pjclem1 - Hilbert Space Explorer

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Theorem pjclem1 31435

Description: Lemma for projection commutation theorem. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)

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

**Assertion**
Ref	Expression
pjclem1	⊢ (𝐺 𝐶_ℋ 𝐻 → ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = (proj_ℎ‘(𝐺 ∩ 𝐻)))

**Proof of Theorem pjclem1**
Step	Hyp	Ref	Expression
1		pjclem1.1	. . . . . 6 ⊢ 𝐺 ∈ C_ℋ
2		pjclem1.2	. . . . . 6 ⊢ 𝐻 ∈ C_ℋ
3	1, 2	cmbri 30830	. . . . 5 ⊢ (𝐺 𝐶_ℋ 𝐻 ↔ 𝐺 = ((𝐺 ∩ 𝐻) ∨_ℋ (𝐺 ∩ (⊥‘𝐻))))
4		fveq2 6888	. . . . 5 ⊢ (𝐺 = ((𝐺 ∩ 𝐻) ∨_ℋ (𝐺 ∩ (⊥‘𝐻))) → (proj_ℎ‘𝐺) = (proj_ℎ‘((𝐺 ∩ 𝐻) ∨_ℋ (𝐺 ∩ (⊥‘𝐻)))))
5	3, 4	sylbi 216	. . . 4 ⊢ (𝐺 𝐶_ℋ 𝐻 → (proj_ℎ‘𝐺) = (proj_ℎ‘((𝐺 ∩ 𝐻) ∨_ℋ (𝐺 ∩ (⊥‘𝐻)))))
6		inss2 4228	. . . . . . . 8 ⊢ (𝐺 ∩ 𝐻) ⊆ 𝐻
7	1	choccli 30547	. . . . . . . . . 10 ⊢ (⊥‘𝐺) ∈ C_ℋ
8	2, 7	chub2i 30710	. . . . . . . . 9 ⊢ 𝐻 ⊆ ((⊥‘𝐺) ∨_ℋ 𝐻)
9	1, 2	chdmm3i 30719	. . . . . . . . 9 ⊢ (⊥‘(𝐺 ∩ (⊥‘𝐻))) = ((⊥‘𝐺) ∨_ℋ 𝐻)
10	8, 9	sseqtrri 4018	. . . . . . . 8 ⊢ 𝐻 ⊆ (⊥‘(𝐺 ∩ (⊥‘𝐻)))
11	6, 10	sstri 3990	. . . . . . 7 ⊢ (𝐺 ∩ 𝐻) ⊆ (⊥‘(𝐺 ∩ (⊥‘𝐻)))
12	1, 2	chincli 30700	. . . . . . . 8 ⊢ (𝐺 ∩ 𝐻) ∈ C_ℋ
13	2	choccli 30547	. . . . . . . . 9 ⊢ (⊥‘𝐻) ∈ C_ℋ
14	1, 13	chincli 30700	. . . . . . . 8 ⊢ (𝐺 ∩ (⊥‘𝐻)) ∈ C_ℋ
15	12, 14	pjscji 31410	. . . . . . 7 ⊢ ((𝐺 ∩ 𝐻) ⊆ (⊥‘(𝐺 ∩ (⊥‘𝐻))) → (proj_ℎ‘((𝐺 ∩ 𝐻) ∨_ℋ (𝐺 ∩ (⊥‘𝐻)))) = ((proj_ℎ‘(𝐺 ∩ 𝐻)) +_op (proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))))
16	11, 15	ax-mp 5	. . . . . 6 ⊢ (proj_ℎ‘((𝐺 ∩ 𝐻) ∨_ℋ (𝐺 ∩ (⊥‘𝐻)))) = ((proj_ℎ‘(𝐺 ∩ 𝐻)) +_op (proj_ℎ‘(𝐺 ∩ (⊥‘𝐻))))
17	16	eqeq2i 2745	. . . . 5 ⊢ ((proj_ℎ‘𝐺) = (proj_ℎ‘((𝐺 ∩ 𝐻) ∨_ℋ (𝐺 ∩ (⊥‘𝐻)))) ↔ (proj_ℎ‘𝐺) = ((proj_ℎ‘(𝐺 ∩ 𝐻)) +_op (proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))))
18		coeq2 5856	. . . . . 6 ⊢ ((proj_ℎ‘𝐺) = ((proj_ℎ‘(𝐺 ∩ 𝐻)) +_op (proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))) → ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) = ((proj_ℎ‘𝐻) ∘ ((proj_ℎ‘(𝐺 ∩ 𝐻)) +_op (proj_ℎ‘(𝐺 ∩ (⊥‘𝐻))))))
19	12	pjfi 30944	. . . . . . . . . 10 ⊢ (proj_ℎ‘(𝐺 ∩ 𝐻)): ℋ⟶ ℋ
20	14	pjfi 30944	. . . . . . . . . 10 ⊢ (proj_ℎ‘(𝐺 ∩ (⊥‘𝐻))): ℋ⟶ ℋ
21	2, 19, 20	pjsdii 31395	. . . . . . . . 9 ⊢ ((proj_ℎ‘𝐻) ∘ ((proj_ℎ‘(𝐺 ∩ 𝐻)) +_op (proj_ℎ‘(𝐺 ∩ (⊥‘𝐻))))) = (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(𝐺 ∩ 𝐻))) +_op ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))))
22	12, 2	pjss1coi 31403	. . . . . . . . . . 11 ⊢ ((𝐺 ∩ 𝐻) ⊆ 𝐻 ↔ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(𝐺 ∩ 𝐻))) = (proj_ℎ‘(𝐺 ∩ 𝐻)))
