Theorem polid 10659

Description: Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of axiom ax-his3 10584.

Assertion

Ref	Expression
polid	|- ((A e. ~H /\ B e. ~H) -> (A .ih B) = (((((normh` (A +h B))^2) - ((normh` (A -h B))^2)) + (_i x. (((normh` (A +h (_i .h B)))^2) - ((normh` (A -h (_i .h B)))^2)))) / 4))

Proof of Theorem polid

Step	Hyp	Ref	Expression
1		opreq1 4889	. . 3 |- (A = if(A e. ~H, A, 0h) -> (A .ih B) = (if(A e. ~H, A, 0h) .ih B))
2		opreq1 4889	. . . . . . . 8 |- (A = if(A e. ~H, A, 0h) -> (A +h B) = (if(A e. ~H, A, 0h) +h B))
3	2	fveq2d 4685	. . . . . . 7 |- (A = if(A e. ~H, A, 0h) -> (normh` (A +h B)) = (normh` (if(A e. ~H, A, 0h) +h B)))
4	3	opreq1d 4897	. . . . . 6 |- (A = if(A e. ~H, A, 0h) -> ((normh` (A +h B))^2) = ((normh` (if(A e. ~H, A, 0h) +h B))^2))
5		opreq1 4889	. . . . . . . 8 |- (A = if(A e. ~H, A, 0h) -> (A -h B) = (if(A e. ~H, A, 0h) -h B))
6	5	fveq2d 4685	. . . . . . 7 |- (A = if(A e. ~H, A, 0h) -> (normh` (A -h B)) = (normh` (if(A e. ~H, A, 0h) -h B)))
7	6	opreq1d 4897	. . . . . 6 |- (A = if(A e. ~H, A, 0h) -> ((normh` (A -h B))^2) = ((normh` (if(A e. ~H, A, 0h) -h B))^2))
8	4, 7	opreq12d 4900	. . . . 5 |- (A = if(A e. ~H, A, 0h) -> (((normh` (A +h B))^2) - ((normh` (A -h B))^2)) = (((normh` (if(A e. ~H, A, 0h) +h B))^2) - ((normh` (if(A e. ~H, A, 0h) -h B))^2)))
9		opreq1 4889	. . . . . . . . 9 |- (A = if(A e. ~H, A, 0h) -> (A +h (_i .h B)) = (if(A e. ~H, A, 0h) +h (_i .h B)))
10	9	fveq2d 4685	. . . . . . . 8 |- (A = if(A e. ~H, A, 0h) -> (normh` (A +h (_i .h B))) = (normh` (if(A e. ~H, A, 0h) +h (_i .h B))))
11	10	opreq1d 4897	. . . . . . 7 |- (A = if(A e. ~H, A, 0h) -> ((normh` (A +h (_i .h B)))^2) = ((normh` (if(A e. ~H, A, 0h) +h (_i .h B)))^2))
12		opreq1 4889	. . . . . . . . 9 |- (A = if(A e. ~H, A, 0h) -> (A -h (_i .h B)) = (if(A e. ~H, A, 0h) -h (_i .h B)))
13	12	fveq2d 4685	. . . . . . . 8 |- (A = if(A e. ~H, A, 0h) -> (normh` (A -h (_i .h B))) = (normh` (if(A e. ~H, A, 0h) -h (_i .h B))))
14	13	opreq1d 4897	. . . . . . 7 |- (A = if(A e. ~H, A, 0h) -> ((normh` (A -h (_i .h B)))^2) = ((normh` (if(A e. ~H, A, 0h) -h (_i .h B)))^2))
15	11, 14	opreq12d 4900	. . . . . 6 |- (A = if(A e. ~H, A, 0h) -> (((normh` (A +h (_i .h B)))^2) - ((normh` (A -h (_i .h B)))^2)) = (((normh` (if(A e. ~H, A, 0h) +h (_i .h B)))^2) - ((normh` (if(A e. ~H, A, 0h) -h (_i .h B)))^2)))
