Theorem mdexchi 26930

Description: An exchange lemma for modular pairs. Lemma 1.6 of [MaedaMaeda] p. 2. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Hypotheses

Ref	Expression
mdexch.1	|- A e. CH
mdexch.2	|- B e. CH
mdexch.3	|- C e. CH

Assertion

Ref	Expression
mdexchi	|- ( ( A MH B /\ C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) -> ( ( C vH A ) MH B /\ ( ( C vH A ) i^i B ) = ( A i^i B ) ) )

Proof of Theorem mdexchi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mdexch.3	. . . . . . . . . . . . . . 15 |- C e. CH
2		mdexch.1	. . . . . . . . . . . . . . 15 |- A e. CH
3		chjass 26127	. . . . . . . . . . . . . . 15 |- ( ( C e. CH /\ A e. CH /\ x e. CH ) -> ( ( C vH A ) vH x ) = ( C vH ( A vH x ) ) )
4	1, 2, 3	mp3an12 1314	. . . . . . . . . . . . . 14 |- ( x e. CH -> ( ( C vH A ) vH x ) = ( C vH ( A vH x ) ) )
5	1, 2	chjcli 26051	. . . . . . . . . . . . . . 15 |- ( C vH A ) e. CH
6		chjcom 26100	. . . . . . . . . . . . . . 15 |- ( ( x e. CH /\ ( C vH A ) e. CH ) -> ( x vH ( C vH A ) ) = ( ( C vH A ) vH x ) )
7	5, 6	mpan2 671	. . . . . . . . . . . . . 14 |- ( x e. CH -> ( x vH ( C vH A ) ) = ( ( C vH A ) vH x ) )
8		chjcl 25951	. . . . . . . . . . . . . . . 16 |- ( ( A e. CH /\ x e. CH ) -> ( A vH x ) e. CH )
9	2, 8	mpan 670	. . . . . . . . . . . . . . 15 |- ( x e. CH -> ( A vH x ) e. CH )
10		chjcom 26100	. . . . . . . . . . . . . . 15 |- ( ( ( A vH x ) e. CH /\ C e. CH ) -> ( ( A vH x ) vH C ) = ( C vH ( A vH x ) ) )
11	9, 1, 10	sylancl 662	. . . . . . . . . . . . . 14 |- ( x e. CH -> ( ( A vH x ) vH C ) = ( C vH ( A vH x ) ) )
12	4, 7, 11	3eqtr4d 2518	. . . . . . . . . . . . 13 |- ( x e. CH -> ( x vH ( C vH A ) ) = ( ( A vH x ) vH C ) )
13	12	ineq1d 3699	. . . . . . . . . . . 12 |- ( x e. CH -> ( ( x vH ( C vH A ) ) i^i B ) = ( ( ( A vH x ) vH C ) i^i B ) )
14		inass 3708	. . . . . . . . . . . . 13 |- ( ( ( ( A vH x ) vH C ) i^i ( A vH B ) ) i^i B ) = ( ( ( A vH x ) vH C ) i^i ( ( A vH B ) i^i B ) )
15		incom 3691	. . . . . . . . . . . . . . 15 |- ( ( A vH B ) i^i B ) = ( B i^i ( A vH B ) )
16		mdexch.2	. . . . . . . . . . . . . . . . . 18 |- B e. CH
17	2, 16	chjcomi 26062	. . . . . . . . . . . . . . . . 17 |- ( A vH B ) = ( B vH A )
18	17	ineq2i 3697	. . . . . . . . . . . . . . . 16 |- ( B i^i ( A vH B ) ) = ( B i^i ( B vH A ) )
19	16, 2	chabs2i 26113	. . . . . . . . . . . . . . . 16 |- ( B i^i ( B vH A ) ) = B
20	18, 19	eqtri 2496	. . . . . . . . . . . . . . 15 |- ( B i^i ( A vH B ) ) = B
21	15, 20	eqtri 2496	. . . . . . . . . . . . . 14 |- ( ( A vH B ) i^i B ) = B
22	21	ineq2i 3697	. . . . . . . . . . . . 13 |- ( ( ( A vH x ) vH C ) i^i ( ( A vH B ) i^i B ) ) = ( ( ( A vH x ) vH C ) i^i B )
23	14, 22	eqtri 2496	. . . . . . . . . . . 12 |- ( ( ( ( A vH x ) vH C ) i^i ( A vH B ) ) i^i B ) = ( ( ( A vH x ) vH C ) i^i B )
