Theorem chscllem3 28498

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 ) ) )
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.

Step	Hyp	Ref	Expression
1		chscllem3.7	. . . . . 6 |- ( ph -> N e. NN )
2		fveq2 6191	. . . . . . . 8 |- ( n = N -> ( H ` n ) = ( H ` N ) )
3	2	fveq2d 6195	. . . . . . 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 6201	. . . . . . 7 |- ( ( proj h ` A ) ` ( H ` N ) ) e. _V
6	3, 4, 5	fvmpt 6282	. . . . . 6 |- ( N e. NN -> ( F ` N ) = ( ( proj h ` A ) ` ( H ` N ) ) )
7	1, 6	syl 17	. . . . 5 |- ( ph -> ( F ` N ) = ( ( proj h ` A ) ` ( H ` N ) ) )
8	7	eqcomd 2628	. . . 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 28081	. . . . . . . . 9 |- ( B e. CH -> B e. SH )
12	10, 11	syl 17	. . . . . . . 8 |- ( ph -> B e. SH )
13		chsh 28081	. . . . . . . . . 10 |- ( A e. CH -> A e. SH )
14	9, 13	syl 17	. . . . . . . . 9 |- ( ph -> A e. SH )
15		shocsh 28143	. . . . . . . . 9 |- ( A e. SH -> ( _|_ ` A ) e. SH )
16	14, 15	syl 17	. . . . . . . 8 |- ( ph -> ( _|_ ` A ) e. SH )
17		chscl.3	. . . . . . . 8 |- ( ph -> B C_ ( _|_ ` A ) )
18		shless 28218	. . . . . . . 8 |- ( ( ( B e. SH /\ ( _|_ ` A ) e. SH /\ A e. SH ) /\ B C_ ( _|_ ` A ) ) -> ( B +H A ) C_ ( ( _|_ ` A ) +H A ) )
19	12, 16, 14, 17, 18	syl31anc 1329	. . . . . . 7 |- ( ph -> ( B +H A ) C_ ( ( _|_ ` A ) +H A ) )
20		shscom 28178	. . . . . . . 8 |- ( ( A e. SH /\ B e. SH ) -> ( A +H B ) = ( B +H A ) )
21	14, 12, 20	syl2anc 693	. . . . . . 7 |- ( ph -> ( A +H B ) = ( B +H A ) )
22		shscom 28178	. . . . . . . 8 |- ( ( A e. SH /\ ( _|_ ` A ) e. SH ) -> ( A +H ( _|_ ` A ) ) = ( ( _|_ ` A ) +H A ) )
23	14, 16, 22	syl2anc 693	. . . . . . 7 |- ( ph -> ( A +H ( _|_ ` A ) ) = ( ( _|_ ` A ) +H A ) )
24	19, 21, 23	3sstr4d 3648	. . . . . 6 |- ( ph -> ( A +H B ) C_ ( A +H ( _|_ ` A ) ) )
25		chscl.4	. . . . . . 7 |- ( ph -> H : NN --> ( A +H B ) )
26	25, 1	ffvelrnd 6360	. . . . . 6 |- ( ph -> ( H ` N ) e. ( A +H B ) )
27	24, 26	sseldd 3604	. . . . 5 |- ( ph -> ( H ` N ) e. ( A +H ( _|_ ` A ) ) )
28		pjpreeq 28257	. . . . 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 ) ) ) )
29	9, 27, 28	syl2anc 693	. . . 4 |- ( ph -> ( ( ( proj h ` A ) ` ( H ` N ) ) = ( F ` N ) <-> ( ( F ` N ) e. A /\ E. z e. ( _|_ ` A ) ( H ` N ) = ( ( F ` N ) +h z ) ) ) )
30	8, 29	mpbid 222	. . 3 |- ( ph -> ( ( F ` N ) e. A /\ E. z e. ( _|_ ` A ) ( H ` N ) = ( ( F ` N ) +h z ) ) )
31	30	simprd 479	. 2 |- ( ph -> E. z e. ( _|_ ` A ) ( H ` N ) = ( ( F ` N ) +h z ) )
