pjclem4 - Hilbert Space Explorer

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Theorem pjclem4 29630

Description: Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
pjclem1.1	⊢ 𝐺 ∈ C_ℋ
pjclem1.2	⊢ 𝐻 ∈ C_ℋ

**Assertion**
Ref	Expression
pjclem4	⊢ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) → ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = (proj_ℎ‘(𝐺 ∩ 𝐻)))

Proof of Theorem pjclem4 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pjclem1.1	. . . . . . . 8 ⊢ 𝐺 ∈ C_ℋ
2		pjclem1.2	. . . . . . . 8 ⊢ 𝐻 ∈ C_ℋ
3	1, 2	pjcocli 29590	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ 𝐺)
4	3	adantl 475	. . . . . 6 ⊢ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) ∧ 𝑥 ∈ ℋ) → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ 𝐺)
5	2, 1	pjcocli 29590	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))‘𝑥) ∈ 𝐻)
6		fveq1 6445	. . . . . . . . 9 ⊢ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) = (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))‘𝑥))
7	6	eleq1d 2843	. . . . . . . 8 ⊢ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) → ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ 𝐻 ↔ (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))‘𝑥) ∈ 𝐻))
8	5, 7	syl5ibr 238	. . . . . . 7 ⊢ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) → (𝑥 ∈ ℋ → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ 𝐻))
9	8	imp 397	. . . . . 6 ⊢ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) ∧ 𝑥 ∈ ℋ) → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ 𝐻)
10	4, 9	elind 4020	. . . . 5 ⊢ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) ∧ 𝑥 ∈ ℋ) → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ (𝐺 ∩ 𝐻))
11	1, 2	pjcohcli 29591	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ)
12		hvsubcl 28446	. . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ) → (𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) ∈ ℋ)
13	11, 12	mpdan 677	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) ∈ ℋ)
14	13	adantl 475	. . . . . 6 ⊢ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) ∧ 𝑥 ∈ ℋ) → (𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) ∈ ℋ)
15		simpl 476	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → 𝑥 ∈ ℋ)
16	11	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ)
17	1, 2	chincli 28891	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∩ 𝐻) ∈ C_ℋ
18	17	cheli 28661	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐺 ∩ 𝐻) → 𝑦 ∈ ℋ)
19	18	adantl 475	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → 𝑦 ∈ ℋ)
20	15, 16, 19	3jca 1119	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → (𝑥 ∈ ℋ ∧ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ))
21	20	adantl 475	. . . . . . . . . 10 ⊢ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻))) → (𝑥 ∈ ℋ ∧ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ))
22		his2sub 28521	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℋ ∧ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) ·_ih 𝑦) = ((𝑥 ·_ih 𝑦) − ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ·_ih 𝑦)))
23	21, 22	syl 17	. . . . . . . . 9 ⊢ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻))) → ((𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) ·_ih 𝑦) = ((𝑥 ·_ih 𝑦) − ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ·_ih 𝑦)))
24	6	oveq1d 6937	. . . . . . . . . . 11 ⊢ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) → ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ·_ih 𝑦) = ((((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))‘𝑥) ·_ih 𝑦))
25	2, 1	pjadjcoi 29592	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑦)))
26	18, 25	sylan2 586	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → ((((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑦)))
27	1, 2	pjclem4a 29629	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝐺 ∩ 𝐻) → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑦) = 𝑦)
28	27	oveq2d 6938	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐺 ∩ 𝐻) → (𝑥 ·_ih (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑦)) = (𝑥 ·_ih 𝑦))
29	28	adantl 475	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → (𝑥 ·_ih (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑦)) = (𝑥 ·_ih 𝑦))
30	26, 29	eqtrd 2813	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → ((((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih 𝑦))
31	24, 30	sylan9eq 2833	. . . . . . . . . 10 ⊢ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻))) → ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih 𝑦))
32	31	oveq1d 6937	. . . . . . . . 9 ⊢ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻))) → (((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ·_ih 𝑦) − ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ·_ih 𝑦)) = ((𝑥 ·_ih 𝑦) − ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ·_ih 𝑦)))
33	11, 18	anim12i 606	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ))
34	33	adantl 475	. . . . . . . . . . 11 ⊢ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻))) → ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ))
35		hicl 28509	. . . . . . . . . . 11 ⊢ (((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ) → ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ·_ih 𝑦) ∈ ℂ)
