adjsym - Hilbert Space Explorer

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Theorem adjsym 31862

Description: Symmetry property of an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
adjsym	⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑆‘𝑥) ·_ih 𝑦)))

Distinct variable groups: 𝑥,𝑦,𝑆 𝑥,𝑇,𝑦

Proof of Theorem adjsym Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ralcom 3287	. . . 4 ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦))
2		fveq2 6907	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑆‘𝑧) = (𝑆‘𝑦))
3	2	oveq2d 7447	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑥 ·_ih (𝑆‘𝑧)) = (𝑥 ·_ih (𝑆‘𝑦)))
4		oveq2 7439	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑇‘𝑥) ·_ih 𝑧) = ((𝑇‘𝑥) ·_ih 𝑦))
5	3, 4	eqeq12d 2751	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝑥 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑥) ·_ih 𝑧) ↔ (𝑥 ·_ih (𝑆‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))
6	5	ralbidv 3176	. . . . 5 ⊢ (𝑧 = 𝑦 → (∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑥) ·_ih 𝑧) ↔ ∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))
7	6	cbvralvw 3235	. . . 4 ⊢ (∀𝑧 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑥) ·_ih 𝑧) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦))
8	1, 7	bitr4i 278	. . 3 ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦) ↔ ∀𝑧 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑥) ·_ih 𝑧))
9		oveq1 7438	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ·_ih (𝑆‘𝑧)) = (𝑦 ·_ih (𝑆‘𝑧)))
10		fveq2 6907	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑇‘𝑥) = (𝑇‘𝑦))
11	10	oveq1d 7446	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑇‘𝑥) ·_ih 𝑧) = ((𝑇‘𝑦) ·_ih 𝑧))
12	9, 11	eqeq12d 2751	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑥) ·_ih 𝑧) ↔ (𝑦 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧)))
13	12	cbvralvw 3235	. . . 4 ⊢ (∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑥) ·_ih 𝑧) ↔ ∀𝑦 ∈ ℋ (𝑦 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧))
14	13	ralbii 3091	. . 3 ⊢ (∀𝑧 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑥) ·_ih 𝑧) ↔ ∀𝑧 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧))
15		fveq2 6907	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑆‘𝑧) = (𝑆‘𝑥))
16	15	oveq2d 7447	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑦 ·_ih (𝑆‘𝑧)) = (𝑦 ·_ih (𝑆‘𝑥)))
17		oveq2 7439	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑇‘𝑦) ·_ih 𝑧) = ((𝑇‘𝑦) ·_ih 𝑥))
18	16, 17	eqeq12d 2751	. . . . 5 ⊢ (𝑧 = 𝑥 → ((𝑦 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧) ↔ (𝑦 ·_ih (𝑆‘𝑥)) = ((𝑇‘𝑦) ·_ih 𝑥)))
19	18	ralbidv 3176	. . . 4 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ ℋ (𝑦 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧) ↔ ∀𝑦 ∈ ℋ (𝑦 ·_ih (𝑆‘𝑥)) = ((𝑇‘𝑦) ·_ih 𝑥)))
20	19	cbvralvw 3235	. . 3 ⊢ (∀𝑧 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·_ih (𝑆‘𝑥)) = ((𝑇‘𝑦) ·_ih 𝑥))
21	8, 14, 20	3bitri 297	. 2 ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·_ih (𝑆‘𝑥)) = ((𝑇‘𝑦) ·_ih 𝑥))
22		ffvelcdm 7101	. . . . . . . . . . . 12 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
23		ax-his1 31111	. . . . . . . . . . . 12 ⊢ (((𝑇‘𝑦) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑇‘𝑦))))
24	22, 23	sylan 580	. . . . . . . . . . 11 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑇‘𝑦))))
25	24	adantrl 716	. . . . . . . . . 10 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝑇‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑇‘𝑦))))
