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Theorem counop 30184

Description: The composition of two unitary operators is unitary. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
counop	⊢ ((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) → (𝑆 ∘ 𝑇) ∈ UniOp)

Proof of Theorem counop Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unopf1o 30179	. . . 4 ⊢ (𝑆 ∈ UniOp → 𝑆: ℋ–1-1-onto→ ℋ)
2		unopf1o 30179	. . . 4 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–1-1-onto→ ℋ)
3		f1oco 6722	. . . 4 ⊢ ((𝑆: ℋ–1-1-onto→ ℋ ∧ 𝑇: ℋ–1-1-onto→ ℋ) → (𝑆 ∘ 𝑇): ℋ–1-1-onto→ ℋ)
4	1, 2, 3	syl2an 595	. . 3 ⊢ ((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) → (𝑆 ∘ 𝑇): ℋ–1-1-onto→ ℋ)
5		f1ofo 6707	. . 3 ⊢ ((𝑆 ∘ 𝑇): ℋ–1-1-onto→ ℋ → (𝑆 ∘ 𝑇): ℋ–onto→ ℋ)
6	4, 5	syl 17	. 2 ⊢ ((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) → (𝑆 ∘ 𝑇): ℋ–onto→ ℋ)
7		f1of 6700	. . . . . . . 8 ⊢ (𝑇: ℋ–1-1-onto→ ℋ → 𝑇: ℋ⟶ ℋ)
8	2, 7	syl 17	. . . . . . 7 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
9	8	adantl 481	. . . . . 6 ⊢ ((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) → 𝑇: ℋ⟶ ℋ)
10		simpl 482	. . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → 𝑥 ∈ ℋ)
11		fvco3 6849	. . . . . 6 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 ∘ 𝑇)‘𝑥) = (𝑆‘(𝑇‘𝑥)))
12	9, 10, 11	syl2an 595	. . . . 5 ⊢ (((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆 ∘ 𝑇)‘𝑥) = (𝑆‘(𝑇‘𝑥)))
13		simpr 484	. . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → 𝑦 ∈ ℋ)
14		fvco3 6849	. . . . . 6 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑆 ∘ 𝑇)‘𝑦) = (𝑆‘(𝑇‘𝑦)))
15	9, 13, 14	syl2an 595	. . . . 5 ⊢ (((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆 ∘ 𝑇)‘𝑦) = (𝑆‘(𝑇‘𝑦)))
16	12, 15	oveq12d 7273	. . . 4 ⊢ (((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆 ∘ 𝑇)‘𝑥) ·_ih ((𝑆 ∘ 𝑇)‘𝑦)) = ((𝑆‘(𝑇‘𝑥)) ·_ih (𝑆‘(𝑇‘𝑦))))
17		ffvelrn 6941	. . . . . . . 8 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
18		ffvelrn 6941	. . . . . . . 8 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
19	17, 18	anim12dan 618	. . . . . . 7 ⊢ ((𝑇: ℋ⟶ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ))
20	8, 19	sylan 579	. . . . . 6 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ))
21		unop 30178	. . . . . . 7 ⊢ ((𝑆 ∈ UniOp ∧ (𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → ((𝑆‘(𝑇‘𝑥)) ·_ih (𝑆‘(𝑇‘𝑦))) = ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)))
22	21	3expb 1118	. . . . . 6 ⊢ ((𝑆 ∈ UniOp ∧ ((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ)) → ((𝑆‘(𝑇‘𝑥)) ·_ih (𝑆‘(𝑇‘𝑦))) = ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)))
23	20, 22	sylan2 592	. . . . 5 ⊢ ((𝑆 ∈ UniOp ∧ (𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ))) → ((𝑆‘(𝑇‘𝑥)) ·_ih (𝑆‘(𝑇‘𝑦))) = ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)))
24	23	anassrs 467	. . . 4 ⊢ (((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆‘(𝑇‘𝑥)) ·_ih (𝑆‘(𝑇‘𝑦))) = ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)))
25		unop 30178	. . . . . 6 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦))
26	25	3expb 1118	. . . . 5 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦))
27	26	adantll 710	. . . 4 ⊢ (((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦))
28	16, 24, 27	3eqtrd 2782	. . 3 ⊢ (((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆 ∘ 𝑇)‘𝑥) ·_ih ((𝑆 ∘ 𝑇)‘𝑦)) = (𝑥 ·_ih 𝑦))
29	28	ralrimivva 3114	. 2 ⊢ ((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (((𝑆 ∘ 𝑇)‘𝑥) ·_ih ((𝑆 ∘ 𝑇)‘𝑦)) = (𝑥 ·_ih 𝑦))
30		elunop 30135	. 2 ⊢ ((𝑆 ∘ 𝑇) ∈ UniOp ↔ ((𝑆 ∘ 𝑇): ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (((𝑆 ∘ 𝑇)‘𝑥) ·_ih ((𝑆 ∘ 𝑇)‘𝑦)) = (𝑥 ·_ih 𝑦)))
31	6, 29, 30	sylanbrc 582	1 ⊢ ((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) → (𝑆 ∘ 𝑇) ∈ UniOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∘ ccom 5584 ⟶wf 6414 –onto→wfo 6416 –1-1-onto→wf1o 6417 ‘cfv 6418 (class class class)co 7255 ℋchba 29182 ·_ih csp 29185 UniOpcuo 29212

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-hilex 29262 ax-hfvadd 29263 ax-hvcom 29264 ax-hvass 29265 ax-hv0cl 29266 ax-hvaddid 29267 ax-hfvmul 29268 ax-hvmulid 29269 ax-hvdistr2 29272 ax-hvmul0 29273 ax-hfi 29342 ax-his1 29345 ax-his2 29346 ax-his3 29347 ax-his4 29348

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-2 11966 df-cj 14738 df-re 14739 df-im 14740 df-hvsub 29234 df-unop 30106

This theorem is referenced by: (None)

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