32	14	adantr 481	. . . 4 |- ( ( ph /\ ( z e. ( _|_ ` A ) /\ ( H ` N ) = ( ( F ` N ) +h z ) ) ) -> A e. SH )
33	16	adantr 481	. . . 4 |- ( ( ph /\ ( z e. ( _|_ ` A ) /\ ( H ` N ) = ( ( F ` N ) +h z ) ) ) -> ( _|_ ` A ) e. SH )
34		ocin 28155	. . . . . 6 |- ( A e. SH -> ( A i^i ( _|_ ` A ) ) = 0H )
35	14, 34	syl 17	. . . . 5 |- ( ph -> ( A i^i ( _|_ ` A ) ) = 0H )
36	35	adantr 481	. . . 4 |- ( ( ph /\ ( z e. ( _|_ ` A ) /\ ( H ` N ) = ( ( F ` N ) +h z ) ) ) -> ( A i^i ( _|_ ` A ) ) = 0H )
37		chscllem3.8	. . . . 5 |- ( ph -> C e. A )
38	37	adantr 481	. . . 4 |- ( ( ph /\ ( z e. ( _|_ ` A ) /\ ( H ` N ) = ( ( F ` N ) +h z ) ) ) -> C e. A )
39	17	adantr 481	. . . . 5 |- ( ( ph /\ ( z e. ( _|_ ` A ) /\ ( H ` N ) = ( ( F ` N ) +h z ) ) ) -> B C_ ( _|_ ` A ) )
40		chscllem3.9	. . . . . 6 |- ( ph -> D e. B )
41	40	adantr 481	. . . . 5 |- ( ( ph /\ ( z e. ( _|_ ` A ) /\ ( H ` N ) = ( ( F ` N ) +h z ) ) ) -> D e. B )
42	39, 41	sseldd 3604	. . . 4 |- ( ( ph /\ ( z e. ( _|_ ` A ) /\ ( H ` N ) = ( ( F ` N ) +h z ) ) ) -> D e. ( _|_ ` A ) )
43		chscl.5	. . . . . . 7 |- ( ph -> H ~~>v u )
44	9, 10, 17, 25, 43, 4	chscllem1 28496	. . . . . 6 |- ( ph -> F : NN --> A )
45	44, 1	ffvelrnd 6360	. . . . 5 |- ( ph -> ( F ` N ) e. A )
46	45	adantr 481	. . . 4 |- ( ( ph /\ ( z e. ( _|_ ` A ) /\ ( H ` N ) = ( ( F ` N ) +h z ) ) ) -> ( F ` N ) e. A )
47		simprl 794	. . . 4 |- ( ( ph /\ ( z e. ( _|_ ` A ) /\ ( H ` N ) = ( ( F ` N ) +h z ) ) ) -> z e. ( _|_ ` A ) )
48		chscllem3.10	. . . . . 6 |- ( ph -> ( H ` N ) = ( C +h D ) )
49	48	adantr 481	. . . . 5 |- ( ( ph /\ ( z e. ( _|_ ` A ) /\ ( H ` N ) = ( ( F ` N ) +h z ) ) ) -> ( H ` N ) = ( C +h D ) )
50		simprr 796	. . . . 5 |- ( ( ph /\ ( z e. ( _|_ ` A ) /\ ( H ` N ) = ( ( F ` N ) +h z ) ) ) -> ( H ` N ) = ( ( F ` N ) +h z ) )
51	49, 50	eqtr3d 2658	. . . 4 |- ( ( ph /\ ( z e. ( _|_ ` A ) /\ ( H ` N ) = ( ( F ` N ) +h z ) ) ) -> ( C +h D ) = ( ( F ` N ) +h z ) )
52	32, 33, 36, 38, 42, 46, 47, 51	shuni 28159	. . 3 |- ( ( ph /\ ( z e. ( _|_ ` A ) /\ ( H ` N ) = ( ( F ` N ) +h z ) ) ) -> ( C = ( F ` N ) /\ D = z ) )
53	52	simpld 475	. 2 |- ( ( ph /\ ( z e. ( _|_ ` A ) /\ ( H ` N ) = ( ( F ` N ) +h z ) ) ) -> C = ( F ` N ) )
54	31, 53	rexlimddv 3035	1 |- ( ph -> C = ( F ` N ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   i^i cin 3573   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   0Hc0h 27792   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:  chscllem4  28499