24	13, 23	syl6eqr 2526	. . . . . . . . . . 11 |- ( x e. CH -> ( ( x vH ( C vH A ) ) i^i B ) = ( ( ( ( A vH x ) vH C ) i^i ( A vH B ) ) i^i B ) )
25	24	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( x e. CH /\ x C_ B ) /\ ( C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) ) -> ( ( x vH ( C vH A ) ) i^i B ) = ( ( ( ( A vH x ) vH C ) i^i ( A vH B ) ) i^i B ) )
26		chlej2 26105	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. CH /\ B e. CH /\ A e. CH ) /\ x C_ B ) -> ( A vH x ) C_ ( A vH B ) )
27	26	ex 434	. . . . . . . . . . . . . . . 16 |- ( ( x e. CH /\ B e. CH /\ A e. CH ) -> ( x C_ B -> ( A vH x ) C_ ( A vH B ) ) )
28	16, 2, 27	mp3an23 1316	. . . . . . . . . . . . . . 15 |- ( x e. CH -> ( x C_ B -> ( A vH x ) C_ ( A vH B ) ) )
29	2, 16	chjcli 26051	. . . . . . . . . . . . . . . . . 18 |- ( A vH B ) e. CH
30		mdi 26890	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( C e. CH /\ ( A vH B ) e. CH /\ ( A vH x ) e. CH ) /\ ( C MH ( A vH B ) /\ ( A vH x ) C_ ( A vH B ) ) ) -> ( ( ( A vH x ) vH C ) i^i ( A vH B ) ) = ( ( A vH x ) vH ( C i^i ( A vH B ) ) ) )
31	30	exp32 605	. . . . . . . . . . . . . . . . . 18 |- ( ( C e. CH /\ ( A vH B ) e. CH /\ ( A vH x ) e. CH ) -> ( C MH ( A vH B ) -> ( ( A vH x ) C_ ( A vH B ) -> ( ( ( A vH x ) vH C ) i^i ( A vH B ) ) = ( ( A vH x ) vH ( C i^i ( A vH B ) ) ) ) ) )
32	1, 29, 31	mp3an12 1314	. . . . . . . . . . . . . . . . 17 |- ( ( A vH x ) e. CH -> ( C MH ( A vH B ) -> ( ( A vH x ) C_ ( A vH B ) -> ( ( ( A vH x ) vH C ) i^i ( A vH B ) ) = ( ( A vH x ) vH ( C i^i ( A vH B ) ) ) ) ) )
33	9, 32	syl 16	. . . . . . . . . . . . . . . 16 |- ( x e. CH -> ( C MH ( A vH B ) -> ( ( A vH x ) C_ ( A vH B ) -> ( ( ( A vH x ) vH C ) i^i ( A vH B ) ) = ( ( A vH x ) vH ( C i^i ( A vH B ) ) ) ) ) )
34	33	com23 78	. . . . . . . . . . . . . . 15 |- ( x e. CH -> ( ( A vH x ) C_ ( A vH B ) -> ( C MH ( A vH B ) -> ( ( ( A vH x ) vH C ) i^i ( A vH B ) ) = ( ( A vH x ) vH ( C i^i ( A vH B ) ) ) ) ) )
35	28, 34	syld 44	. . . . . . . . . . . . . 14 |- ( x e. CH -> ( x C_ B -> ( C MH ( A vH B ) -> ( ( ( A vH x ) vH C ) i^i ( A vH B ) ) = ( ( A vH x ) vH ( C i^i ( A vH B ) ) ) ) ) )
36	35	imp31 432	. . . . . . . . . . . . 13 |- ( ( ( x e. CH /\ x C_ B ) /\ C MH ( A vH B ) ) -> ( ( ( A vH x ) vH C ) i^i ( A vH B ) ) = ( ( A vH x ) vH ( C i^i ( A vH B ) ) ) )
37	36	adantrr 716	. . . . . . . . . . . 12 |- ( ( ( x e. CH /\ x C_ B ) /\ ( C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) ) -> ( ( ( A vH x ) vH C ) i^i ( A vH B ) ) = ( ( A vH x ) vH ( C i^i ( A vH B ) ) ) )
38	1, 29	chincli 26054	. . . . . . . . . . . . . . . . 17 |- ( C i^i ( A vH B ) ) e. CH
39		chlej2 26105	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( C i^i ( A vH B ) ) e. CH /\ A e. CH /\ ( A vH x ) e. CH ) /\ ( C i^i ( A vH B ) ) C_ A ) -> ( ( A vH x ) vH ( C i^i ( A vH B ) ) ) C_ ( ( A vH x ) vH A ) )