36	34, 35	syl 17	. . . . . . . . . 10 ⊢ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻))) → ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ·_ih 𝑦) ∈ ℂ)
37	36	subidd 10722	. . . . . . . . 9 ⊢ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻))) → (((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ·_ih 𝑦) − ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ·_ih 𝑦)) = 0)
38	23, 32, 37	3eqtr2d 2819	. . . . . . . 8 ⊢ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻))) → ((𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) ·_ih 𝑦) = 0)
39	38	expr 450	. . . . . . 7 ⊢ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) ∧ 𝑥 ∈ ℋ) → (𝑦 ∈ (𝐺 ∩ 𝐻) → ((𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) ·_ih 𝑦) = 0))
40	39	ralrimiv 3146	. . . . . 6 ⊢ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) ∧ 𝑥 ∈ ℋ) → ∀𝑦 ∈ (𝐺 ∩ 𝐻)((𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) ·_ih 𝑦) = 0)
41	17	chshii 28656	. . . . . . 7 ⊢ (𝐺 ∩ 𝐻) ∈ S_ℋ
42		shocel 28713	. . . . . . 7 ⊢ ((𝐺 ∩ 𝐻) ∈ S_ℋ → ((𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) ∈ (⊥‘(𝐺 ∩ 𝐻)) ↔ ((𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) ∈ ℋ ∧ ∀𝑦 ∈ (𝐺 ∩ 𝐻)((𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) ·_ih 𝑦) = 0)))
43	41, 42	ax-mp 5	. . . . . 6 ⊢ ((𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) ∈ (⊥‘(𝐺 ∩ 𝐻)) ↔ ((𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) ∈ ℋ ∧ ∀𝑦 ∈ (𝐺 ∩ 𝐻)((𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) ·_ih 𝑦) = 0))
44	14, 40, 43	sylanbrc 578	. . . . 5 ⊢ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) ∧ 𝑥 ∈ ℋ) → (𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) ∈ (⊥‘(𝐺 ∩ 𝐻)))
45	17	pjvi 29136	. . . . 5 ⊢ (((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ (𝐺 ∩ 𝐻) ∧ (𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) ∈ (⊥‘(𝐺 ∩ 𝐻))) → ((proj_ℎ‘(𝐺 ∩ 𝐻))‘((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) +_ℎ (𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)))) = (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥))
46	10, 44, 45	syl2anc 579	. . . 4 ⊢ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) ∧ 𝑥 ∈ ℋ) → ((proj_ℎ‘(𝐺 ∩ 𝐻))‘((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) +_ℎ (𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)))) = (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥))
47		id 22	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → 𝑥 ∈ ℋ)
48		hvaddsub12 28467	. . . . . . . 8 ⊢ (((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ ∧ 𝑥 ∈ ℋ ∧ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ) → ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) +_ℎ (𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥))) = (𝑥 +_ℎ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥))))
49	11, 47, 11, 48	syl3anc 1439	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) +_ℎ (𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥))) = (𝑥 +_ℎ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥))))
50		hvsubid 28455	. . . . . . . . 9 ⊢ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ → ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) = 0_ℎ)
51	11, 50	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) = 0_ℎ)
52	51	oveq2d 6938	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑥 +_ℎ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥))) = (𝑥 +_ℎ 0_ℎ))
53		ax-hvaddid 28433	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑥 +_ℎ 0_ℎ) = 𝑥)
54	49, 52, 53	3eqtrd 2817	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) +_ℎ (𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥))) = 𝑥)
55	54	fveq2d 6450	. . . . 5 ⊢ (𝑥 ∈ ℋ → ((proj_ℎ‘(𝐺 ∩ 𝐻))‘((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) +_ℎ (𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)))) = ((proj_ℎ‘(𝐺 ∩ 𝐻))‘𝑥))
56	55	adantl 475	. . . 4 ⊢ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) ∧ 𝑥 ∈ ℋ) → ((proj_ℎ‘(𝐺 ∩ 𝐻))‘((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) +_ℎ (𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)))) = ((proj_ℎ‘(𝐺 ∩ 𝐻))‘𝑥))
57	46, 56	eqtr3d 2815	. . 3 ⊢ ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) ∧ 𝑥 ∈ ℋ) → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) = ((proj_ℎ‘(𝐺 ∩ 𝐻))‘𝑥))
58	57	ralrimiva 3147	. 2 ⊢ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) → ∀𝑥 ∈ ℋ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) = ((proj_ℎ‘(𝐺 ∩ 𝐻))‘𝑥))
59	1	pjfi 29135	. . . 4 ⊢ (proj_ℎ‘𝐺): ℋ⟶ ℋ
60	2	pjfi 29135	. . . 4 ⊢ (proj_ℎ‘𝐻): ℋ⟶ ℋ
61	59, 60	hocofi 29197	. . 3 ⊢ ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)): ℋ⟶ ℋ
62	17	pjfi 29135	. . 3 ⊢ (proj_ℎ‘(𝐺 ∩ 𝐻)): ℋ⟶ ℋ
63	61, 62	hoeqi 29192	. 2 ⊢ (∀𝑥 ∈ ℋ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) = ((proj_ℎ‘(𝐺 ∩ 𝐻))‘𝑥) ↔ ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = (proj_ℎ‘(𝐺 ∩ 𝐻)))
64	58, 63	sylib 210	1 ⊢ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) → ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = (proj_ℎ‘(𝐺 ∩ 𝐻)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ∀wral 3089 ∩ cin 3790 ∘ ccom 5359 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 0cc0 10272 − cmin 10606 ℋchba 28348 +_ℎ cva 28349 ·_ih csp 28351 0_ℎc0v 28353 −_ℎ cmv 28354 S_ℋ csh 28357 C_ℋ cch 28358 ⊥cort 28359 proj_ℎcpjh 28366

