Theorem lnophmlem2 11579

Description: Lemma for lnophmi 11580. Warning: The HTML proof page is 1/2 megabyte in size.

Hypotheses

Ref	Expression
lnophmlem.1	|- A e. ~H
lnophmlem.2	|- B e. ~H
lnophmlem.3	|- T e. LinOp
lnophmlem.4	|- A.x e. ~H (x .ih (T` x)) e. RR

Assertion

Ref	Expression
lnophmlem2	|- (A .ih (T` B)) = ((T` A) .ih B)

Distinct variable groups:   x,A   x,B   x,T

Proof of Theorem lnophmlem2

Step	Hyp	Ref	Expression
1		lnophmlem.2	. . . . . 6 |- B e. ~H
2		lnophmlem.1	. . . . . . 7 |- A e. ~H
3		lnophmlem.3	. . . . . . . . 9 |- T e. LinOp
4	3	lnopfi 11530	. . . . . . . 8 |- T:~H-->~H
5	4	ffvelrni 4788	. . . . . . 7 |- (A e. ~H -> (T` A) e. ~H)
6	2, 5	ax-mp 7	. . . . . 6 |- (T` A) e. ~H
7	4	ffvelrni 4788	. . . . . . 7 |- (B e. ~H -> (T` B) e. ~H)
8	1, 7	ax-mp 7	. . . . . 6 |- (T` B) e. ~H
9	1, 6, 2, 8	polid2i 10657	. . . . 5 |- (B .ih (T` A)) = (((((B +h A) .ih ((T` B) +h (T` A))) - ((B -h A) .ih ((T` B) -h (T` A)))) + (_i x. (((B +h (_i .h A)) .ih ((T` B) +h (_i .h (T` A)))) - ((B -h (_i .h A)) .ih ((T` B) -h (_i .h (T` A))))))) / 4)
10	1, 2	hvcomi 10521	. . . . . . . . 9 |- (B +h A) = (A +h B)
11	8, 6	hvcomi 10521	. . . . . . . . . 10 |- ((T` B) +h (T` A)) = ((T` A) +h (T` B))
12	3	lnopaddi 11532	. . . . . . . . . . 11 |- ((A e. ~H /\ B e. ~H) -> (T` (A +h B)) = ((T` A) +h (T` B)))
13	2, 1, 12	mp2an 761	. . . . . . . . . 10 |- (T` (A +h B)) = ((T` A) +h (T` B))
14	11, 13	eqtr4i 1911	. . . . . . . . 9 |- ((T` B) +h (T` A)) = (T` (A +h B))
15	10, 14	opreq12i 4894	. . . . . . . 8 |- ((B +h A) .ih ((T` B) +h (T` A))) = ((A +h B) .ih (T` (A +h B)))
16	1, 2, 8, 6	hisubcomi 10603	. . . . . . . . 9 |- ((B -h A) .ih ((T` B) -h (T` A))) = ((A -h B) .ih ((T` A) -h (T` B)))
17	3	lnopsubi 11535	. . . . . . . . . . 11 |- ((A e. ~H /\ B e. ~H) -> (T` (A -h B)) = ((T` A) -h (T` B)))
18	2, 1, 17	mp2an 761	. . . . . . . . . 10 |- (T` (A -h B)) = ((T` A) -h (T` B))
19	18	opreq2i 4893	. . . . . . . . 9 |- ((A -h B) .ih (T` (A -h B))) = ((A -h B) .ih ((T` A) -h (T` B)))
20	16, 19	eqtr4i 1911	. . . . . . . 8 |- ((B -h A) .ih ((T` B) -h (T` A))) = ((A -h B) .ih (T` (A -h B)))
21	15, 20	opreq12i 4894	. . . . . . 7 |- (((B +h A) .ih ((T` B) +h (T` A))) - ((B -h A) .ih ((T` B) -h (T` A)))) = (((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B))))
22		axicn 6423	. . . . . . . . . . . . 13 |- _i e. CC
23	22, 1	hvmulcli 10516	. . . . . . . . . . . . 13 |- (_i .h B) e. ~H
24	22, 2, 23	hvsubdistr1i 10551	. . . . . . . . . . . 12 |- (_i .h (A -h (_i .h B))) = ((_i .h A) -h (_i .h (_i .h B)))
25	22, 2	hvmulcli 10516	. . . . . . . . . . . . . 14 |- (_i .h A) e. ~H
26	22, 23	hvmulcli 10516	. . . . . . . . . . . . . 14 |- (_i .h (_i .h B)) e. ~H
27	25, 26	hvsubvali 10522	. . . . . . . . . . . . 13 |- ((_i .h A) -h (_i .h (_i .h B))) = ((_i .h A) +h (-u1 .h (_i .h (_i .h B))))
28	22, 22, 1	hvmulassi 10545	. . . . . . . . . . . . . . . 16 |- ((_i x. _i) .h B) = (_i .h (_i .h B))
29	28	opreq2i 4893	. . . . . . . . . . . . . . 15 |- (-u1 .h ((_i x. _i) .h B)) = (-u1 .h (_i .h (_i .h B)))
