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Theorem pjclem4 32359

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 32319	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ 𝐺)
4	3	adantl 485	. . . . . 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 32319	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))‘𝑥) ∈ 𝐻)
6		fveq1 6861	. . . . . . . . 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 2846	. . . . . . . 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 248	. . . . . . 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 410	. . . . . 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 4150	. . . . 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 32320	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ)
12		hvsubcl 31177	. . . . . . . 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 697	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥)) ∈ ℋ)
14	13	adantl 485	. . . . . 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 486	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → 𝑥 ∈ ℋ)
16	11	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ)
17	1, 2	chincli 31620	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∩ 𝐻) ∈ C_ℋ
18	17	cheli 31392	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐺 ∩ 𝐻) → 𝑦 ∈ ℋ)
19	18	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → 𝑦 ∈ ℋ)
20	15, 16, 19	3jca 1140	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → (𝑥 ∈ ℋ ∧ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ))
21	20	adantl 485	. . . . . . . . . 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 31252	. . . . . . . . . 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 7406	. . . . . . . . . . 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 32321	. . . . . . . . . . . . 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 602	. . . . . . . . . . . 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 32358	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝐺 ∩ 𝐻) → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑦) = 𝑦)
28	27	oveq2d 7407	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐺 ∩ 𝐻) → (𝑥 ·_ih (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑦)) = (𝑥 ·_ih 𝑦))
29	28	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → (𝑥 ·_ih (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑦)) = (𝑥 ·_ih 𝑦))
30	26, 29	eqtrd 2796	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → ((((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih 𝑦))
31	24, 30	sylan9eq 2816	. . . . . . . . . 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 7406	. . . . . . . . 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 622	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ ∧ 𝑦 ∈ ℋ))
34	33	adantl 485	. . . . . . . . . . 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 31240	. . . . . . . . . . 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 11524	. . . . . . . . 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 2802	. . . . . . . 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 460	. . . . . . 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 3152	. . . . . 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 31387	. . . . . . 7 ⊢ (𝐺 ∩ 𝐻) ∈ S_ℋ
42		shocel 31442	. . . . . . 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 592	. . . . 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 31865	. . . . 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 593	. . . 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 31198	. . . . . . . 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 1389	. . . . . . 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 31186	. . . . . . . . 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 7407	. . . . . . 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 31164	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑥 +_ℎ 0_ℎ) = 𝑥)
54	49, 52, 53	3eqtrd 2800	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) +_ℎ (𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥))) = 𝑥)
55	54	fveq2d 6866	. . . . 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 485	. . . 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 2798	. . 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 3153	. 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 31864	. . . 4 ⊢ (proj_ℎ‘𝐺): ℋ⟶ ℋ
60	2	pjfi 31864	. . . 4 ⊢ (proj_ℎ‘𝐻): ℋ⟶ ℋ
61	59, 60	hocofi 31926	. . 3 ⊢ ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)): ℋ⟶ ℋ
62	17	pjfi 31864	. . 3 ⊢ (proj_ℎ‘(𝐺 ∩ 𝐻)): ℋ⟶ ℋ
63	61, 62	hoeqi 31921	. 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 220	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 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ∀wral 3075 ∩ cin 3901 ∘ ccom 5647 ‘cfv 6516 (class class class)co 7391 ℂcc 11065 0cc0 11067 − cmin 11408 ℋchba 31079 +_ℎ cva 31080 ·_ih csp 31082 0_ℎc0v 31084 −_ℎ cmv 31085 S_ℋ csh 31088 C_ℋ cch 31089 ⊥cort 31090 proj_ℎcpjh 31097

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-inf2 9590 ax-cc 10386 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145 ax-addf 11146 ax-mulf 11147 ax-hilex 31159 ax-hfvadd 31160 ax-hvcom 31161 ax-hvass 31162 ax-hv0cl 31163 ax-hvaddid 31164 ax-hfvmul 31165 ax-hvmulid 31166 ax-hvmulass 31167 ax-hvdistr1 31168 ax-hvdistr2 31169 ax-hvmul0 31170 ax-hfi 31239 ax-his1 31242 ax-his2 31243 ax-his3 31244 ax-his4 31245 ax-hcompl 31362

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-of 7655 df-om 7842 df-1st 7965 df-2nd 7966 df-supp 8135 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-2o 8432 df-oadd 8435 df-omul 8436 df-er 8672 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9302 df-fi 9351 df-sup 9382 df-inf 9383 df-oi 9452 df-card 9891 df-acn 9894 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12476 df-z 12563 df-dec 12683 df-uz 12834 df-q 12944 df-rp 12988 df-xneg 13108 df-xadd 13109 df-xmul 13110 df-ioo 13347 df-ico 13349 df-icc 13350 df-fz 13507 df-fzo 13654 df-fl 13796 df-seq 14009 df-exp 14069 df-hash 14338 df-cj 15117 df-re 15118 df-im 15119 df-sqrt 15253 df-abs 15254 df-clim 15506 df-rlim 15507 df-sum 15705 df-struct 17174 df-sets 17191 df-slot 17209 df-ndx 17221 df-base 17237 df-ress 17258 df-plusg 17290 df-mulr 17291 df-starv 17292 df-sca 17293 df-vsca 17294 df-ip 17295 df-tset 17296 df-ple 17297 df-ds 17299 df-unif 17300 df-hom 17301 df-cco 17302 df-rest 17442 df-topn 17443 df-0g 17461 df-gsum 17462 df-topgen 17463 df-pt 17464 df-prds 17467 df-xrs 17523 df-qtop 17528 df-imas 17529 df-xps 17531 df-mre 17605 df-mrc 17606 df-acs 17608 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19101 df-cntz 19348 df-cmn 19813 df-psmet 21404 df-xmet 21405 df-met 21406 df-bl 21407 df-mopn 21408 df-fbas 21409 df-fg 21410 df-cnfld 21413 df-top 22942 df-topon 22959 df-topsp 22981 df-bases 22994 df-cld 23067 df-ntr 23068 df-cls 23069 df-nei 23146 df-cn 23275 df-cnp 23276 df-lm 23277 df-haus 23363 df-tx 23610 df-hmeo 23803 df-fil 23894 df-fm 23986 df-flim 23987 df-flf 23988 df-xms 24368 df-ms 24369 df-tms 24370 df-cfil 25305 df-cau 25306 df-cmet 25307 df-grpo 30653 df-gid 30654 df-ginv 30655 df-gdiv 30656 df-ablo 30705 df-vc 30719 df-nv 30752 df-va 30755 df-ba 30756 df-sm 30757 df-0v 30758 df-vs 30759 df-nmcv 30760 df-ims 30761 df-dip 30861 df-ssp 30882 df-ph 30973 df-cbn 31023 df-hnorm 31128 df-hba 31129 df-hvsub 31131 df-hlim 31132 df-hcau 31133 df-sh 31367 df-ch 31381 df-oc 31412 df-ch0 31413 df-shs 31468 df-pjh 31555

This theorem is referenced by: pjci 32360 pjcmul1i 32361 pjcmul2i 32362

W3C validator