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Theorem counop 29377

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 29372	. . . 4 ⊢ (𝑆 ∈ UniOp → 𝑆: ℋ–1-1-onto→ ℋ)
2		unopf1o 29372	. . . 4 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–1-1-onto→ ℋ)
3		f1oco 6497	. . . 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 6482	. . 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 6475	. . . . . . . 8 ⊢ (𝑇: ℋ–1-1-onto→ ℋ → 𝑇: ℋ⟶ ℋ)
8	2, 7	syl 17	. . . . . . 7 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ)
9	8	adantl 482	. . . . . 6 ⊢ ((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) → 𝑇: ℋ⟶ ℋ)
10		simpl 483	. . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → 𝑥 ∈ ℋ)
11		fvco3 6619	. . . . . 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 485	. . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → 𝑦 ∈ ℋ)
14		fvco3 6619	. . . . . 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 7025	. . . 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 6705	. . . . . . . 8 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
18		ffvelrn 6705	. . . . . . . 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 580	. . . . . 6 ⊢ ((𝑇 ∈ UniOp ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ))
21		unop 29371	. . . . . . 7 ⊢ ((𝑆 ∈ UniOp ∧ (𝑇‘𝑥) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → ((𝑆‘(𝑇‘𝑥)) ·_ih (𝑆‘(𝑇‘𝑦))) = ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)))
22	21	3expb 1111	. . . . . 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 468	. . . 4 ⊢ (((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → ((𝑆‘(𝑇‘𝑥)) ·_ih (𝑆‘(𝑇‘𝑦))) = ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)))
25		unop 29371	. . . . . 6 ⊢ ((𝑇 ∈ UniOp ∧ 𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑇‘𝑥) ·_ih (𝑇‘𝑦)) = (𝑥 ·_ih 𝑦))
26	25	3expb 1111	. . . . 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 2833	. . 3 ⊢ (((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑆 ∘ 𝑇)‘𝑥) ·_ih ((𝑆 ∘ 𝑇)‘𝑦)) = (𝑥 ·_ih 𝑦))
29	28	ralrimivva 3156	. 2 ⊢ ((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (((𝑆 ∘ 𝑇)‘𝑥) ·_ih ((𝑆 ∘ 𝑇)‘𝑦)) = (𝑥 ·_ih 𝑦))
30		elunop 29328	. 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 583	1 ⊢ ((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) → (𝑆 ∘ 𝑇) ∈ UniOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1520 ∈ wcel 2079 ∀wral 3103 ∘ ccom 5439 ⟶wf 6213 –onto→wfo 6215 –1-1-onto→wf1o 6216 ‘cfv 6217 (class class class)co 7007 ℋchba 28375 ·_ih csp 28378 UniOpcuo 28405

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-hilex 28455 ax-hfvadd 28456 ax-hvcom 28457 ax-hvass 28458 ax-hv0cl 28459 ax-hvaddid 28460 ax-hfvmul 28461 ax-hvmulid 28462 ax-hvdistr2 28465 ax-hvmul0 28466 ax-hfi 28535 ax-his1 28538 ax-his2 28539 ax-his3 28540 ax-his4 28541

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-po 5354 df-so 5355 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-2 11537 df-cj 14280 df-re 14281 df-im 14282 df-hvsub 28427 df-unop 29299

This theorem is referenced by: (None)

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