Theorem chscllem4 28499

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 ) ) )
chscl.7	|- G = ( n e. NN |-> ( ( proj h ` B ) ` ( H ` n ) ) )

Assertion

Ref	Expression
chscllem4	|- ( ph -> u e. ( A +H B ) )

Distinct variable groups:   u, n, A   ph, n   B, n, u   n, H, u

Allowed substitution hints:   ph( u)   F( u, n)   G( u, n)

Proof of Theorem chscllem4

Dummy variables x y k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hlimf 28094	. . . . 5 |- ~~>v : dom ~~>v --> ~H
2		ffun 6048	. . . . 5 |- ( ~~>v : dom ~~>v --> ~H -> Fun ~~>v )
3	1, 2	ax-mp 5	. . . 4 |- Fun ~~>v
4		chscl.5	. . . 4 |- ( ph -> H ~~>v u )
5		funbrfv 6234	. . . 4 |- ( Fun ~~>v -> ( H ~~>v u -> ( ~~>v ` H ) = u ) )
6	3, 4, 5	mpsyl 68	. . 3 |- ( ph -> ( ~~>v ` H ) = u )
7		chscl.4	. . . . . . 7 |- ( ph -> H : NN --> ( A +H B ) )
8	7	feqmptd 6249	. . . . . 6 |- ( ph -> H = ( k e. NN |-> ( H ` k ) ) )
9	7	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ph /\ k e. NN ) -> ( H ` k ) e. ( A +H B ) )
10		chscl.1	. . . . . . . . . . . 12 |- ( ph -> A e. CH )
11		chsh 28081	. . . . . . . . . . . 12 |- ( A e. CH -> A e. SH )
12	10, 11	syl 17	. . . . . . . . . . 11 |- ( ph -> A e. SH )
13		chscl.2	. . . . . . . . . . . 12 |- ( ph -> B e. CH )
14		chsh 28081	. . . . . . . . . . . 12 |- ( B e. CH -> B e. SH )
15	13, 14	syl 17	. . . . . . . . . . 11 |- ( ph -> B e. SH )
16		shsel 28173	. . . . . . . . . . 11 |- ( ( A e. SH /\ B e. SH ) -> ( ( H ` k ) e. ( A +H B ) <-> E. x e. A E. y e. B ( H ` k ) = ( x +h y ) ) )
17	12, 15, 16	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( ( H ` k ) e. ( A +H B ) <-> E. x e. A E. y e. B ( H ` k ) = ( x +h y ) ) )
18	17	biimpa 501	. . . . . . . . 9 |- ( ( ph /\ ( H ` k ) e. ( A +H B ) ) -> E. x e. A E. y e. B ( H ` k ) = ( x +h y ) )
19	9, 18	syldan 487	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> E. x e. A E. y e. B ( H ` k ) = ( x +h y ) )
20		simp3 1063	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> ( H ` k ) = ( x +h y ) )
21		simp1l 1085	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> ph )
22	21, 10	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> A e. CH )
23	21, 13	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> B e. CH )
24		chscl.3	. . . . . . . . . . . . . 14 |- ( ph -> B C_ ( _|_ ` A ) )
25	21, 24	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> B C_ ( _|_ ` A ) )
26	21, 7	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> H : NN --> ( A +H B ) )
27	21, 4	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> H ~~>v u )
28		chscl.6	. . . . . . . . . . . . 13 |- F = ( n e. NN |-> ( ( proj h ` A ) ` ( H ` n ) ) )
29		simp1r 1086	. . . . . . . . . . . . 13 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> k e. NN )
30		simp2l 1087	. . . . . . . . . . . . 13 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> x e. A )
31		simp2r 1088	. . . . . . . . . . . . 13 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> y e. B )
32	22, 23, 25, 26, 27, 28, 29, 30, 31, 20	chscllem3 28498	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> x = ( F ` k ) )
