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Theorem adjsym 28692

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		ffvelrn 6357	. . . . . . . . . . . 12 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
2		ax-his1 27939	. . . . . . . . . . . 12 ⊢ (((𝑇‘𝑦) ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑇‘𝑦))))
3	1, 2	sylan 488	. . . . . . . . . . 11 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑇‘𝑦))))
4	3	adantrl 752	. . . . . . . . . 10 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → ((𝑇‘𝑦) ·_ih 𝑥) = (∗‘(𝑥 ·_ih (𝑇‘𝑦))))
5		ffvelrn 6357	. . . . . . . . . . . 12 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑆‘𝑥) ∈ ℋ)
6		ax-his1 27939	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℋ ∧ (𝑆‘𝑥) ∈ ℋ) → (𝑦 ·_ih (𝑆‘𝑥)) = (∗‘((𝑆‘𝑥) ·_ih 𝑦)))
7	5, 6	sylan2 491	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℋ ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·_ih (𝑆‘𝑥)) = (∗‘((𝑆‘𝑥) ·_ih 𝑦)))
8	7	adantll 750	. . . . . . . . . 10 ⊢ (((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝑦 ·_ih (𝑆‘𝑥)) = (∗‘((𝑆‘𝑥) ·_ih 𝑦)))
9	4, 8	eqeq12d 2637	. . . . . . . . 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))))
10	9	ancoms 469	. . . . . . . 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))))
11		hicl 27937	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → (𝑥 ·_ih (𝑇‘𝑦)) ∈ ℂ)
12	1, 11	sylan2 491	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℋ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑇‘𝑦)) ∈ ℂ)
13	12	adantll 750	. . . . . . . . 9 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (𝑥 ·_ih (𝑇‘𝑦)) ∈ ℂ)
14		hicl 27937	. . . . . . . . . . 11 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆‘𝑥) ·_ih 𝑦) ∈ ℂ)
15	5, 14	sylan 488	. . . . . . . . . 10 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ 𝑦 ∈ ℋ) → ((𝑆‘𝑥) ·_ih 𝑦) ∈ ℂ)
16	15	adantrl 752	. . . . . . . . 9 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆‘𝑥) ·_ih 𝑦) ∈ ℂ)
17		cj11 13902	. . . . . . . . 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)))
18	13, 16, 17	syl2anc 693	. . . . . . . 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)))
19	10, 18	bitr2d 269	. . . . . . 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))))
20	19	an4s 869	. . . . . 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))))
21	20	anassrs 680	. . . . 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))))
22		eqcom 2629	. . . . 5 ⊢ (((𝑇‘𝑦) ·_ih 𝑥) = (𝑦 ·_ih (𝑆‘𝑥)) ↔ (𝑦 ·_ih (𝑆‘𝑥)) = ((𝑇‘𝑦) ·_ih 𝑥))
23	21, 22	syl6bb 276	. . . 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)))
24	23	ralbidva 2985	. . 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)))
25	24	ralbidva 2985	. 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)))
26		ralcom 3098	. . . 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))
27		fveq2 6191	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑆‘𝑧) = (𝑆‘𝑦))
28	27	oveq2d 6666	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑥 ·_ih (𝑆‘𝑧)) = (𝑥 ·_ih (𝑆‘𝑦)))
29		oveq2 6658	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑇‘𝑥) ·_ih 𝑧) = ((𝑇‘𝑥) ·_ih 𝑦))
30	28, 29	eqeq12d 2637	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝑥 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑥) ·_ih 𝑧) ↔ (𝑥 ·_ih (𝑆‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))
31	30	ralbidv 2986	. . . . 5 ⊢ (𝑧 = 𝑦 → (∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑥) ·_ih 𝑧) ↔ ∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = ((𝑇‘𝑥) ·_ih 𝑦)))
32	31	cbvralv 3171	. . . 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))
33	26, 32	bitr4i 267	. . 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))
34		oveq1 6657	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ·_ih (𝑆‘𝑧)) = (𝑦 ·_ih (𝑆‘𝑧)))
35		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑇‘𝑥) = (𝑇‘𝑦))
36	35	oveq1d 6665	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑇‘𝑥) ·_ih 𝑧) = ((𝑇‘𝑦) ·_ih 𝑧))
37	34, 36	eqeq12d 2637	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑥) ·_ih 𝑧) ↔ (𝑦 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧)))
38	37	cbvralv 3171	. . . 4 ⊢ (∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑥) ·_ih 𝑧) ↔ ∀𝑦 ∈ ℋ (𝑦 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧))
39	38	ralbii 2980	. . 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))
40		fveq2 6191	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑆‘𝑧) = (𝑆‘𝑥))
41	40	oveq2d 6666	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑦 ·_ih (𝑆‘𝑧)) = (𝑦 ·_ih (𝑆‘𝑥)))
42		oveq2 6658	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑇‘𝑦) ·_ih 𝑧) = ((𝑇‘𝑦) ·_ih 𝑥))
43	41, 42	eqeq12d 2637	. . . . 5 ⊢ (𝑧 = 𝑥 → ((𝑦 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧) ↔ (𝑦 ·_ih (𝑆‘𝑥)) = ((𝑇‘𝑦) ·_ih 𝑥)))
44	43	ralbidv 2986	. . . 4 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ ℋ (𝑦 ·_ih (𝑆‘𝑧)) = ((𝑇‘𝑦) ·_ih 𝑧) ↔ ∀𝑦 ∈ ℋ (𝑦 ·_ih (𝑆‘𝑥)) = ((𝑇‘𝑦) ·_ih 𝑥)))
45	44	cbvralv 3171	. . 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))
46	33, 39, 45	3bitri 286	. 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))
47	25, 46	syl6rbbr 279	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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ∗ccj 13836 ℋchil 27776 ·_ih csp 27779

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-hfi 27936 ax-his1 27939

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-2 11079 df-cj 13839 df-re 13840 df-im 13841

This theorem is referenced by: dfadj2 28744 adjval2 28750 cnlnadjeui 28936 cnlnssadj 28939 adjbdln 28942

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