40	39	ex 434	. . . . . . . . . . . . . . . . 17 |- ( ( ( C i^i ( A vH B ) ) e. CH /\ A e. CH /\ ( A vH x ) e. CH ) -> ( ( C i^i ( A vH B ) ) C_ A -> ( ( A vH x ) vH ( C i^i ( A vH B ) ) ) C_ ( ( A vH x ) vH A ) ) )
41	38, 2, 40	mp3an12 1314	. . . . . . . . . . . . . . . 16 |- ( ( A vH x ) e. CH -> ( ( C i^i ( A vH B ) ) C_ A -> ( ( A vH x ) vH ( C i^i ( A vH B ) ) ) C_ ( ( A vH x ) vH A ) ) )
42	9, 41	syl 16	. . . . . . . . . . . . . . 15 |- ( x e. CH -> ( ( C i^i ( A vH B ) ) C_ A -> ( ( A vH x ) vH ( C i^i ( A vH B ) ) ) C_ ( ( A vH x ) vH A ) ) )
43	42	imp 429	. . . . . . . . . . . . . 14 |- ( ( x e. CH /\ ( C i^i ( A vH B ) ) C_ A ) -> ( ( A vH x ) vH ( C i^i ( A vH B ) ) ) C_ ( ( A vH x ) vH A ) )
44		chjcom 26100	. . . . . . . . . . . . . . . . 17 |- ( ( ( A vH x ) e. CH /\ A e. CH ) -> ( ( A vH x ) vH A ) = ( A vH ( A vH x ) ) )
45	9, 2, 44	sylancl 662	. . . . . . . . . . . . . . . 16 |- ( x e. CH -> ( ( A vH x ) vH A ) = ( A vH ( A vH x ) ) )
46	2	chjidmi 26115	. . . . . . . . . . . . . . . . . 18 |- ( A vH A ) = A
47	46	oveq1i 6292	. . . . . . . . . . . . . . . . 17 |- ( ( A vH A ) vH x ) = ( A vH x )
48		chjass 26127	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. CH /\ A e. CH /\ x e. CH ) -> ( ( A vH A ) vH x ) = ( A vH ( A vH x ) ) )
49	2, 2, 48	mp3an12 1314	. . . . . . . . . . . . . . . . 17 |- ( x e. CH -> ( ( A vH A ) vH x ) = ( A vH ( A vH x ) ) )
50		chjcom 26100	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. CH /\ x e. CH ) -> ( A vH x ) = ( x vH A ) )
51	2, 50	mpan 670	. . . . . . . . . . . . . . . . 17 |- ( x e. CH -> ( A vH x ) = ( x vH A ) )
52	47, 49, 51	3eqtr3a 2532	. . . . . . . . . . . . . . . 16 |- ( x e. CH -> ( A vH ( A vH x ) ) = ( x vH A ) )
53	45, 52	eqtrd 2508	. . . . . . . . . . . . . . 15 |- ( x e. CH -> ( ( A vH x ) vH A ) = ( x vH A ) )
54	53	adantr 465	. . . . . . . . . . . . . 14 |- ( ( x e. CH /\ ( C i^i ( A vH B ) ) C_ A ) -> ( ( A vH x ) vH A ) = ( x vH A ) )
55	43, 54	sseqtrd 3540	. . . . . . . . . . . . 13 |- ( ( x e. CH /\ ( C i^i ( A vH B ) ) C_ A ) -> ( ( A vH x ) vH ( C i^i ( A vH B ) ) ) C_ ( x vH A ) )
56	55	ad2ant2rl 748	. . . . . . . . . . . 12 |- ( ( ( x e. CH /\ x C_ B ) /\ ( C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) ) -> ( ( A vH x ) vH ( C i^i ( A vH B ) ) ) C_ ( x vH A ) )
57	37, 56	eqsstrd 3538	. . . . . . . . . . 11 |- ( ( ( x e. CH /\ x C_ B ) /\ ( C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) ) -> ( ( ( A vH x ) vH C ) i^i ( A vH B ) ) C_ ( x vH A ) )
58		ssrin 3723	. . . . . . . . . . 11 |- ( ( ( ( A vH x ) vH C ) i^i ( A vH B ) ) C_ ( x vH A ) -> ( ( ( ( A vH x ) vH C ) i^i ( A vH B ) ) i^i B ) C_ ( ( x vH A ) i^i B ) )
