adjsym - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > adjsym		Structured version Visualization version GIF version

Theorem adjsym 32222

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 3296	. . . 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 6888	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑆‘𝑧) = (𝑆‘𝑦))
3	2	oveq2d 7439	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑥 ·_ih (𝑆‘𝑧)) = (𝑥 ·_ih (𝑆‘𝑦)))
4		oveq2 7431	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑇‘𝑥) ·_ih 𝑧) = ((𝑇‘𝑥) ·_ih 𝑦))
5	3, 4	eqeq12d 2782	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝑥 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑥) ·_ih 𝑧) ↔ (𝑥 ·_ih (𝑆‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))
6	5	ralbidv 3191	. . . . 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 3246	. . . 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 281	. . 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 7430	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ·_ih (𝑆‘𝑧)) = (𝑦 ·_ih (𝑆‘𝑧)))
10		fveq2 6888	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑇‘𝑥) = (𝑇‘𝑦))
11	10	oveq1d 7438	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑇‘𝑥) ·_ih 𝑧) = ((𝑇‘𝑦) ·_ih 𝑧))
12	9, 11	eqeq12d 2782	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑥) ·_ih 𝑧) ↔ (𝑦 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧)))
13	12	cbvralvw 3246	. . . 4 ⊢ (∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑥) ·_ih 𝑧) ↔ ∀𝑦 ∈ ℋ (𝑦 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧))
14	13	ralbii 3114	. . 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 6888	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑆‘𝑧) = (𝑆‘𝑥))
16	15	oveq2d 7439	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑦 ·_ih (𝑆‘𝑧)) = (𝑦 ·_ih (𝑆‘𝑥)))
17		oveq2 7431	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑇‘𝑦) ·_ih 𝑧) = ((𝑇‘𝑦) ·_ih 𝑥))
18	16, 17	eqeq12d 2782	. . . . 5 ⊢ (𝑧 = 𝑥 → ((𝑦 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧) ↔ (𝑦 ·_ih (𝑆‘𝑥)) = ((𝑇‘𝑦) ·_ih 𝑥)))
19	18	ralbidv 3191	. . . 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 3246	. . 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 300	. 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 7083	. . . . . . . . . . . 12 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
23		ax-his1 31471	. . . . . . . . . . . 12 ⊢ (((𝑇‘𝑦) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑇‘𝑦))))
24	22, 23	sylan 592	. . . . . . . . . . 11 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑇‘𝑦))))
25	24	adantrl 729	. . . . . . . . . 10 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝑇‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑇‘𝑦))))
26		ffvelcdm 7083	. . . . . . . . . . . 12 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑆‘𝑥) ∈ ℋ)
27		ax-his1 31471	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℋ ∧ (𝑆‘𝑥) ∈ ℋ) → (𝑦 ·_ih (𝑆‘𝑥)) = (∗‘((𝑆‘𝑥) ·_ih 𝑦)))
28	26, 27	sylan2 605	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℋ ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·_ih (𝑆‘𝑥)) = (∗‘((𝑆‘𝑥) ·_ih 𝑦)))
29	28	adantll 727	. . . . . . . . . 10 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·_ih (𝑆‘𝑥)) = (∗‘((𝑆‘𝑥) ·_ih 𝑦)))
30	25, 29	eqeq12d 2782	. . . . . . . . 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 464	. . . . . . . 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 31469	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝑥 ·_ih (𝑇‘𝑦)) ∈ ℂ)
33	22, 32	sylan2 605	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑇‘𝑦)) ∈ ℂ)
34	33	adantll 727	. . . . . . . . 9 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑇‘𝑦)) ∈ ℂ)
35		hicl 31469	. . . . . . . . . . 11 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆‘𝑥) ·_ih 𝑦) ∈ ℂ)
36	26, 35	sylan 592	. . . . . . . . . 10 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑆‘𝑥) ·_ih 𝑦) ∈ ℂ)
37	36	adantrl 729	. . . . . . . . 9 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆‘𝑥) ·_ih 𝑦) ∈ ℂ)
38		cj11 15239	. . . . . . . . 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 596	. . . . . . . 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 283	. . . . . . 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 673	. . . . . 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 473	. . . . 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 2773	. . . . 5 ⊢ (((𝑇‘𝑦) ·_ih 𝑥) = (𝑦 ·_ih (𝑆‘𝑥)) ↔ (𝑦 ·_ih (𝑆‘𝑥)) = ((𝑇‘𝑦) ·_ih 𝑥))
44	42, 43	bitrdi 290	. . . 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 3189	. . 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 3189	. 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 293	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∀wral 3082 ⟶wf 6539 ‘cfv 6543 (class class class)co 7423 ℂcc 11116 ∗ccj 15173 ℋchba 31308 ·_ih csp 31311

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-hfi 31468 ax-his1 31471

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-cj 15176 df-re 15177 df-im 15178

This theorem is used by: dfadj2 32274 adjval2 32280 cnlnadjeui 32466 cnlnssadj 32469 adjbdln 32472

W3C validator