adjsym - Hilbert Space Explorer

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Theorem adjsym 31992

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 3289	. . . 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 6861	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑆‘𝑧) = (𝑆‘𝑦))
3	2	oveq2d 7406	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑥 ·_ih (𝑆‘𝑧)) = (𝑥 ·_ih (𝑆‘𝑦)))
4		oveq2 7398	. . . . . . 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 3184	. . . . 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 3239	. . . 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 280	. . 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 7397	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ·_ih (𝑆‘𝑧)) = (𝑦 ·_ih (𝑆‘𝑧)))
10		fveq2 6861	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑇‘𝑥) = (𝑇‘𝑦))
11	10	oveq1d 7405	. . . . . 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 3239	. . . 4 ⊢ (∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑥) ·_ih 𝑧) ↔ ∀𝑦 ∈ ℋ (𝑦 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧))
14	13	ralbii 3107	. . 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 6861	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑆‘𝑧) = (𝑆‘𝑥))
16	15	oveq2d 7406	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑦 ·_ih (𝑆‘𝑧)) = (𝑦 ·_ih (𝑆‘𝑥)))
17		oveq2 7398	. . . . . 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 3184	. . . 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 3239	. . 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 299	. 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 7056	. . . . . . . . . . . 12 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
23		ax-his1 31241	. . . . . . . . . . . 12 ⊢ (((𝑇‘𝑦) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑇‘𝑦))))
24	22, 23	sylan 589	. . . . . . . . . . 11 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑇‘𝑦))))
25	24	adantrl 726	. . . . . . . . . 10 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝑇‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑇‘𝑦))))
26		ffvelcdm 7056	. . . . . . . . . . . 12 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑆‘𝑥) ∈ ℋ)
27		ax-his1 31241	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℋ ∧ (𝑆‘𝑥) ∈ ℋ) → (𝑦 ·_ih (𝑆‘𝑥)) = (∗‘((𝑆‘𝑥) ·_ih 𝑦)))
28	26, 27	sylan2 602	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℋ ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·_ih (𝑆‘𝑥)) = (∗‘((𝑆‘𝑥) ·_ih 𝑦)))
29	28	adantll 724	. . . . . . . . . 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 462	. . . . . . . 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 31239	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝑥 ·_ih (𝑇‘𝑦)) ∈ ℂ)
33	22, 32	sylan2 602	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑇‘𝑦)) ∈ ℂ)
34	33	adantll 724	. . . . . . . . 9 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑇‘𝑦)) ∈ ℂ)
35		hicl 31239	. . . . . . . . . . 11 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆‘𝑥) ·_ih 𝑦) ∈ ℂ)
36	26, 35	sylan 589	. . . . . . . . . 10 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑆‘𝑥) ·_ih 𝑦) ∈ ℂ)
37	36	adantrl 726	. . . . . . . . 9 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆‘𝑥) ·_ih 𝑦) ∈ ℂ)
38		cj11 15179	. . . . . . . . 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 593	. . . . . . . 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 282	. . . . . . 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 670	. . . . . 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 471	. . . . 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 289	. . . 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 3182	. . 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 3182	. 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 292	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 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ∀wral 3075 ⟶wf 6511 ‘cfv 6515 (class class class)co 7390 ℂcc 11064 ∗ccj 15113 ℋchba 31078 ·_ih csp 31081

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143 ax-hfi 31238 ax-his1 31241

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7841 df-2nd 7965 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-er 8671 df-en 8921 df-dom 8922 df-sdom 8923 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-div 11838 df-nn 12204 df-2 12273 df-cj 15116 df-re 15117 df-im 15118

This theorem is referenced by: dfadj2 32044 adjval2 32050 cnlnadjeui 32236 cnlnssadj 32239 adjbdln 32242

W3C validator