26		ffvelcdm 7101	. . . . . . . . . . . 12 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑆‘𝑥) ∈ ℋ)
27		ax-his1 31111	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℋ ∧ (𝑆‘𝑥) ∈ ℋ) → (𝑦 ·_ih (𝑆‘𝑥)) = (∗‘((𝑆‘𝑥) ·_ih 𝑦)))
28	26, 27	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℋ ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·_ih (𝑆‘𝑥)) = (∗‘((𝑆‘𝑥) ·_ih 𝑦)))
29	28	adantll 714	. . . . . . . . . 10 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·_ih (𝑆‘𝑥)) = (∗‘((𝑆‘𝑥) ·_ih 𝑦)))
30	25, 29	eqeq12d 2751	. . . . . . . . 9 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (((𝑇‘𝑦) ·_ih 𝑥) = (𝑦 ·_ih (𝑆‘𝑥)) ↔ (∗‘(𝑥 ·_ih (𝑇‘𝑦))) = (∗‘((𝑆‘𝑥) ·_ih 𝑦))))
31	30	ancoms 458	. . . . . . . 8 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑇‘𝑦) ·_ih 𝑥) = (𝑦 ·_ih (𝑆‘𝑥)) ↔ (∗‘(𝑥 ·_ih (𝑇‘𝑦))) = (∗‘((𝑆‘𝑥) ·_ih 𝑦))))
32		hicl 31109	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝑥 ·_ih (𝑇‘𝑦)) ∈ ℂ)
33	22, 32	sylan2 593	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑇‘𝑦)) ∈ ℂ)
34	33	adantll 714	. . . . . . . . 9 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑇‘𝑦)) ∈ ℂ)
35		hicl 31109	. . . . . . . . . . 11 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆‘𝑥) ·_ih 𝑦) ∈ ℂ)
36	26, 35	sylan 580	. . . . . . . . . 10 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑆‘𝑥) ·_ih 𝑦) ∈ ℂ)
37	36	adantrl 716	. . . . . . . . 9 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆‘𝑥) ·_ih 𝑦) ∈ ℂ)
38		cj11 15198	. . . . . . . . 9 ⊢ (((𝑥 ·_ih (𝑇‘𝑦)) ∈ ℂ ∧ ((𝑆‘𝑥) ·_ih 𝑦) ∈ ℂ) → ((∗‘(𝑥 ·_ih (𝑇‘𝑦))) = (∗‘((𝑆‘𝑥) ·_ih 𝑦)) ↔ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑆‘𝑥) ·_ih 𝑦)))
39	34, 37, 38	syl2anc 584	. . . . . . . 8 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((∗‘(𝑥 ·_ih (𝑇‘𝑦))) = (∗‘((𝑆‘𝑥) ·_ih 𝑦)) ↔ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑆‘𝑥) ·_ih 𝑦)))
40	31, 39	bitr2d 280	. . . . . . 7 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 ·_ih (𝑇‘𝑦)) = ((𝑆‘𝑥) ·_ih 𝑦) ↔ ((𝑇‘𝑦) ·_ih 𝑥) = (𝑦 ·_ih (𝑆‘𝑥))))
41	40	an4s 660	. . . . . 6 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑥 ·_ih (𝑇‘𝑦)) = ((𝑆‘𝑥) ·_ih 𝑦) ↔ ((𝑇‘𝑦) ·_ih 𝑥) = (𝑦 ·_ih (𝑆‘𝑥))))
42	41	anassrs 467	. . . . 5 ⊢ ((((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑥 ·_ih (𝑇‘𝑦)) = ((𝑆‘𝑥) ·_ih 𝑦) ↔ ((𝑇‘𝑦) ·_ih 𝑥) = (𝑦 ·_ih (𝑆‘𝑥))))
43		eqcom 2742	. . . . 5 ⊢ (((𝑇‘𝑦) ·_ih 𝑥) = (𝑦 ·_ih (𝑆‘𝑥)) ↔ (𝑦 ·_ih (𝑆‘𝑥)) = ((𝑇‘𝑦) ·_ih 𝑥))
44	42, 43	bitrdi 287	. . . 4 ⊢ ((((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑥 ·_ih (𝑇‘𝑦)) = ((𝑆‘𝑥) ·_ih 𝑦) ↔ (𝑦 ·_ih (𝑆‘𝑥)) = ((𝑇‘𝑦) ·_ih 𝑥)))
45	44	ralbidva 3174	. . 3 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑆‘𝑥) ·_ih 𝑦) ↔ ∀𝑦 ∈ ℋ (𝑦 ·_ih (𝑆‘𝑥)) = ((𝑇‘𝑦) ·_ih 𝑥)))
46	45	ralbidva 3174	. 2 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑆‘𝑥) ·_ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑦 ·_ih (𝑆‘𝑥)) = ((𝑇‘𝑦) ·_ih 𝑥)))
47	21, 46	bitr4id 290	1 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑆‘𝑥) ·_ih 𝑦)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ∗ccj 15132 ℋchba 30948 ·_ih csp 30951

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-hfi 31108 ax-his1 31111

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-2 12327 df-cj 15135 df-re 15136 df-im 15137

This theorem is referenced by: dfadj2 31914 adjval2 31920 cnlnadjeui 32106 cnlnssadj 32109 adjbdln 32112

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