Theorem polid2i 10657

Description: Generalized polarization identity. Generalization of Exercise 4(a) of [ReedSimon] p. 63.

Hypotheses

Ref	Expression
polid2.1	|- A e. ~H
polid2.2	|- B e. ~H
polid2.3	|- C e. ~H
polid2.4	|- D e. ~H

Assertion

Ref	Expression
polid2i	|- (A .ih B) = (((((A +h C) .ih (D +h B)) - ((A -h C) .ih (D -h B))) + (_i x. (((A +h (_i .h C)) .ih (D +h (_i .h B))) - ((A -h (_i .h C)) .ih (D -h (_i .h B)))))) / 4)

Proof of Theorem polid2i

Step	Hyp	Ref	Expression
1		polid2.1	. . . 4 |- A e. ~H
2		polid2.2	. . . 4 |- B e. ~H
3	1, 2	hicli 10581	. . 3 |- (A .ih B) e. CC
4		4re 7166	. . . 4 |- 4 e. RR
5	4	recni 6467	. . 3 |- 4 e. CC
6		4pos 7176	. . . 4 |- 0 < 4
7	4, 6	gt0ne0ii 6799	. . 3 |- 4 =/= 0
8	3, 5, 7	divcan3i 6934	. 2 |- ((4 x. (A .ih B)) / 4) = (A .ih B)
9		2cn 7164	. . . . 5 |- 2 e. CC
10		polid2.3	. . . . . . 7 |- C e. ~H
11		polid2.4	. . . . . . 7 |- D e. ~H
12	10, 11	hicli 10581	. . . . . 6 |- (C .ih D) e. CC
13	3, 12	addcli 6473	. . . . 5 |- ((A .ih B) + (C .ih D)) e. CC
14	3, 12	subcli 6523	. . . . 5 |- ((A .ih B) - (C .ih D)) e. CC
15	9, 13, 14	adddii 6479	. . . 4 |- (2 x. (((A .ih B) + (C .ih D)) + ((A .ih B) - (C .ih D)))) = ((2 x. ((A .ih B) + (C .ih D))) + (2 x. ((A .ih B) - (C .ih D))))
16		ppncan 6648	. . . . . . . 8 |- (((A .ih B) e. CC /\ (C .ih D) e. CC /\ (A .ih B) e. CC) -> (((A .ih B) + (C .ih D)) + ((A .ih B) - (C .ih D))) = ((A .ih B) + (A .ih B)))
17	3, 12, 3, 16	mp3an 1191	. . . . . . 7 |- (((A .ih B) + (C .ih D)) + ((A .ih B) - (C .ih D))) = ((A .ih B) + (A .ih B))
18	3	2timesi 7187	. . . . . . 7 |- (2 x. (A .ih B)) = ((A .ih B) + (A .ih B))
19	17, 18	eqtr4i 1911	. . . . . 6 |- (((A .ih B) + (C .ih D)) + ((A .ih B) - (C .ih D))) = (2 x. (A .ih B))
20	19	opreq2i 4893	. . . . 5 |- (2 x. (((A .ih B) + (C .ih D)) + ((A .ih B) - (C .ih D)))) = (2 x. (2 x. (A .ih B)))
21	9, 9, 3	mulassi 6478	. . . . 5 |- ((2 x. 2) x. (A .ih B)) = (2 x. (2 x. (A .ih B)))
22		2t2e4 7206	. . . . . 6 |- (2 x. 2) = 4
23	22	opreq1i 4892	. . . . 5 |- ((2 x. 2) x. (A .ih B)) = (4 x. (A .ih B))
24	20, 21, 23	3eqtr2ri 1916	. . . 4 |- (4 x. (A .ih B)) = (2 x. (((A .ih B) + (C .ih D)) + ((A .ih B) - (C .ih D))))
25	1, 11	hicli 10581	. . . . . . . 8 |- (A .ih D) e. CC
26	10, 2	hicli 10581	. . . . . . . 8 |- (C .ih B) e. CC
27	25, 26	addcli 6473	. . . . . . 7 |- ((A .ih D) + (C .ih B)) e. CC
28	27, 13, 13	pnncani 6649	. . . . . 6 |- ((((A .ih D) + (C .ih B)) + ((A .ih B) + (C .ih D))) - (((A .ih D) + (C .ih B)) - ((A .ih B) + (C .ih D)))) = (((A .ih B) + (C .ih D)) + ((A .ih B) + (C .ih D)))