30		ixi 6872	. . . . . . . . . . . . . . . . . . 19 |- (_i x. _i) = -u1
31	30	opreq2i 4893	. . . . . . . . . . . . . . . . . 18 |- (-u1 x. (_i x. _i)) = (-u1 x. -u1)
32		ax1cn 6422	. . . . . . . . . . . . . . . . . . 19 |- 1 e. CC
33	32, 32	mul2negi 6610	. . . . . . . . . . . . . . . . . 18 |- (-u1 x. -u1) = (1 x. 1)
34	32	mulid1i 6485	. . . . . . . . . . . . . . . . . 18 |- (1 x. 1) = 1
35	31, 33, 34	3eqtri 1912	. . . . . . . . . . . . . . . . 17 |- (-u1 x. (_i x. _i)) = 1
36	35	opreq1i 4892	. . . . . . . . . . . . . . . 16 |- ((-u1 x. (_i x. _i)) .h B) = (1 .h B)
37	32	negcli 6526	. . . . . . . . . . . . . . . . 17 |- -u1 e. CC
38	22, 22	mulcli 6474	. . . . . . . . . . . . . . . . 17 |- (_i x. _i) e. CC
39	37, 38, 1	hvmulassi 10545	. . . . . . . . . . . . . . . 16 |- ((-u1 x. (_i x. _i)) .h B) = (-u1 .h ((_i x. _i) .h B))
40		ax-hvmulid 10508	. . . . . . . . . . . . . . . . 17 |- (B e. ~H -> (1 .h B) = B)
41	1, 40	ax-mp 7	. . . . . . . . . . . . . . . 16 |- (1 .h B) = B
42	36, 39, 41	3eqtr3i 1918	. . . . . . . . . . . . . . 15 |- (-u1 .h ((_i x. _i) .h B)) = B
43	29, 42	eqtr3i 1910	. . . . . . . . . . . . . 14 |- (-u1 .h (_i .h (_i .h B))) = B
44	43	opreq2i 4893	. . . . . . . . . . . . 13 |- ((_i .h A) +h (-u1 .h (_i .h (_i .h B)))) = ((_i .h A) +h B)
45	27, 44	eqtri 1908	. . . . . . . . . . . 12 |- ((_i .h A) -h (_i .h (_i .h B))) = ((_i .h A) +h B)
46	25, 1	hvcomi 10521	. . . . . . . . . . . 12 |- ((_i .h A) +h B) = (B +h (_i .h A))
47	24, 45, 46	3eqtri 1912	. . . . . . . . . . 11 |- (_i .h (A -h (_i .h B))) = (B +h (_i .h A))
48	47	fveq2i 4684	. . . . . . . . . . . 12 |- (T` (_i .h (A -h (_i .h B)))) = (T` (B +h (_i .h A)))
49	2, 23	hvsubcli 10523	. . . . . . . . . . . . 13 |- (A -h (_i .h B)) e. ~H
50	3	lnopmuli 11533	. . . . . . . . . . . . 13 |- ((_i e. CC /\ (A -h (_i .h B)) e. ~H) -> (T` (_i .h (A -h (_i .h B)))) = (_i .h (T` (A -h (_i .h B)))))
51	22, 49, 50	mp2an 761	. . . . . . . . . . . 12 |- (T` (_i .h (A -h (_i .h B)))) = (_i .h (T` (A -h (_i .h B))))
52	3	lnopaddmuli 11534	. . . . . . . . . . . . 13 |- ((_i e. CC /\ B e. ~H /\ A e. ~H) -> (T` (B +h (_i .h A))) = ((T` B) +h (_i .h (T` A))))
53	22, 1, 2, 52	mp3an 1191	. . . . . . . . . . . 12 |- (T` (B +h (_i .h A))) = ((T` B) +h (_i .h (T` A)))
54	48, 51, 53	3eqtr3i 1918	. . . . . . . . . . 11 |- (_i .h (T` (A -h (_i .h B)))) = ((T` B) +h (_i .h (T` A)))
55	47, 54	opreq12i 4894	. . . . . . . . . 10 |- ((_i .h (A -h (_i .h B))) .ih (_i .h (T` (A -h (_i .h B))))) = ((B +h (_i .h A)) .ih ((T` B) +h (_i .h (T` A))))
56	4	ffvelrni 4788	. . . . . . . . . . . . 13 |- ((A -h (_i .h B)) e. ~H -> (T` (A -h (_i .h B))) e. ~H)
57	49, 56	ax-mp 7	. . . . . . . . . . . 12 |- (T` (A -h (_i .h B))) e. ~H
58	22, 22, 49, 57	his35i 10588	. . . . . . . . . . 11 |- ((_i .h (A -h (_i .h B))) .ih (_i .h (T` (A -h (_i .h B))))) = ((_i x. (*` _i)) x. ((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))))
59		cji 8077	. . . . . . . . . . . . . 14 |- (*` _i) = -u_i