33		chsscon2 28361	. . . . . . . . . . . . . . . 16 |- ( ( B e. CH /\ A e. CH ) -> ( B C_ ( _|_ ` A ) <-> A C_ ( _|_ ` B ) ) )
34	13, 10, 33	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ph -> ( B C_ ( _|_ ` A ) <-> A C_ ( _|_ ` B ) ) )
35	24, 34	mpbid 222	. . . . . . . . . . . . . 14 |- ( ph -> A C_ ( _|_ ` B ) )
36	21, 35	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> A C_ ( _|_ ` B ) )
37		shscom 28178	. . . . . . . . . . . . . . . . 17 |- ( ( A e. SH /\ B e. SH ) -> ( A +H B ) = ( B +H A ) )
38	12, 15, 37	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( ph -> ( A +H B ) = ( B +H A ) )
39	38	feq3d 6032	. . . . . . . . . . . . . . 15 |- ( ph -> ( H : NN --> ( A +H B ) <-> H : NN --> ( B +H A ) ) )
40	7, 39	mpbid 222	. . . . . . . . . . . . . 14 |- ( ph -> H : NN --> ( B +H A ) )
41	21, 40	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> H : NN --> ( B +H A ) )
42		chscl.7	. . . . . . . . . . . . 13 |- G = ( n e. NN |-> ( ( proj h ` B ) ` ( H ` n ) ) )
43		shss 28067	. . . . . . . . . . . . . . . . . 18 |- ( A e. SH -> A C_ ~H )
44	12, 43	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ph -> A C_ ~H )
45	21, 44	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> A C_ ~H )
46	45, 30	sseldd 3604	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> x e. ~H )
47		shss 28067	. . . . . . . . . . . . . . . . . 18 |- ( B e. SH -> B C_ ~H )
48	15, 47	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ph -> B C_ ~H )
49	21, 48	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> B C_ ~H )
50	49, 31	sseldd 3604	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> y e. ~H )
51		ax-hvcom 27858	. . . . . . . . . . . . . . 15 |- ( ( x e. ~H /\ y e. ~H ) -> ( x +h y ) = ( y +h x ) )
52	46, 50, 51	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> ( x +h y ) = ( y +h x ) )
53	20, 52	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> ( H ` k ) = ( y +h x ) )
54	23, 22, 36, 41, 27, 42, 29, 31, 30, 53	chscllem3 28498	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> y = ( G ` k ) )
55	32, 54	oveq12d 6668	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> ( x +h y ) = ( ( F ` k ) +h ( G ` k ) ) )
56	20, 55	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ph /\ k e. NN ) /\ ( x e. A /\ y e. B ) /\ ( H ` k ) = ( x +h y ) ) -> ( H ` k ) = ( ( F ` k ) +h ( G ` k ) ) )
57	56	3exp 1264	. . . . . . . . 9 |- ( ( ph /\ k e. NN ) -> ( ( x e. A /\ y e. B ) -> ( ( H ` k ) = ( x +h y ) -> ( H ` k ) = ( ( F ` k ) +h ( G ` k ) ) ) ) )
58	57	rexlimdvv 3037	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> ( E. x e. A E. y e. B ( H ` k ) = ( x +h y ) -> ( H ` k ) = ( ( F ` k ) +h ( G ` k ) ) ) )
59	19, 58	mpd 15	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( H ` k ) = ( ( F ` k ) +h ( G ` k ) ) )
60	59	mpteq2dva 4744	. . . . . 6 |- ( ph -> ( k e. NN |-> ( H ` k ) ) = ( k e. NN |-> ( ( F ` k ) +h ( G ` k ) ) ) )
61	8, 60	eqtrd 2656	. . . . 5 |- ( ph -> H = ( k e. NN |-> ( ( F ` k ) +h ( G ` k ) ) ) )
62	10, 13, 24, 7, 4, 28	chscllem1 28496	. . . . . . 7 |- ( ph -> F : NN --> A )
63	62, 44	fssd 6057	. . . . . 6 |- ( ph -> F : NN --> ~H )
64	13, 10, 35, 40, 4, 42	chscllem1 28496	. . . . . . 7 |- ( ph -> G : NN --> B )