16	15	opreq2d 4898	. . . . 5 |- (A = if(A e. ~H, A, 0h) -> (_i x. (((normh` (A +h (_i .h B)))^2) - ((normh` (A -h (_i .h B)))^2))) = (_i x. (((normh` (if(A e. ~H, A, 0h) +h (_i .h B)))^2) - ((normh` (if(A e. ~H, A, 0h) -h (_i .h B)))^2))))
17	8, 16	opreq12d 4900	. . . 4 |- (A = if(A e. ~H, A, 0h) -> ((((normh` (A +h B))^2) - ((normh` (A -h B))^2)) + (_i x. (((normh` (A +h (_i .h B)))^2) - ((normh` (A -h (_i .h B)))^2)))) = ((((normh` (if(A e. ~H, A, 0h) +h B))^2) - ((normh` (if(A e. ~H, A, 0h) -h B))^2)) + (_i x. (((normh` (if(A e. ~H, A, 0h) +h (_i .h B)))^2) - ((normh` (if(A e. ~H, A, 0h) -h (_i .h B)))^2)))))
18	17	opreq1d 4897	. . 3 |- (A = if(A e. ~H, A, 0h) -> (((((normh` (A +h B))^2) - ((normh` (A -h B))^2)) + (_i x. (((normh` (A +h (_i .h B)))^2) - ((normh` (A -h (_i .h B)))^2)))) / 4) = (((((normh` (if(A e. ~H, A, 0h) +h B))^2) - ((normh` (if(A e. ~H, A, 0h) -h B))^2)) + (_i x. (((normh` (if(A e. ~H, A, 0h) +h (_i .h B)))^2) - ((normh` (if(A e. ~H, A, 0h) -h (_i .h B)))^2)))) / 4))
19	1, 18	eqeq12d 1899	. 2 |- (A = if(A e. ~H, A, 0h) -> ((A .ih B) = (((((normh` (A +h B))^2) - ((normh` (A -h B))^2)) + (_i x. (((normh` (A +h (_i .h B)))^2) - ((normh` (A -h (_i .h B)))^2)))) / 4) <-> (if(A e. ~H, A, 0h) .ih B) = (((((normh` (if(A e. ~H, A, 0h) +h B))^2) - ((normh` (if(A e. ~H, A, 0h) -h B))^2)) + (_i x. (((normh` (if(A e. ~H, A, 0h) +h (_i .h B)))^2) - ((normh` (if(A e. ~H, A, 0h) -h (_i .h B)))^2)))) / 4)))
20		opreq2 4890	. . 3 |- (B = if(B e. ~H, B, 0h) -> (if(A e. ~H, A, 0h) .ih B) = (if(A e. ~H, A, 0h) .ih if(B e. ~H, B, 0h)))
21		opreq2 4890	. . . . . . . 8 |- (B = if(B e. ~H, B, 0h) -> (if(A e. ~H, A, 0h) +h B) = (if(A e. ~H, A, 0h) +h if(B e. ~H, B, 0h)))
22	21	fveq2d 4685	. . . . . . 7 |- (B = if(B e. ~H, B, 0h) -> (normh` (if(A e. ~H, A, 0h) +h B)) = (normh` (if(A e. ~H, A, 0h) +h if(B e. ~H, B, 0h))))
23	22	opreq1d 4897	. . . . . 6 |- (B = if(B e. ~H, B, 0h) -> ((normh` (if(A e. ~H, A, 0h) +h B))^2) = ((normh` (if(A e. ~H, A, 0h) +h if(B e. ~H, B, 0h)))^2))
24		opreq2 4890	. . . . . . . 8 |- (B = if(B e. ~H, B, 0h) -> (if(A e. ~H, A, 0h) -h B) = (if(A e. ~H, A, 0h) -h if(B e. ~H, B, 0h)))
25	24	fveq2d 4685	. . . . . . 7 |- (B = if(B e. ~H, B, 0h) -> (normh` (if(A e. ~H, A, 0h) -h B)) = (normh` (if(A e. ~H, A, 0h) -h if(B e. ~H, B, 0h))))
26	25	opreq1d 4897	. . . . . 6 |- (B = if(B e. ~H, B, 0h) -> ((normh` (if(A e. ~H, A, 0h) -h B))^2) = ((normh` (if(A e. ~H, A, 0h) -h if(B e. ~H, B, 0h)))^2))
