pjclem4 - Hilbert Space Explorer

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Theorem pjclem4 32274

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 32234	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ 𝐺)
4	3	adantl 481	. . . . . 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 32234	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))‘𝑥) ∈ 𝐻)
6		fveq1 6833	. . . . . . . . 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 2821	. . . . . . . 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	imbitrrid 246	. . . . . . 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 406	. . . . . 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 4152	. . . . 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 32235	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ)
12		hvsubcl 31092	. . . . . . . 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 687	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) ∈ ℋ)
14	13	adantl 481	. . . . . 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 482	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → 𝑥 ∈ ℋ)
16	11	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ)
17	1, 2	chincli 31535	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∩ 𝐻) ∈ C_ℋ
18	17	cheli 31307	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐺 ∩ 𝐻) → 𝑦 ∈ ℋ)
19	18	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → 𝑦 ∈ ℋ)
20	15, 16, 19	3jca 1128	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → (𝑥 ∈ ℋ ∧ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ))
21	20	adantl 481	. . . . . . . . . 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 31167	. . . . . . . . . 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 7373	. . . . . . . . . . 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 32236	. . . . . . . . . . . . 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 593	. . . . . . . . . . . 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 32273	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝐺 ∩ 𝐻) → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑦) = 𝑦)
28	27	oveq2d 7374	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐺 ∩ 𝐻) → (𝑥 ·_ih (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑦)) = (𝑥 ·_ih 𝑦))
29	28	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → (𝑥 ·_ih (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑦)) = (𝑥 ·_ih 𝑦))
30	26, 29	eqtrd 2771	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → ((((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih 𝑦))
31	24, 30	sylan9eq 2791	. . . . . . . . . 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 7373	. . . . . . . . 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 613	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ))
34	33	adantl 481	. . . . . . . . . . 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 31155	. . . . . . . . . . 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 11480	. . . . . . . . 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 2777	. . . . . . . 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 456	. . . . . . 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 3127	. . . . . 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 31302	. . . . . . 7 ⊢ (𝐺 ∩ 𝐻) ∈ S_ℋ
42		shocel 31357	. . . . . . 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 583	. . . . 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 31780	. . . . 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 584	. . . 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 31113	. . . . . . . 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 1373	. . . . . . 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 31101	. . . . . . . . 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 7374	. . . . . . 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 31079	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑥 +_ℎ 0_ℎ) = 𝑥)
54	49, 52, 53	3eqtrd 2775	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) +_ℎ (𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥))) = 𝑥)
55	54	fveq2d 6838	. . . . 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 481	. . . 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 2773	. . 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 3128	. 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 31779	. . . 4 ⊢ (proj_ℎ‘𝐺): ℋ⟶ ℋ
60	2	pjfi 31779	. . . 4 ⊢ (proj_ℎ‘𝐻): ℋ⟶ ℋ
61	59, 60	hocofi 31841	. . 3 ⊢ ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)): ℋ⟶ ℋ
62	17	pjfi 31779	. . 3 ⊢ (proj_ℎ‘(𝐺 ∩ 𝐻)): ℋ⟶ ℋ
63	61, 62	hoeqi 31836	. 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 218	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∀wral 3051 ∩ cin 3900 ∘ ccom 5628 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 0cc0 11026 − cmin 11364 ℋchba 30994 +_ℎ cva 30995 ·_ih csp 30997 0_ℎc0v 30999 −_ℎ cmv 31000 S_ℋ csh 31003 C_ℋ cch 31004 ⊥cort 31005 proj_ℎcpjh 31012

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cc 10345 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105 ax-mulf 11106 ax-hilex 31074 ax-hfvadd 31075 ax-hvcom 31076 ax-hvass 31077 ax-hv0cl 31078 ax-hvaddid 31079 ax-hfvmul 31080 ax-hvmulid 31081 ax-hvmulass 31082 ax-hvdistr1 31083 ax-hvdistr2 31084 ax-hvmul0 31085 ax-hfi 31154 ax-his1 31157 ax-his2 31158 ax-his3 31159 ax-his4 31160 ax-hcompl 31277

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-omul 8402 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-acn 9854 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-rlim 15412 df-sum 15610 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-cn 23171 df-cnp 23172 df-lm 23173 df-haus 23259 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cfil 25211 df-cau 25212 df-cmet 25213 df-grpo 30568 df-gid 30569 df-ginv 30570 df-gdiv 30571 df-ablo 30620 df-vc 30634 df-nv 30667 df-va 30670 df-ba 30671 df-sm 30672 df-0v 30673 df-vs 30674 df-nmcv 30675 df-ims 30676 df-dip 30776 df-ssp 30797 df-ph 30888 df-cbn 30938 df-hnorm 31043 df-hba 31044 df-hvsub 31046 df-hlim 31047 df-hcau 31048 df-sh 31282 df-ch 31296 df-oc 31327 df-ch0 31328 df-shs 31383 df-pjh 31470

This theorem is referenced by: pjci 32275 pjcmul1i 32276 pjcmul2i 32277

W3C validator