pjclem4 - Hilbert Space Explorer

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Theorem pjclem4 32288

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 32248	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ 𝐺)
4	3	adantl 482	. . . . . 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 32248	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))‘𝑥) ∈ 𝐻)
6		fveq1 6826	. . . . . . . . 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 2824	. . . . . . . 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 247	. . . . . . 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 407	. . . . . 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 4129	. . . . 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 32249	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ)
12		hvsubcl 31106	. . . . . . . 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 693	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) ∈ ℋ)
14	13	adantl 482	. . . . . 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 483	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → 𝑥 ∈ ℋ)
16	11	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ)
17	1, 2	chincli 31549	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∩ 𝐻) ∈ C_ℋ
18	17	cheli 31321	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐺 ∩ 𝐻) → 𝑦 ∈ ℋ)
19	18	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → 𝑦 ∈ ℋ)
20	15, 16, 19	3jca 1134	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → (𝑥 ∈ ℋ ∧ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ))
21	20	adantl 482	. . . . . . . . . 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 31181	. . . . . . . . . 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 7371	. . . . . . . . . . 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 32250	. . . . . . . . . . . . 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 599	. . . . . . . . . . . 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 32287	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝐺 ∩ 𝐻) → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑦) = 𝑦)
28	27	oveq2d 7372	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐺 ∩ 𝐻) → (𝑥 ·_ih (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑦)) = (𝑥 ·_ih 𝑦))
29	28	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → (𝑥 ·_ih (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑦)) = (𝑥 ·_ih 𝑦))
30	26, 29	eqtrd 2774	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → ((((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih 𝑦))
31	24, 30	sylan9eq 2794	. . . . . . . . . 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 7371	. . . . . . . . 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 619	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ))
34	33	adantl 482	. . . . . . . . . . 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 31169	. . . . . . . . . . 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 11484	. . . . . . . . 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 2780	. . . . . . . 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 457	. . . . . . 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 3130	. . . . . 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 31316	. . . . . . 7 ⊢ (𝐺 ∩ 𝐻) ∈ S_ℋ
42		shocel 31371	. . . . . . 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 589	. . . . 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 31794	. . . . 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 590	. . . 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 31127	. . . . . . . 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 1379	. . . . . . 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 31115	. . . . . . . . 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 7372	. . . . . . 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 31093	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑥 +_ℎ 0_ℎ) = 𝑥)
54	49, 52, 53	3eqtrd 2778	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) +_ℎ (𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥))) = 𝑥)
55	54	fveq2d 6831	. . . . 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 482	. . . 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 2776	. . 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 3131	. 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 31793	. . . 4 ⊢ (proj_ℎ‘𝐺): ℋ⟶ ℋ
60	2	pjfi 31793	. . . 4 ⊢ (proj_ℎ‘𝐻): ℋ⟶ ℋ
61	59, 60	hocofi 31855	. . 3 ⊢ ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)): ℋ⟶ ℋ
62	17	pjfi 31793	. . 3 ⊢ (proj_ℎ‘(𝐺 ∩ 𝐻)): ℋ⟶ ℋ
63	61, 62	hoeqi 31850	. 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 219	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 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ∀wral 3053 ∩ cin 3882 ∘ ccom 5622 ‘cfv 6485 (class class class)co 7356 ℂcc 11027 0cc0 11029 − cmin 11368 ℋchba 31008 +_ℎ cva 31009 ·_ih csp 31011 0_ℎc0v 31013 −_ℎ cmv 31014 S_ℋ csh 31017 C_ℋ cch 31018 ⊥cort 31019 proj_ℎcpjh 31026

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cc 10348 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 11109 ax-hilex 31088 ax-hfvadd 31089 ax-hvcom 31090 ax-hvass 31091 ax-hv0cl 31092 ax-hvaddid 31093 ax-hfvmul 31094 ax-hvmulid 31095 ax-hvmulass 31096 ax-hvdistr1 31097 ax-hvdistr2 31098 ax-hvmul0 31099 ax-hfi 31168 ax-his1 31171 ax-his2 31172 ax-his3 31173 ax-his4 31174 ax-hcompl 31291

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8633 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 9854 df-acn 9857 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-rlim 15442 df-sum 15640 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-cn 23210 df-cnp 23211 df-lm 23212 df-haus 23298 df-tx 23545 df-hmeo 23738 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-xms 24303 df-ms 24304 df-tms 24305 df-cfil 25240 df-cau 25241 df-cmet 25242 df-grpo 30582 df-gid 30583 df-ginv 30584 df-gdiv 30585 df-ablo 30634 df-vc 30648 df-nv 30681 df-va 30684 df-ba 30685 df-sm 30686 df-0v 30687 df-vs 30688 df-nmcv 30689 df-ims 30690 df-dip 30790 df-ssp 30811 df-ph 30902 df-cbn 30952 df-hnorm 31057 df-hba 31058 df-hvsub 31060 df-hlim 31061 df-hcau 31062 df-sh 31296 df-ch 31310 df-oc 31341 df-ch0 31342 df-shs 31397 df-pjh 31484

This theorem is referenced by: pjci 32289 pjcmul1i 32290 pjcmul2i 32291

W3C validator