60	59	opreq2i 4893	. . . . . . . . . . . . 13 |- (_i x. (*` _i)) = (_i x. -u_i)
61	22, 22	mulneg2i 6609	. . . . . . . . . . . . 13 |- (_i x. -u_i) = -u(_i x. _i)
62	30	negeqi 6515	. . . . . . . . . . . . . 14 |- -u(_i x. _i) = -u-u1
63	32	negnegi 6549	. . . . . . . . . . . . . 14 |- -u-u1 = 1
64	62, 63	eqtri 1908	. . . . . . . . . . . . 13 |- -u(_i x. _i) = 1
65	60, 61, 64	3eqtri 1912	. . . . . . . . . . . 12 |- (_i x. (*` _i)) = 1
66	65	opreq1i 4892	. . . . . . . . . . 11 |- ((_i x. (*` _i)) x. ((A -h (_i .h B)) .ih (T` (A -h (_i .h B))))) = (1 x. ((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))))
67		lnophmlem.4	. . . . . . . . . . . . . 14 |- A.x e. ~H (x .ih (T` x)) e. RR
68	49, 2, 3, 67	lnophmlem1 11578	. . . . . . . . . . . . 13 |- ((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) e. RR
69	68	recni 6467	. . . . . . . . . . . 12 |- ((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) e. CC
70	69	mulid2i 6486	. . . . . . . . . . 11 |- (1 x. ((A -h (_i .h B)) .ih (T` (A -h (_i .h B))))) = ((A -h (_i .h B)) .ih (T` (A -h (_i .h B))))
71	58, 66, 70	3eqtri 1912	. . . . . . . . . 10 |- ((_i .h (A -h (_i .h B))) .ih (_i .h (T` (A -h (_i .h B))))) = ((A -h (_i .h B)) .ih (T` (A -h (_i .h B))))
72	55, 71	eqtr3i 1910	. . . . . . . . 9 |- ((B +h (_i .h A)) .ih ((T` B) +h (_i .h (T` A)))) = ((A -h (_i .h B)) .ih (T` (A -h (_i .h B))))
73	22, 6	hvmulcli 10516	. . . . . . . . . . 11 |- (_i .h (T` A)) e. ~H
74	1, 25, 8, 73	hisubcomi 10603	. . . . . . . . . 10 |- ((B -h (_i .h A)) .ih ((T` B) -h (_i .h (T` A)))) = (((_i .h A) -h B) .ih ((_i .h (T` A)) -h (T` B)))
75	30	opreq1i 4892	. . . . . . . . . . . . . 14 |- ((_i x. _i) .h B) = (-u1 .h B)
76	28, 75	eqtr3i 1910	. . . . . . . . . . . . 13 |- (_i .h (_i .h B)) = (-u1 .h B)
77	76	opreq2i 4893	. . . . . . . . . . . 12 |- ((_i .h A) +h (_i .h (_i .h B))) = ((_i .h A) +h (-u1 .h B))
78	22, 2, 23	hvdistr1i 10550	. . . . . . . . . . . 12 |- (_i .h (A +h (_i .h B))) = ((_i .h A) +h (_i .h (_i .h B)))
79	25, 1	hvsubvali 10522	. . . . . . . . . . . 12 |- ((_i .h A) -h B) = ((_i .h A) +h (-u1 .h B))
80	77, 78, 79	3eqtr4i 1921	. . . . . . . . . . 11 |- (_i .h (A +h (_i .h B))) = ((_i .h A) -h B)
81	80	fveq2i 4684	. . . . . . . . . . . 12 |- (T` (_i .h (A +h (_i .h B)))) = (T` ((_i .h A) -h B))
82	2, 23	hvaddcli 10520	. . . . . . . . . . . . 13 |- (A +h (_i .h B)) e. ~H
83	3	lnopmuli 11533	. . . . . . . . . . . . 13 |- ((_i e. CC /\ (A +h (_i .h B)) e. ~H) -> (T` (_i .h (A +h (_i .h B)))) = (_i .h (T` (A +h (_i .h B)))))
84	22, 82, 83	mp2an 761	. . . . . . . . . . . 12 |- (T` (_i .h (A +h (_i .h B)))) = (_i .h (T` (A +h (_i .h B))))
85	3	lnopmulsubi 11537	. . . . . . . . . . . . 13 |- ((_i e. CC /\ A e. ~H /\ B e. ~H) -> (T` ((_i .h A) -h B)) = ((_i .h (T` A)) -h (T` B)))
86	22, 2, 1, 85	mp3an 1191	. . . . . . . . . . . 12 |- (T` ((_i .h A) -h B)) = ((_i .h (T` A)) -h (T` B))
87	81, 84, 86	3eqtr3i 1918	. . . . . . . . . . 11 |- (_i .h (T` (A +h (_i .h B)))) = ((_i .h (T` A)) -h (T` B))