29	1, 10, 11, 2	normlem8 10616	. . . . . . 7 |- ((A +h C) .ih (D +h B)) = (((A .ih D) + (C .ih B)) + ((A .ih B) + (C .ih D)))
30	1, 10, 11, 2	normlem9 10617	. . . . . . 7 |- ((A -h C) .ih (D -h B)) = (((A .ih D) + (C .ih B)) - ((A .ih B) + (C .ih D)))
31	29, 30	opreq12i 4894	. . . . . 6 |- (((A +h C) .ih (D +h B)) - ((A -h C) .ih (D -h B))) = ((((A .ih D) + (C .ih B)) + ((A .ih B) + (C .ih D))) - (((A .ih D) + (C .ih B)) - ((A .ih B) + (C .ih D))))
32	13	2timesi 7187	. . . . . 6 |- (2 x. ((A .ih B) + (C .ih D))) = (((A .ih B) + (C .ih D)) + ((A .ih B) + (C .ih D)))
33	28, 31, 32	3eqtr4i 1921	. . . . 5 |- (((A +h C) .ih (D +h B)) - ((A -h C) .ih (D -h B))) = (2 x. ((A .ih B) + (C .ih D)))
34		axicn 6423	. . . . . . . . . . 11 |- _i e. CC
35	34, 10	hvmulcli 10516	. . . . . . . . . 10 |- (_i .h C) e. ~H
36	34, 2	hvmulcli 10516	. . . . . . . . . 10 |- (_i .h B) e. ~H
37	1, 35, 11, 36	normlem8 10616	. . . . . . . . 9 |- ((A +h (_i .h C)) .ih (D +h (_i .h B))) = (((A .ih D) + ((_i .h C) .ih (_i .h B))) + ((A .ih (_i .h B)) + ((_i .h C) .ih D)))
38	1, 35, 11, 36	normlem9 10617	. . . . . . . . 9 |- ((A -h (_i .h C)) .ih (D -h (_i .h B))) = (((A .ih D) + ((_i .h C) .ih (_i .h B))) - ((A .ih (_i .h B)) + ((_i .h C) .ih D)))
39	37, 38	opreq12i 4894	. . . . . . . 8 |- (((A +h (_i .h C)) .ih (D +h (_i .h B))) - ((A -h (_i .h C)) .ih (D -h (_i .h B)))) = ((((A .ih D) + ((_i .h C) .ih (_i .h B))) + ((A .ih (_i .h B)) + ((_i .h C) .ih D))) - (((A .ih D) + ((_i .h C) .ih (_i .h B))) - ((A .ih (_i .h B)) + ((_i .h C) .ih D))))
40	35, 36	hicli 10581	. . . . . . . . . 10 |- ((_i .h C) .ih (_i .h B)) e. CC
41	25, 40	addcli 6473	. . . . . . . . 9 |- ((A .ih D) + ((_i .h C) .ih (_i .h B))) e. CC
42	1, 36	hicli 10581	. . . . . . . . . 10 |- (A .ih (_i .h B)) e. CC
43	35, 11	hicli 10581	. . . . . . . . . 10 |- ((_i .h C) .ih D) e. CC
44	42, 43	addcli 6473	. . . . . . . . 9 |- ((A .ih (_i .h B)) + ((_i .h C) .ih D)) e. CC
45	41, 44, 44	pnncani 6649	. . . . . . . 8 |- ((((A .ih D) + ((_i .h C) .ih (_i .h B))) + ((A .ih (_i .h B)) + ((_i .h C) .ih D))) - (((A .ih D) + ((_i .h C) .ih (_i .h B))) - ((A .ih (_i .h B)) + ((_i .h C) .ih D)))) = (((A .ih (_i .h B)) + ((_i .h C) .ih D)) + ((A .ih (_i .h B)) + ((_i .h C) .ih D)))
46	44	2timesi 7187	. . . . . . . . 9 |- (2 x. ((A .ih (_i .h B)) + ((_i .h C) .ih D))) = (((A .ih (_i .h B)) + ((_i .h C) .ih D)) + ((A .ih (_i .h B)) + ((_i .h C) .ih D)))
47		his5 10586	. . . . . . . . . . . . 13 |- ((_i e. CC /\ A e. ~H /\ B e. ~H) -> (A .ih (_i .h B)) = ((*` _i) x. (A .ih B)))
48	34, 1, 2, 47	mp3an 1191	. . . . . . . . . . . 12 |- (A .ih (_i .h B)) = ((*` _i) x. (A .ih B))
