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Theorem pjclem4 30561

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 30521	. . . . . . 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 30521	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))‘𝑥) ∈ 𝐻)
6		fveq1 6773	. . . . . . . . 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 2823	. . . . . . . 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 245	. . . . . . 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 4128	. . . . 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 30522	. . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) ∈ ℋ)
12		hvsubcl 29379	. . . . . . . 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 684	. . . . . . 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 29822	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∩ 𝐻) ∈ C_ℋ
18	17	cheli 29594	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐺 ∩ 𝐻) → 𝑦 ∈ ℋ)
19	18	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → 𝑦 ∈ ℋ)
20	15, 16, 19	3jca 1127	. . . . . . . . . . 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 29454	. . . . . . . . . 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 7290	. . . . . . . . . . 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 30523	. . . . . . . . . . . . 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 30560	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝐺 ∩ 𝐻) → (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑦) = 𝑦)
28	27	oveq2d 7291	. . . . . . . . . . . . 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 2778	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ (𝐺 ∩ 𝐻)) → ((((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))‘𝑥) ·_ih 𝑦) = (𝑥 ·_ih 𝑦))
31	24, 30	sylan9eq 2798	. . . . . . . . . 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 7290	. . . . . . . . 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 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 29442	. . . . . . . . . . 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 11320	. . . . . . . . 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 2784	. . . . . . . 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 3102	. . . . . 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 29589	. . . . . . 7 ⊢ (𝐺 ∩ 𝐻) ∈ S_ℋ
42		shocel 29644	. . . . . . 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 30067	. . . . 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 29400	. . . . . . . 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 1370	. . . . . . 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 29388	. . . . . . . . 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 7291	. . . . . . 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 29366	. . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑥 +_ℎ 0_ℎ) = 𝑥)
54	49, 52, 53	3eqtrd 2782	. . . . . 6 ⊢ (𝑥 ∈ ℋ → ((((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥) +_ℎ (𝑥 −_ℎ (((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))‘𝑥))) = 𝑥)
55	54	fveq2d 6778	. . . . 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 2780	. . 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 3103	. 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 30066	. . . 4 ⊢ (proj_ℎ‘𝐺): ℋ⟶ ℋ
60	2	pjfi 30066	. . . 4 ⊢ (proj_ℎ‘𝐻): ℋ⟶ ℋ
61	59, 60	hocofi 30128	. . 3 ⊢ ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)): ℋ⟶ ℋ
62	17	pjfi 30066	. . 3 ⊢ (proj_ℎ‘(𝐺 ∩ 𝐻)): ℋ⟶ ℋ
63	61, 62	hoeqi 30123	. 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 217	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∩ cin 3886 ∘ ccom 5593 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 0cc0 10871 − cmin 11205 ℋchba 29281 +_ℎ cva 29282 ·_ih csp 29284 0_ℎc0v 29286 −_ℎ cmv 29287 S_ℋ csh 29290 C_ℋ cch 29291 ⊥cort 29292 proj_ℎcpjh 29299

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cc 10191 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 ax-hilex 29361 ax-hfvadd 29362 ax-hvcom 29363 ax-hvass 29364 ax-hv0cl 29365 ax-hvaddid 29366 ax-hfvmul 29367 ax-hvmulid 29368 ax-hvmulass 29369 ax-hvdistr1 29370 ax-hvdistr2 29371 ax-hvmul0 29372 ax-hfi 29441 ax-his1 29444 ax-his2 29445 ax-his3 29446 ax-his4 29447 ax-hcompl 29564

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-omul 8302 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-acn 9700 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-cn 22378 df-cnp 22379 df-lm 22380 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cfil 24419 df-cau 24420 df-cmet 24421 df-grpo 28855 df-gid 28856 df-ginv 28857 df-gdiv 28858 df-ablo 28907 df-vc 28921 df-nv 28954 df-va 28957 df-ba 28958 df-sm 28959 df-0v 28960 df-vs 28961 df-nmcv 28962 df-ims 28963 df-dip 29063 df-ssp 29084 df-ph 29175 df-cbn 29225 df-hnorm 29330 df-hba 29331 df-hvsub 29333 df-hlim 29334 df-hcau 29335 df-sh 29569 df-ch 29583 df-oc 29614 df-ch0 29615 df-shs 29670 df-pjh 29757

This theorem is referenced by: pjci 30562 pjcmul1i 30563 pjcmul2i 30564

W3C validator