27	23, 26	opreq12d 4900	. . . . 5 |- (B = if(B e. ~H, B, 0h) -> (((normh` (if(A e. ~H, A, 0h) +h B))^2) - ((normh` (if(A e. ~H, A, 0h) -h B))^2)) = (((normh` (if(A e. ~H, A, 0h) +h if(B e. ~H, B, 0h)))^2) - ((normh` (if(A e. ~H, A, 0h) -h if(B e. ~H, B, 0h)))^2)))
28		opreq2 4890	. . . . . . . . . 10 |- (B = if(B e. ~H, B, 0h) -> (_i .h B) = (_i .h if(B e. ~H, B, 0h)))
29	28	opreq2d 4898	. . . . . . . . 9 |- (B = if(B e. ~H, B, 0h) -> (if(A e. ~H, A, 0h) +h (_i .h B)) = (if(A e. ~H, A, 0h) +h (_i .h if(B e. ~H, B, 0h))))
30	29	fveq2d 4685	. . . . . . . 8 |- (B = if(B e. ~H, B, 0h) -> (normh` (if(A e. ~H, A, 0h) +h (_i .h B))) = (normh` (if(A e. ~H, A, 0h) +h (_i .h if(B e. ~H, B, 0h)))))
31	30	opreq1d 4897	. . . . . . 7 |- (B = if(B e. ~H, B, 0h) -> ((normh` (if(A e. ~H, A, 0h) +h (_i .h B)))^2) = ((normh` (if(A e. ~H, A, 0h) +h (_i .h if(B e. ~H, B, 0h))))^2))
32	28	opreq2d 4898	. . . . . . . . 9 |- (B = if(B e. ~H, B, 0h) -> (if(A e. ~H, A, 0h) -h (_i .h B)) = (if(A e. ~H, A, 0h) -h (_i .h if(B e. ~H, B, 0h))))
33	32	fveq2d 4685	. . . . . . . 8 |- (B = if(B e. ~H, B, 0h) -> (normh` (if(A e. ~H, A, 0h) -h (_i .h B))) = (normh` (if(A e. ~H, A, 0h) -h (_i .h if(B e. ~H, B, 0h)))))
34	33	opreq1d 4897	. . . . . . 7 |- (B = if(B e. ~H, B, 0h) -> ((normh` (if(A e. ~H, A, 0h) -h (_i .h B)))^2) = ((normh` (if(A e. ~H, A, 0h) -h (_i .h if(B e. ~H, B, 0h))))^2))
35	31, 34	opreq12d 4900	. . . . . 6 |- (B = if(B e. ~H, B, 0h) -> (((normh` (if(A e. ~H, A, 0h) +h (_i .h B)))^2) - ((normh` (if(A e. ~H, A, 0h) -h (_i .h B)))^2)) = (((normh` (if(A e. ~H, A, 0h) +h (_i .h if(B e. ~H, B, 0h))))^2) - ((normh` (if(A e. ~H, A, 0h) -h (_i .h if(B e. ~H, B, 0h))))^2)))
36	35	opreq2d 4898	. . . . 5 |- (B = if(B e. ~H, B, 0h) -> (_i x. (((normh` (if(A e. ~H, A, 0h) +h (_i .h B)))^2) - ((normh` (if(A e. ~H, A, 0h) -h (_i .h B)))^2))) = (_i x. (((normh` (if(A e. ~H, A, 0h) +h (_i .h if(B e. ~H, B, 0h))))^2) - ((normh` (if(A e. ~H, A, 0h) -h (_i .h if(B e. ~H, B, 0h))))^2))))
37	27, 36	opreq12d 4900	. . . 4 |- (B = if(B e. ~H, B, 0h) -> ((((normh` (if(A e. ~H, A, 0h) +h B))^2) - ((normh` (if(A e. ~H, A, 0h) -h B))^2)) + (_i x. (((normh` (if(A e. ~H, A, 0h) +h (_i .h B)))^2) - ((normh` (if(A e. ~H, A, 0h) -h (_i .h B)))^2)))) = ((((normh` (if(A e. ~H, A, 0h) +h if(B e. ~H, B, 0h)))^2) - ((normh` (if(A e. ~H, A, 0h) -h if(B e. ~H, B, 0h)))^2)) + (_i x. (((normh` (if(A e. ~H, A, 0h) +h (_i .h if(B e. ~H, B, 0h))))^2) - ((normh` (if(A e. ~H, A, 0h) -h (_i .h if(B e. ~H, B, 0h))))^2)))))
