adjsym - Hilbert Space Explorer

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Theorem adjsym 32417

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 3291	. . . 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 6877	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑆‘𝑧) = (𝑆‘𝑦))
3	2	oveq2d 7428	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑥 ·_ih (𝑆‘𝑧)) = (𝑥 ·_ih (𝑆‘𝑦)))
4		oveq2 7420	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑇‘𝑥) ·_ih 𝑧) = ((𝑇‘𝑥) ·_ih 𝑦))
5	3, 4	eqeq12d 2777	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝑥 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑥) ·_ih 𝑧) ↔ (𝑥 ·_ih (𝑆‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))
6	5	ralbidv 3186	. . . . 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 3241	. . . 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 7419	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ·_ih (𝑆‘𝑧)) = (𝑦 ·_ih (𝑆‘𝑧)))
10		fveq2 6877	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑇‘𝑥) = (𝑇‘𝑦))
11	10	oveq1d 7427	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑇‘𝑥) ·_ih 𝑧) = ((𝑇‘𝑦) ·_ih 𝑧))
12	9, 11	eqeq12d 2777	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑥) ·_ih 𝑧) ↔ (𝑦 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧)))
13	12	cbvralvw 3241	. . . 4 ⊢ (∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑥) ·_ih 𝑧) ↔ ∀𝑦 ∈ ℋ (𝑦 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧))
14	13	ralbii 3109	. . 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 6877	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑆‘𝑧) = (𝑆‘𝑥))
16	15	oveq2d 7428	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑦 ·_ih (𝑆‘𝑧)) = (𝑦 ·_ih (𝑆‘𝑥)))
17		oveq2 7420	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑇‘𝑦) ·_ih 𝑧) = ((𝑇‘𝑦) ·_ih 𝑥))
18	16, 17	eqeq12d 2777	. . . . 5 ⊢ (𝑧 = 𝑥 → ((𝑦 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧) ↔ (𝑦 ·_ih (𝑆‘𝑥)) = ((𝑇‘𝑦) ·_ih 𝑥)))
19	18	ralbidv 3186	. . . 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 3241	. . 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 7073	. . . . . . . . . . . 12 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
23		ax-his1 31666	. . . . . . . . . . . 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 7073	. . . . . . . . . . . 12 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑆‘𝑥) ∈ ℋ)
27		ax-his1 31666	. . . . . . . . . . . 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 2777	. . . . . . . . 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 31664	. . . . . . . . . . 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 31664	. . . . . . . . . . 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 15309	. . . . . . . . 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 2768	. . . . 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 3184	. . 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 3184	. 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 2145 ∀wral 3077 ⟶wf 6527 ‘cfv 6531 (class class class)co 7412 ℂcc 11179 ∗ccj 15243 ℋchba 31503 ·_ih csp 31506

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-hfi 31663 ax-his1 31666

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-er 8701 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-cj 15246 df-re 15247 df-im 15248

This theorem is used by: dfadj2 32469 adjval2 32475 cnlnadjeui 32661 cnlnssadj 32664 adjbdln 32667

W3C validator