59	57, 58	syl 16	. . . . . . . . . 10 |- ( ( ( x e. CH /\ x C_ B ) /\ ( C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) ) -> ( ( ( ( A vH x ) vH C ) i^i ( A vH B ) ) i^i B ) C_ ( ( x vH A ) i^i B ) )
60	25, 59	eqsstrd 3538	. . . . . . . . 9 |- ( ( ( x e. CH /\ x C_ B ) /\ ( C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) ) -> ( ( x vH ( C vH A ) ) i^i B ) C_ ( ( x vH A ) i^i B ) )
61	60	adantrl 715	. . . . . . . 8 |- ( ( ( x e. CH /\ x C_ B ) /\ ( A MH B /\ ( C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) ) ) -> ( ( x vH ( C vH A ) ) i^i B ) C_ ( ( x vH A ) i^i B ) )
62		mdi 26890	. . . . . . . . . . . . . 14 |- ( ( ( A e. CH /\ B e. CH /\ x e. CH ) /\ ( A MH B /\ x C_ B ) ) -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) )
63	62	exp32 605	. . . . . . . . . . . . 13 |- ( ( A e. CH /\ B e. CH /\ x e. CH ) -> ( A MH B -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) )
64	2, 16, 63	mp3an12 1314	. . . . . . . . . . . 12 |- ( x e. CH -> ( A MH B -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) )
65	64	com23 78	. . . . . . . . . . 11 |- ( x e. CH -> ( x C_ B -> ( A MH B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) )
66	65	imp31 432	. . . . . . . . . 10 |- ( ( ( x e. CH /\ x C_ B ) /\ A MH B ) -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) )
67	2, 1	chub2i 26064	. . . . . . . . . . . . 13 |- A C_ ( C vH A )
68		ssrin 3723	. . . . . . . . . . . . 13 |- ( A C_ ( C vH A ) -> ( A i^i B ) C_ ( ( C vH A ) i^i B ) )
69	67, 68	ax-mp 5	. . . . . . . . . . . 12 |- ( A i^i B ) C_ ( ( C vH A ) i^i B )
70	2, 16	chincli 26054	. . . . . . . . . . . . 13 |- ( A i^i B ) e. CH
71	5, 16	chincli 26054	. . . . . . . . . . . . 13 |- ( ( C vH A ) i^i B ) e. CH
72		chlej2 26105	. . . . . . . . . . . . . 14 |- ( ( ( ( A i^i B ) e. CH /\ ( ( C vH A ) i^i B ) e. CH /\ x e. CH ) /\ ( A i^i B ) C_ ( ( C vH A ) i^i B ) ) -> ( x vH ( A i^i B ) ) C_ ( x vH ( ( C vH A ) i^i B ) ) )
73	72	ex 434	. . . . . . . . . . . . 13 |- ( ( ( A i^i B ) e. CH /\ ( ( C vH A ) i^i B ) e. CH /\ x e. CH ) -> ( ( A i^i B ) C_ ( ( C vH A ) i^i B ) -> ( x vH ( A i^i B ) ) C_ ( x vH ( ( C vH A ) i^i B ) ) ) )
74	70, 71, 73	mp3an12 1314	. . . . . . . . . . . 12 |- ( x e. CH -> ( ( A i^i B ) C_ ( ( C vH A ) i^i B ) -> ( x vH ( A i^i B ) ) C_ ( x vH ( ( C vH A ) i^i B ) ) ) )
75	69, 74	mpi 17	. . . . . . . . . . 11 |- ( x e. CH -> ( x vH ( A i^i B ) ) C_ ( x vH ( ( C vH A ) i^i B ) ) )
76	75	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( x e. CH /\ x C_ B ) /\ A MH B ) -> ( x vH ( A i^i B ) ) C_ ( x vH ( ( C vH A ) i^i B ) ) )
77	66, 76	eqsstrd 3538	. . . . . . . . 9 |- ( ( ( x e. CH /\ x C_ B ) /\ A MH B ) -> ( ( x vH A ) i^i B ) C_ ( x vH ( ( C vH A ) i^i B ) ) )