88	80, 87	opreq12i 4894	. . . . . . . . . 10 |- ((_i .h (A +h (_i .h B))) .ih (_i .h (T` (A +h (_i .h B))))) = (((_i .h A) -h B) .ih ((_i .h (T` A)) -h (T` B)))
89	4	ffvelrni 4788	. . . . . . . . . . . . 13 |- ((A +h (_i .h B)) e. ~H -> (T` (A +h (_i .h B))) e. ~H)
90	82, 89	ax-mp 7	. . . . . . . . . . . 12 |- (T` (A +h (_i .h B))) e. ~H
91	22, 22, 82, 90	his35i 10588	. . . . . . . . . . 11 |- ((_i .h (A +h (_i .h B))) .ih (_i .h (T` (A +h (_i .h B))))) = ((_i x. (*` _i)) x. ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))))
92	65	opreq1i 4892	. . . . . . . . . . 11 |- ((_i x. (*` _i)) x. ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))) = (1 x. ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))))
93	82, 2, 3, 67	lnophmlem1 11578	. . . . . . . . . . . . 13 |- ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))) e. RR
94	93	recni 6467	. . . . . . . . . . . 12 |- ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))) e. CC
95	94	mulid2i 6486	. . . . . . . . . . 11 |- (1 x. ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))) = ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))
96	91, 92, 95	3eqtri 1912	. . . . . . . . . 10 |- ((_i .h (A +h (_i .h B))) .ih (_i .h (T` (A +h (_i .h B))))) = ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))
97	74, 88, 96	3eqtr2i 1915	. . . . . . . . 9 |- ((B -h (_i .h A)) .ih ((T` B) -h (_i .h (T` A)))) = ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))
98	72, 97	opreq12i 4894	. . . . . . . 8 |- (((B +h (_i .h A)) .ih ((T` B) +h (_i .h (T` A)))) - ((B -h (_i .h A)) .ih ((T` B) -h (_i .h (T` A))))) = (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))))
99	98	opreq2i 4893	. . . . . . 7 |- (_i x. (((B +h (_i .h A)) .ih ((T` B) +h (_i .h (T` A)))) - ((B -h (_i .h A)) .ih ((T` B) -h (_i .h (T` A)))))) = (_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))))
100	21, 99	opreq12i 4894	. . . . . 6 |- ((((B +h A) .ih ((T` B) +h (T` A))) - ((B -h A) .ih ((T` B) -h (T` A)))) + (_i x. (((B +h (_i .h A)) .ih ((T` B) +h (_i .h (T` A)))) - ((B -h (_i .h A)) .ih ((T` B) -h (_i .h (T` A))))))) = ((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) + (_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))))))
101	100	opreq1i 4892	. . . . 5 |- (((((B +h A) .ih ((T` B) +h (T` A))) - ((B -h A) .ih ((T` B) -h (T` A)))) + (_i x. (((B +h (_i .h A)) .ih ((T` B) +h (_i .h (T` A)))) - ((B -h (_i .h A)) .ih ((T` B) -h (_i .h (T` A))))))) / 4) = (((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) + (_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))))) / 4)
102	9, 101	eqtri 1908	. . . 4 |- (B .ih (T` A)) = (((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) + (_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))))) / 4)
103	102	fveq2i 4684	. . 3 |- (*` (B .ih (T` A))) = (*` (((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) + (_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))))) / 4))
104		4re 7166	. . . . 5 |- 4 e. RR
105		4pos 7176	. . . . 5 |- 0 < 4