65	64, 48	fssd 6057	. . . . . 6 |- ( ph -> G : NN --> ~H )
66	10, 13, 24, 7, 4, 28	chscllem2 28497	. . . . . . 7 |- ( ph -> F e. dom ~~>v )
67		funfvbrb 6330	. . . . . . . 8 |- ( Fun ~~>v -> ( F e. dom ~~>v <-> F ~~>v ( ~~>v ` F ) ) )
68	3, 67	ax-mp 5	. . . . . . 7 |- ( F e. dom ~~>v <-> F ~~>v ( ~~>v ` F ) )
69	66, 68	sylib 208	. . . . . 6 |- ( ph -> F ~~>v ( ~~>v ` F ) )
70	13, 10, 35, 40, 4, 42	chscllem2 28497	. . . . . . 7 |- ( ph -> G e. dom ~~>v )
71		funfvbrb 6330	. . . . . . . 8 |- ( Fun ~~>v -> ( G e. dom ~~>v <-> G ~~>v ( ~~>v ` G ) ) )
72	3, 71	ax-mp 5	. . . . . . 7 |- ( G e. dom ~~>v <-> G ~~>v ( ~~>v ` G ) )
73	70, 72	sylib 208	. . . . . 6 |- ( ph -> G ~~>v ( ~~>v ` G ) )
74		eqid 2622	. . . . . 6 |- ( k e. NN |-> ( ( F ` k ) +h ( G ` k ) ) ) = ( k e. NN |-> ( ( F ` k ) +h ( G ` k ) ) )
75	63, 65, 69, 73, 74	hlimadd 28050	. . . . 5 |- ( ph -> ( k e. NN |-> ( ( F ` k ) +h ( G ` k ) ) ) ~~>v ( ( ~~>v ` F ) +h ( ~~>v ` G ) ) )
76	61, 75	eqbrtrd 4675	. . . 4 |- ( ph -> H ~~>v ( ( ~~>v ` F ) +h ( ~~>v ` G ) ) )
77		funbrfv 6234	. . . 4 |- ( Fun ~~>v -> ( H ~~>v ( ( ~~>v ` F ) +h ( ~~>v ` G ) ) -> ( ~~>v ` H ) = ( ( ~~>v ` F ) +h ( ~~>v ` G ) ) ) )
78	3, 76, 77	mpsyl 68	. . 3 |- ( ph -> ( ~~>v ` H ) = ( ( ~~>v ` F ) +h ( ~~>v ` G ) ) )
79	6, 78	eqtr3d 2658	. 2 |- ( ph -> u = ( ( ~~>v ` F ) +h ( ~~>v ` G ) ) )
80		fvex 6201	. . . . 5 |- ( ~~>v ` F ) e. _V
81	80	chlimi 28091	. . . 4 |- ( ( A e. CH /\ F : NN --> A /\ F ~~>v ( ~~>v ` F ) ) -> ( ~~>v ` F ) e. A )
82	10, 62, 69, 81	syl3anc 1326	. . 3 |- ( ph -> ( ~~>v ` F ) e. A )
83		fvex 6201	. . . . 5 |- ( ~~>v ` G ) e. _V
84	83	chlimi 28091	. . . 4 |- ( ( B e. CH /\ G : NN --> B /\ G ~~>v ( ~~>v ` G ) ) -> ( ~~>v ` G ) e. B )
85	13, 64, 73, 84	syl3anc 1326	. . 3 |- ( ph -> ( ~~>v ` G ) e. B )
86		shsva 28179	. . . 4 |- ( ( A e. SH /\ B e. SH ) -> ( ( ( ~~>v ` F ) e. A /\ ( ~~>v ` G ) e. B ) -> ( ( ~~>v ` F ) +h ( ~~>v ` G ) ) e. ( A +H B ) ) )
87	12, 15, 86	syl2anc 693	. . 3 |- ( ph -> ( ( ( ~~>v ` F ) e. A /\ ( ~~>v ` G ) e. B ) -> ( ( ~~>v ` F ) +h ( ~~>v ` G ) ) e. ( A +H B ) ) )
88	82, 85, 87	mp2and 715	. 2 |- ( ph -> ( ( ~~>v ` F ) +h ( ~~>v ` G ) ) e. ( A +H B ) )
89	79, 88	eqeltrd 2701	1 |- ( ph -> u e. ( A +H B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   NNcn 11020   ~Hchil 27776   +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-inf2 8538  ax-cnex 9992  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-pre-sup 10014  ax-addf 10015  ax-mulf 10016  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-his1 27939  ax-his2 27940  ax-his3 27941  ax-his4 27942  ax-hcompl 28059

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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-lm 21033  df-haus 21119  df-tx 21365  df-hmeo 21558  df-xms 22125  df-tms 22127  df-cau 23054  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-vs 27454  df-nmcv 27455  df-ims 27456  df-hnorm 27825  df-hba 27826  df-hvsub 27828  df-hlim 27829  df-hcau 27830  df-sh 28064  df-ch 28078  df-oc 28109  df-ch0 28110  df-shs 28167  df-pjh 28254

This theorem is referenced by:  chscl  28500