78	77	adantrr 716	. . . . . . . 8 |- ( ( ( x e. CH /\ x C_ B ) /\ ( A MH B /\ ( C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) ) ) -> ( ( x vH A ) i^i B ) C_ ( x vH ( ( C vH A ) i^i B ) ) )
79	61, 78	sstrd 3514	. . . . . . 7 |- ( ( ( x e. CH /\ x C_ B ) /\ ( A MH B /\ ( C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) ) ) -> ( ( x vH ( C vH A ) ) i^i B ) C_ ( x vH ( ( C vH A ) i^i B ) ) )
80	79	exp31 604	. . . . . 6 |- ( x e. CH -> ( x C_ B -> ( ( A MH B /\ ( C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) ) -> ( ( x vH ( C vH A ) ) i^i B ) C_ ( x vH ( ( C vH A ) i^i B ) ) ) ) )
81	80	com3r 79	. . . . 5 |- ( ( A MH B /\ ( C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) ) -> ( x e. CH -> ( x C_ B -> ( ( x vH ( C vH A ) ) i^i B ) C_ ( x vH ( ( C vH A ) i^i B ) ) ) ) )
82	81	3impb 1192	. . . 4 |- ( ( A MH B /\ C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) -> ( x e. CH -> ( x C_ B -> ( ( x vH ( C vH A ) ) i^i B ) C_ ( x vH ( ( C vH A ) i^i B ) ) ) ) )
83	82	ralrimiv 2876	. . 3 |- ( ( A MH B /\ C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) -> A. x e. CH ( x C_ B -> ( ( x vH ( C vH A ) ) i^i B ) C_ ( x vH ( ( C vH A ) i^i B ) ) ) )
84		mdbr2 26891	. . . 4 |- ( ( ( C vH A ) e. CH /\ B e. CH ) -> ( ( C vH A ) MH B <-> A. x e. CH ( x C_ B -> ( ( x vH ( C vH A ) ) i^i B ) C_ ( x vH ( ( C vH A ) i^i B ) ) ) ) )
85	5, 16, 84	mp2an 672	. . 3 |- ( ( C vH A ) MH B <-> A. x e. CH ( x C_ B -> ( ( x vH ( C vH A ) ) i^i B ) C_ ( x vH ( ( C vH A ) i^i B ) ) ) )
86	83, 85	sylibr 212	. 2 |- ( ( A MH B /\ C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) -> ( C vH A ) MH B )
87	1, 2	chjcomi 26062	. . . . 5 |- ( C vH A ) = ( A vH C )
88		incom 3691	. . . . . 6 |- ( B i^i ( A vH B ) ) = ( ( A vH B ) i^i B )
89	18, 88, 19	3eqtr3ri 2505	. . . . 5 |- B = ( ( A vH B ) i^i B )
90	87, 89	ineq12i 3698	. . . 4 |- ( ( C vH A ) i^i B ) = ( ( A vH C ) i^i ( ( A vH B ) i^i B ) )
91		inass 3708	. . . . 5 |- ( ( ( A vH C ) i^i ( A vH B ) ) i^i B ) = ( ( A vH C ) i^i ( ( A vH B ) i^i B ) )
92	2, 16	chub1i 26063	. . . . . . . 8 |- A C_ ( A vH B )
93		mdi 26890	. . . . . . . . . 10 |- ( ( ( C e. CH /\ ( A vH B ) e. CH /\ A e. CH ) /\ ( C MH ( A vH B ) /\ A C_ ( A vH B ) ) ) -> ( ( A vH C ) i^i ( A vH B ) ) = ( A vH ( C i^i ( A vH B ) ) ) )
94	93	exp32 605	. . . . . . . . 9 |- ( ( C e. CH /\ ( A vH B ) e. CH /\ A e. CH ) -> ( C MH ( A vH B ) -> ( A C_ ( A vH B ) -> ( ( A vH C ) i^i ( A vH B ) ) = ( A vH ( C i^i ( A vH B ) ) ) ) ) )
95	1, 29, 2, 94	mp3an 1324	. . . . . . . 8 |- ( C MH ( A vH B ) -> ( A C_ ( A vH B ) -> ( ( A vH C ) i^i ( A vH B ) ) = ( A vH ( C i^i ( A vH B ) ) ) ) )
96	92, 95	mpi 17	. . . . . . 7 |- ( C MH ( A vH B ) -> ( ( A vH C ) i^i ( A vH B ) ) = ( A vH ( C i^i ( A vH B ) ) ) )
97	2, 38	chjcomi 26062	. . . . . . . 8 |- ( A vH ( C i^i ( A vH B ) ) ) = ( ( C i^i ( A vH B ) ) vH A )