106	104, 105	gt0ne0ii 6799	. . . 4 |- 4 =/= 0
107	2, 1	hvaddcli 10520	. . . . . . . . 9 |- (A +h B) e. ~H
108	107, 2, 3, 67	lnophmlem1 11578	. . . . . . . 8 |- ((A +h B) .ih (T` (A +h B))) e. RR
109	2, 1	hvsubcli 10523	. . . . . . . . 9 |- (A -h B) e. ~H
110	109, 2, 3, 67	lnophmlem1 11578	. . . . . . . 8 |- ((A -h B) .ih (T` (A -h B))) e. RR
111	108, 110	resubcli 6602	. . . . . . 7 |- (((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) e. RR
112	111	recni 6467	. . . . . 6 |- (((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) e. CC
113	68, 93	resubcli 6602	. . . . . . . 8 |- (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))) e. RR
114	113	recni 6467	. . . . . . 7 |- (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))) e. CC
115	22, 114	mulcli 6474	. . . . . 6 |- (_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))))) e. CC
116	112, 115	addcli 6473	. . . . 5 |- ((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) + (_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))))) e. CC
117	104	recni 6467	. . . . 5 |- 4 e. CC
118	116, 117	cjdivi 8140	. . . 4 |- (4 =/= 0 -> (*` (((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) + (_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))))) / 4)) = ((*` ((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) + (_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))))))) / (*` 4)))
119	106, 118	ax-mp 7	. . 3 |- (*` (((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) + (_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))))) / 4)) = ((*` ((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) + (_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))))))) / (*` 4))
120	82, 1, 3, 67	lnophmlem1 11578	. . . . . . . . 9 |- ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))) e. RR
121	68, 120	resubcli 6602	. . . . . . . 8 |- (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))) e. RR
122	121	recni 6467	. . . . . . 7 |- (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))) e. CC
123	22, 122	mulcli 6474	. . . . . 6 |- (_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))))) e. CC
124	112, 123	negsubi 6538	. . . . 5 |- ((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) + -u(_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))))) = ((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) - (_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))))))
125	22, 114	mulneg2i 6609	. . . . . . . 8 |- (_i x. -u(((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))))) = -u(_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))))
126	69, 94	negsubdi2i 6614	. . . . . . . . 9 |- -u(((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))) = (((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))) - ((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))))
127	126	opreq2i 4893	. . . . . . . 8 |- (_i x. -u(((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))))) = (_i x. (((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))) - ((A -h (_i .h B)) .ih (T` (A -h (_i .h B))))))