49		cji 8077	. . . . . . . . . . . . 13 |- (*` _i) = -u_i
50	49	opreq1i 4892	. . . . . . . . . . . 12 |- ((*` _i) x. (A .ih B)) = (-u_i x. (A .ih B))
51	48, 50	eqtri 1908	. . . . . . . . . . 11 |- (A .ih (_i .h B)) = (-u_i x. (A .ih B))
52		ax-his3 10584	. . . . . . . . . . . 12 |- ((_i e. CC /\ C e. ~H /\ D e. ~H) -> ((_i .h C) .ih D) = (_i x. (C .ih D)))
53	34, 10, 11, 52	mp3an 1191	. . . . . . . . . . 11 |- ((_i .h C) .ih D) = (_i x. (C .ih D))
54	51, 53	opreq12i 4894	. . . . . . . . . 10 |- ((A .ih (_i .h B)) + ((_i .h C) .ih D)) = ((-u_i x. (A .ih B)) + (_i x. (C .ih D)))
55	54	opreq2i 4893	. . . . . . . . 9 |- (2 x. ((A .ih (_i .h B)) + ((_i .h C) .ih D))) = (2 x. ((-u_i x. (A .ih B)) + (_i x. (C .ih D))))
56	46, 55	eqtr3i 1910	. . . . . . . 8 |- (((A .ih (_i .h B)) + ((_i .h C) .ih D)) + ((A .ih (_i .h B)) + ((_i .h C) .ih D))) = (2 x. ((-u_i x. (A .ih B)) + (_i x. (C .ih D))))
57	39, 45, 56	3eqtri 1912	. . . . . . 7 |- (((A +h (_i .h C)) .ih (D +h (_i .h B))) - ((A -h (_i .h C)) .ih (D -h (_i .h B)))) = (2 x. ((-u_i x. (A .ih B)) + (_i x. (C .ih D))))
58	57	opreq2i 4893	. . . . . 6 |- (_i x. (((A +h (_i .h C)) .ih (D +h (_i .h B))) - ((A -h (_i .h C)) .ih (D -h (_i .h B))))) = (_i x. (2 x. ((-u_i x. (A .ih B)) + (_i x. (C .ih D)))))
59	34	negcli 6526	. . . . . . . . 9 |- -u_i e. CC
60	59, 3	mulcli 6474	. . . . . . . 8 |- (-u_i x. (A .ih B)) e. CC
61	34, 12	mulcli 6474	. . . . . . . 8 |- (_i x. (C .ih D)) e. CC
62	60, 61	addcli 6473	. . . . . . 7 |- ((-u_i x. (A .ih B)) + (_i x. (C .ih D))) e. CC
63	9, 34, 62	mul12i 6585	. . . . . 6 |- (2 x. (_i x. ((-u_i x. (A .ih B)) + (_i x. (C .ih D))))) = (_i x. (2 x. ((-u_i x. (A .ih B)) + (_i x. (C .ih D)))))
64	34, 60, 61	adddii 6479	. . . . . . . 8 |- (_i x. ((-u_i x. (A .ih B)) + (_i x. (C .ih D)))) = ((_i x. (-u_i x. (A .ih B))) + (_i x. (_i x. (C .ih D))))
65	34, 34	mulneg2i 6609	. . . . . . . . . . . 12 |- (_i x. -u_i) = -u(_i x. _i)
66		ixi 6872	. . . . . . . . . . . . 13 |- (_i x. _i) = -u1
67	66	negeqi 6515	. . . . . . . . . . . 12 |- -u(_i x. _i) = -u-u1
68		ax1cn 6422	. . . . . . . . . . . . 13 |- 1 e. CC
69	68	negnegi 6549	. . . . . . . . . . . 12 |- -u-u1 = 1
70	65, 67, 69	3eqtri 1912	. . . . . . . . . . 11 |- (_i x. -u_i) = 1
71	70	opreq1i 4892	. . . . . . . . . 10 |- ((_i x. -u_i) x. (A .ih B)) = (1 x. (A .ih B))
72	34, 59, 3	mulassi 6478	. . . . . . . . . 10 |- ((_i x. -u_i) x. (A .ih B)) = (_i x. (-u_i x. (A .ih B)))
73	3	mulid2i 6486	. . . . . . . . . 10 |- (1 x. (A .ih B)) = (A .ih B)
74	71, 72, 73	3eqtr3i 1918	. . . . . . . . 9 |- (_i x. (-u_i x. (A .ih B))) = (A .ih B)
