 3648	. . . . . . 7 |- ( ph -> ( A +H B ) C_ ( A +H ( _|_ ` A ) ) )
21	20	sselda 3603	. . . . . 6 |- ( ( ph /\ ( H ` n ) e. ( A +H B ) ) -> ( H ` n ) e. ( A +H ( _|_ ` A ) ) )
22	5, 21	syldan 487	. . . . 5 |- ( ( ph /\ n e. NN ) -> ( H ` n ) e. ( A +H ( _|_ ` A ) ) )
23		pjpreeq 28257	. . . . 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 ) ) ) )
24	3, 22, 23	syl2anc 693	. . . 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 ) ) ) )
25	1, 24	mpbii 223	. . 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 ) ) )
26	25	simpld 475	. 2 |- ( ( ph /\ n e. NN ) -> ( ( proj h ` A ) ` ( H ` n ) ) e. A )
27		chscl.6	. 2 |- F = ( n e. NN |-> ( ( proj h ` A ) ` ( H ` n ) ) )
28	26, 27	fmptd 6385	1 |- ( ph -> F : NN --> A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   NNcn 11020   +h cva 27777   ~~>v chli 27784   SHcsh 27785   CHcch 27786   _|_cort 27787   +H cph 27788   proj hcpjh 27794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-hilex 27856  ax-hfvadd 27857  ax-hvcom 27858  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvmulass 27864  ax-hvdistr1 27865  ax-hvdistr2 27866  ax-hvmul0 27867  ax-hfi 27936  ax-his2 27940  ax-his3 27941  ax-his4 27942

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-grpo 27347  df-ablo 27399  df-hvsub 27828  df-sh 28064  df-ch 28078  df-oc 28109  df-ch0 28110  df-shs 28167  df-pjh 28254

This theorem is referenced by:  chscllem2  28497  chscllem3  28498  chscllem4  28499