128	125, 127	eqtr3i 1910	. . . . . . 7 |- -u(_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))))) = (_i x. (((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))) - ((A -h (_i .h B)) .ih (T` (A -h (_i .h B))))))
129	128	opreq2i 4893	. . . . . 6 |- ((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) + -u(_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))))) = ((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) + (_i x. (((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))) - ((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))))))
130	13	opreq2i 4893	. . . . . . . 8 |- ((A +h B) .ih (T` (A +h B))) = ((A +h B) .ih ((T` A) +h (T` B)))
131	130, 19	opreq12i 4894	. . . . . . 7 |- (((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) = (((A +h B) .ih ((T` A) +h (T` B))) - ((A -h B) .ih ((T` A) -h (T` B))))
132	3	lnopaddmuli 11534	. . . . . . . . . . 11 |- ((_i e. CC /\ A e. ~H /\ B e. ~H) -> (T` (A +h (_i .h B))) = ((T` A) +h (_i .h (T` B))))
133	22, 2, 1, 132	mp3an 1191	. . . . . . . . . 10 |- (T` (A +h (_i .h B))) = ((T` A) +h (_i .h (T` B)))
134	133	opreq2i 4893	. . . . . . . . 9 |- ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))) = ((A +h (_i .h B)) .ih ((T` A) +h (_i .h (T` B))))
135	3	lnopsubmuli 11536	. . . . . . . . . . 11 |- ((_i e. CC /\ A e. ~H /\ B e. ~H) -> (T` (A -h (_i .h B))) = ((T` A) -h (_i .h (T` B))))
136	22, 2, 1, 135	mp3an 1191	. . . . . . . . . 10 |- (T` (A -h (_i .h B))) = ((T` A) -h (_i .h (T` B)))
137	136	opreq2i 4893	. . . . . . . . 9 |- ((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) = ((A -h (_i .h B)) .ih ((T` A) -h (_i .h (T` B))))
138	134, 137	opreq12i 4894	. . . . . . . 8 |- (((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))) - ((A -h (_i .h B)) .ih (T` (A -h (_i .h B))))) = (((A +h (_i .h B)) .ih ((T` A) +h (_i .h (T` B)))) - ((A -h (_i .h B)) .ih ((T` A) -h (_i .h (T` B)))))
139	138	opreq2i 4893	. . . . . . 7 |- (_i x. (((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))) - ((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))))) = (_i x. (((A +h (_i .h B)) .ih ((T` A) +h (_i .h (T` B)))) - ((A -h (_i .h B)) .ih ((T` A) -h (_i .h (T` B))))))
140	131, 139	opreq12i 4894	. . . . . 6 |- ((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) + (_i x. (((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))) - ((A -h (_i .h B)) .ih (T` (A -h (_i .h B))))))) = ((((A +h B) .ih ((T` A) +h (T` B))) - ((A -h B) .ih ((T` A) -h (T` B)))) + (_i x. (((A +h (_i .h B)) .ih ((T` A) +h (_i .h (T` B)))) - ((A -h (_i .h B)) .ih ((T` A) -h (_i .h (T` B)))))))
141	129, 140	eqtr2i 1909	. . . . 5 |- ((((A +h B) .ih ((T` A) +h (T` B))) - ((A -h B) .ih ((T` A) -h (T` B)))) + (_i x. (((A +h (_i .h B)) .ih ((T` A) +h (_i .h (T` B)))) - ((A -h (_i .h B)) .ih ((T` A) -h (_i .h (T` B))))))) = ((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) + -u(_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))))))