75	66	opreq1i 4892	. . . . . . . . . 10 |- ((_i x. _i) x. (C .ih D)) = (-u1 x. (C .ih D))
76	34, 34, 12	mulassi 6478	. . . . . . . . . 10 |- ((_i x. _i) x. (C .ih D)) = (_i x. (_i x. (C .ih D)))
77	12	mulm1i 6639	. . . . . . . . . 10 |- (-u1 x. (C .ih D)) = -u(C .ih D)
78	75, 76, 77	3eqtr3i 1918	. . . . . . . . 9 |- (_i x. (_i x. (C .ih D))) = -u(C .ih D)
79	74, 78	opreq12i 4894	. . . . . . . 8 |- ((_i x. (-u_i x. (A .ih B))) + (_i x. (_i x. (C .ih D)))) = ((A .ih B) + -u(C .ih D))
80	3, 12	negsubi 6538	. . . . . . . 8 |- ((A .ih B) + -u(C .ih D)) = ((A .ih B) - (C .ih D))
81	64, 79, 80	3eqtri 1912	. . . . . . 7 |- (_i x. ((-u_i x. (A .ih B)) + (_i x. (C .ih D)))) = ((A .ih B) - (C .ih D))
82	81	opreq2i 4893	. . . . . 6 |- (2 x. (_i x. ((-u_i x. (A .ih B)) + (_i x. (C .ih D))))) = (2 x. ((A .ih B) - (C .ih D)))
83	58, 63, 82	3eqtr2i 1915	. . . . 5 |- (_i x. (((A +h (_i .h C)) .ih (D +h (_i .h B))) - ((A -h (_i .h C)) .ih (D -h (_i .h B))))) = (2 x. ((A .ih B) - (C .ih D)))
84	33, 83	opreq12i 4894	. . . 4 |- ((((A +h C) .ih (D +h B)) - ((A -h C) .ih (D -h B))) + (_i x. (((A +h (_i .h C)) .ih (D +h (_i .h B))) - ((A -h (_i .h C)) .ih (D -h (_i .h B)))))) = ((2 x. ((A .ih B) + (C .ih D))) + (2 x. ((A .ih B) - (C .ih D))))
85	15, 24, 84	3eqtr4i 1921	. . 3 |- (4 x. (A .ih B)) = ((((A +h C) .ih (D +h B)) - ((A -h C) .ih (D -h B))) + (_i x. (((A +h (_i .h C)) .ih (D +h (_i .h B))) - ((A -h (_i .h C)) .ih (D -h (_i .h B))))))
86	85	opreq1i 4892	. 2 |- ((4 x. (A .ih B)) / 4) = (((((A +h C) .ih (D +h B)) - ((A -h C) .ih (D -h B))) + (_i x. (((A +h (_i .h C)) .ih (D +h (_i .h B))) - ((A -h (_i .h C)) .ih (D -h (_i .h B)))))) / 4)
87	8, 86	eqtr3i 1910	1 |- (A .ih B) = (((((A +h C) .ih (D +h B)) - ((A -h C) .ih (D -h B))) + (_i x. (((A +h (_i .h C)) .ih (D +h (_i .h B))) - ((A -h (_i .h C)) .ih (D -h (_i .h B)))))) / 4)

Syntax hints:   = wceq 1298   e. wcel 1300  ` cfv 3998  (class class class)co 4884  CCcc 6384  1c1 6387  _ici 6388   + caddc 6389   x. cmul 6391   - cmin 6445  -ucneg 6446   / cdiv 6447  2c2 7145  4c4 7147  *ccj 7999  ~Hchil 10420   +h cva 10421   .h csm 10422   -h cmv 10424   .ih csp 10425

This theorem is referenced by:  polidi 10658  lnopeq0lem1 11567  lnophmlem2 11579

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-hfvadd 10502  ax-hfvmul 10507  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-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-2 7154  df-3 7155  df-4 7156  df-re 8001  df-im 8002  df-cj 8003  df-hvsub 10472

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