98	38, 2	chlejb1i 26070	. . . . . . . . 9 |- ( ( C i^i ( A vH B ) ) C_ A <-> ( ( C i^i ( A vH B ) ) vH A ) = A )
99	98	biimpi 194	. . . . . . . 8 |- ( ( C i^i ( A vH B ) ) C_ A -> ( ( C i^i ( A vH B ) ) vH A ) = A )
100	97, 99	syl5eq 2520	. . . . . . 7 |- ( ( C i^i ( A vH B ) ) C_ A -> ( A vH ( C i^i ( A vH B ) ) ) = A )
101	96, 100	sylan9eq 2528	. . . . . 6 |- ( ( C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) -> ( ( A vH C ) i^i ( A vH B ) ) = A )
102	101	ineq1d 3699	. . . . 5 |- ( ( C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) -> ( ( ( A vH C ) i^i ( A vH B ) ) i^i B ) = ( A i^i B ) )
103	91, 102	syl5eqr 2522	. . . 4 |- ( ( C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) -> ( ( A vH C ) i^i ( ( A vH B ) i^i B ) ) = ( A i^i B ) )
104	90, 103	syl5eq 2520	. . 3 |- ( ( C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) -> ( ( C vH A ) i^i B ) = ( A i^i B ) )
105	104	3adant1 1014	. 2 |- ( ( A MH B /\ C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) -> ( ( C vH A ) i^i B ) = ( A i^i B ) )
106	86, 105	jca 532	1 |- ( ( A MH B /\ C MH ( A vH B ) /\ ( C i^i ( A vH B ) ) C_ A ) -> ( ( C vH A ) MH B /\ ( ( C vH A ) i^i B ) = ( A i^i B ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   A.wral 2814   i^i cin 3475   C_ wss 3476   class class class wbr 4447  (class class class)co 6282   CHcch 25522   vH chj 25526   MH cmd 25559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cc 8811  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568  ax-hilex 25592  ax-hfvadd 25593  ax-hvcom 25594  ax-hvass 25595  ax-hv0cl 25596  ax-hvaddid 25597  ax-hfvmul 25598  ax-hvmulid 25599  ax-hvmulass 25600  ax-hvdistr1 25601  ax-hvdistr2 25602  ax-hvmul0 25603  ax-hfi 25672  ax-his1 25675  ax-his2 25676  ax-his3 25677  ax-his4 25678  ax-hcompl 25795

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-omul 7132  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-acn 8319  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-rlim 13271  df-sum 13468  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-cn 19494  df-cnp 19495  df-lm 19496  df-haus 19582  df-tx 19798  df-hmeo 19991  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-tms 20560  df-cfil 21429  df-cau 21430  df-cmet 21431  df-grpo 24869  df-gid 24870  df-ginv 24871  df-gdiv 24872  df-ablo 24960  df-subgo 24980  df-vc 25115  df-nv 25161  df-va 25164  df-ba 25165  df-sm 25166  df-0v 25167  df-vs 25168  df-nmcv 25169  df-ims 25170  df-dip 25287  df-ssp 25311  df-ph 25404  df-cbn 25455  df-hnorm 25561  df-hba 25562  df-hvsub 25564  df-hlim 25565  df-hcau 25566  df-sh 25800  df-ch 25815  df-oc 25846  df-ch0 25847  df-shs 25902  df-chj 25904  df-md 26875

This theorem is referenced by: (None)