142		cjreim 8078	. . . . . 6 |- (((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) e. RR /\ (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))) e. RR) -> (*` ((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) + (_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))))))) = ((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) - (_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B))))))))
143	111, 113, 142	mp2an 761	. . . . 5 |- (*` ((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) + (_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))))))) = ((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) - (_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))))))
144	124, 141, 143	3eqtr4ri 1923	. . . 4 |- (*` ((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) + (_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))))))) = ((((A +h B) .ih ((T` A) +h (T` B))) - ((A -h B) .ih ((T` A) -h (T` B)))) + (_i x. (((A +h (_i .h B)) .ih ((T` A) +h (_i .h (T` B)))) - ((A -h (_i .h B)) .ih ((T` A) -h (_i .h (T` B)))))))
145		cjre 8060	. . . . 5 |- (4 e. RR -> (*` 4) = 4)
146	104, 145	ax-mp 7	. . . 4 |- (*` 4) = 4
147	144, 146	opreq12i 4894	. . 3 |- ((*` ((((A +h B) .ih (T` (A +h B))) - ((A -h B) .ih (T` (A -h B)))) + (_i x. (((A -h (_i .h B)) .ih (T` (A -h (_i .h B)))) - ((A +h (_i .h B)) .ih (T` (A +h (_i .h B)))))))) / (*` 4)) = (((((A +h B) .ih ((T` A) +h (T` B))) - ((A -h B) .ih ((T` A) -h (T` B)))) + (_i x. (((A +h (_i .h B)) .ih ((T` A) +h (_i .h (T` B)))) - ((A -h (_i .h B)) .ih ((T` A) -h (_i .h (T` B))))))) / 4)
148	103, 119, 147	3eqtrri 1913	. 2 |- (((((A +h B) .ih ((T` A) +h (T` B))) - ((A -h B) .ih ((T` A) -h (T` B)))) + (_i x. (((A +h (_i .h B)) .ih ((T` A) +h (_i .h (T` B)))) - ((A -h (_i .h B)) .ih ((T` A) -h (_i .h (T` B))))))) / 4) = (*` (B .ih (T` A)))
149	2, 8, 1, 6	polid2i 10657	. 2 |- (A .ih (T` B)) = (((((A +h B) .ih ((T` A) +h (T` B))) - ((A -h B) .ih ((T` A) -h (T` B)))) + (_i x. (((A +h (_i .h B)) .ih ((T` A) +h (_i .h (T` B)))) - ((A -h (_i .h B)) .ih ((T` A) -h (_i .h (T` B))))))) / 4)
150	6, 1	his1i 10599	. 2 |- ((T` A) .ih B) = (*` (B .ih (T` A)))
151	148, 149, 150	3eqtr4i 1921	1 |- (A .ih (T` B)) = ((T` A) .ih B)

Syntax hints:   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387  _ici 6388   + caddc 6389   x. cmul 6391   - cmin 6445  -ucneg 6446   / cdiv 6447  4c4 7147  *ccj 7999  ~Hchil 10420   +h cva 10421   .h csm 10422   -h cmv 10424   .ih csp 10425  LinOpclo 10448

This theorem is referenced by:  lnophmi 11580

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731  ax-hilex 10501  ax-hfvadd 10502  ax-hvcom 10503  ax-hvass 10504  ax-hv0cl 10505  ax-hvaddid 10506  ax-hfvmul 10507  ax-hvmulid 10508  ax-hvmulass 10509  ax-hvdistr1 10510  ax-hvdistr2 10511  ax-hvmul0 10512  ax-hfi 10579  ax-his1 10582  ax-his2 10583  ax-his3 10584

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-hvsub 10472  df-lnop 11404

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