38	37	opreq1d 4897	. . 3 |- (B = if(B e. ~H, B, 0h) -> (((((normh` (if(A e. ~H, A, 0h) +h B))^2) - ((normh` (if(A e. ~H, A, 0h) -h B))^2)) + (_i x. (((normh` (if(A e. ~H, A, 0h) +h (_i .h B)))^2) - ((normh` (if(A e. ~H, A, 0h) -h (_i .h B)))^2)))) / 4) = (((((normh` (if(A e. ~H, A, 0h) +h if(B e. ~H, B, 0h)))^2) - ((normh` (if(A e. ~H, A, 0h) -h if(B e. ~H, B, 0h)))^2)) + (_i x. (((normh` (if(A e. ~H, A, 0h) +h (_i .h if(B e. ~H, B, 0h))))^2) - ((normh` (if(A e. ~H, A, 0h) -h (_i .h if(B e. ~H, B, 0h))))^2)))) / 4))
39	20, 38	eqeq12d 1899	. 2 |- (B = if(B e. ~H, B, 0h) -> ((if(A e. ~H, A, 0h) .ih B) = (((((normh` (if(A e. ~H, A, 0h) +h B))^2) - ((normh` (if(A e. ~H, A, 0h) -h B))^2)) + (_i x. (((normh` (if(A e. ~H, A, 0h) +h (_i .h B)))^2) - ((normh` (if(A e. ~H, A, 0h) -h (_i .h B)))^2)))) / 4) <-> (if(A e. ~H, A, 0h) .ih if(B e. ~H, B, 0h)) = (((((normh` (if(A e. ~H, A, 0h) +h if(B e. ~H, B, 0h)))^2) - ((normh` (if(A e. ~H, A, 0h) -h if(B e. ~H, B, 0h)))^2)) + (_i x. (((normh` (if(A e. ~H, A, 0h) +h (_i .h if(B e. ~H, B, 0h))))^2) - ((normh` (if(A e. ~H, A, 0h) -h (_i .h if(B e. ~H, B, 0h))))^2)))) / 4)))
40		ax-hv0cl 10505	. . . 4 |- 0h e. ~H
41	40	elimel 3025	. . 3 |- if(A e. ~H, A, 0h) e. ~H
42	40	elimel 3025	. . 3 |- if(B e. ~H, B, 0h) e. ~H
43	41, 42	polidi 10658	. 2 |- (if(A e. ~H, A, 0h) .ih if(B e. ~H, B, 0h)) = (((((normh` (if(A e. ~H, A, 0h) +h if(B e. ~H, B, 0h)))^2) - ((normh` (if(A e. ~H, A, 0h) -h if(B e. ~H, B, 0h)))^2)) + (_i x. (((normh` (if(A e. ~H, A, 0h) +h (_i .h if(B e. ~H, B, 0h))))^2) - ((normh` (if(A e. ~H, A, 0h) -h (_i .h if(B e. ~H, B, 0h))))^2)))) / 4)
44	19, 39, 43	dedth2h 3015	1 |- ((A e. ~H /\ B e. ~H) -> (A .ih B) = (((((normh` (A +h B))^2) - ((normh` (A -h B))^2)) + (_i x. (((normh` (A +h (_i .h B)))^2) - ((normh` (A -h (_i .h B)))^2)))) / 4))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  ifcif 2982  ` cfv 3998  (class class class)co 4884  _ici 6388   + caddc 6389   x. cmul 6391   - cmin 6445   / cdiv 6447  2c2 7145  4c4 7147  ^cexp 7811  ~Hchil 10420   +h cva 10421   .h csm 10422  0hc0v 10423   -h cmv 10424   .ih csp 10425  normhcno 10426

This theorem is referenced by:  hhip 10677

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-hv0cl 10505  ax-hfvmul 10507  ax-hvmul0 10512  ax-hfi 10579  ax-his1 10582  ax-his2 10583  ax-his3 10584  ax-his4 10585

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-hnorm 10469  df-hvsub 10472

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