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cc 9592 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 ax-hilex 28428 ax-hfvadd 28429 ax-hvcom 28430 ax-hvass 28431 ax-hv0cl 28432 ax-hvaddid 28433 ax-hfvmul 28434 ax-hvmulid 28435 ax-hvmulass 28436 ax-hvdistr1 28437 ax-hvdistr2 28438 ax-hvmul0 28439 ax-hfi 28508 ax-his1 28511 ax-his2 28512 ax-his3 28513 ax-his4 28514 ax-hcompl 28631

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-omul 7848 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-acn 9101 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-clim 14627 df-rlim 14628 df-sum 14825 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-cn 21439 df-cnp 21440 df-lm 21441 df-haus 21527 df-tx 21774 df-hmeo 21967 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-xms 22533 df-ms 22534 df-tms 22535 df-cfil 23461 df-cau 23462 df-cmet 23463 df-grpo 27920 df-gid 27921 df-ginv 27922 df-gdiv 27923 df-ablo 27972 df-vc 27986 df-nv 28019 df-va 28022 df-ba 28023 df-sm 28024 df-0v 28025 df-vs 28026 df-nmcv 28027 df-ims 28028 df-dip 28128 df-ssp 28149 df-ph 28240 df-cbn 28291 df-hnorm 28397 df-hba 28398 df-hvsub 28400 df-hlim 28401 df-hcau 28402 df-sh 28636 df-ch 28650 df-oc 28681 df-ch0 28682 df-shs 28739 df-pjh 28826

This theorem is referenced by: pjci 29631 pjcmul1i 29632 pjcmul2i 29633

W3C validator