23	6, 22	mpbi 229	. . . . . . . . . 10 ⊢ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(𝐺 ∩ 𝐻))) = (proj_ℎ‘(𝐺 ∩ 𝐻))
24	2, 14	pjorthcoi 31409	. . . . . . . . . . 11 ⊢ (𝐻 ⊆ (⊥‘(𝐺 ∩ (⊥‘𝐻))) → ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))) = 0_hop )
25	10, 24	ax-mp 5	. . . . . . . . . 10 ⊢ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))) = 0_hop
26	23, 25	oveq12i 7417	. . . . . . . . 9 ⊢ (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(𝐺 ∩ 𝐻))) +_op ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(𝐺 ∩ (⊥‘𝐻))))) = ((proj_ℎ‘(𝐺 ∩ 𝐻)) +_op 0_hop )
27	19	hoaddridi 31026	. . . . . . . . 9 ⊢ ((proj_ℎ‘(𝐺 ∩ 𝐻)) +_op 0_hop ) = (proj_ℎ‘(𝐺 ∩ 𝐻))
28	21, 26, 27	3eqtri 2764	. . . . . . . 8 ⊢ ((proj_ℎ‘𝐻) ∘ ((proj_ℎ‘(𝐺 ∩ 𝐻)) +_op (proj_ℎ‘(𝐺 ∩ (⊥‘𝐻))))) = (proj_ℎ‘(𝐺 ∩ 𝐻))
29	28	eqeq2i 2745	. . . . . . 7 ⊢ (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) = ((proj_ℎ‘𝐻) ∘ ((proj_ℎ‘(𝐺 ∩ 𝐻)) +_op (proj_ℎ‘(𝐺 ∩ (⊥‘𝐻))))) ↔ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) = (proj_ℎ‘(𝐺 ∩ 𝐻)))
30		coeq2 5856	. . . . . . . 8 ⊢ (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) = (proj_ℎ‘(𝐺 ∩ 𝐻)) → ((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))) = ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘(𝐺 ∩ 𝐻))))
31		inss1 4227	. . . . . . . . 9 ⊢ (𝐺 ∩ 𝐻) ⊆ 𝐺
32	12, 1	pjss1coi 31403	. . . . . . . . 9 ⊢ ((𝐺 ∩ 𝐻) ⊆ 𝐺 ↔ ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘(𝐺 ∩ 𝐻))) = (proj_ℎ‘(𝐺 ∩ 𝐻)))
33	31, 32	mpbi 229	. . . . . . . 8 ⊢ ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘(𝐺 ∩ 𝐻))) = (proj_ℎ‘(𝐺 ∩ 𝐻))
34	30, 33	eqtrdi 2788	. . . . . . 7 ⊢ (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) = (proj_ℎ‘(𝐺 ∩ 𝐻)) → ((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))) = (proj_ℎ‘(𝐺 ∩ 𝐻)))
35	29, 34	sylbi 216	. . . . . 6 ⊢ (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) = ((proj_ℎ‘𝐻) ∘ ((proj_ℎ‘(𝐺 ∩ 𝐻)) +_op (proj_ℎ‘(𝐺 ∩ (⊥‘𝐻))))) → ((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))) = (proj_ℎ‘(𝐺 ∩ 𝐻)))
36	18, 35	syl 17	. . . . 5 ⊢ ((proj_ℎ‘𝐺) = ((proj_ℎ‘(𝐺 ∩ 𝐻)) +_op (proj_ℎ‘(𝐺 ∩ (⊥‘𝐻)))) → ((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))) = (proj_ℎ‘(𝐺 ∩ 𝐻)))
37	17, 36	sylbi 216	. . . 4 ⊢ ((proj_ℎ‘𝐺) = (proj_ℎ‘((𝐺 ∩ 𝐻) ∨_ℋ (𝐺 ∩ (⊥‘𝐻)))) → ((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))) = (proj_ℎ‘(𝐺 ∩ 𝐻)))
38	5, 37	syl 17	. . 3 ⊢ (𝐺 𝐶_ℋ 𝐻 → ((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))) = (proj_ℎ‘(𝐺 ∩ 𝐻)))
39	1, 2	cmcm3i 30834	. . . . 5 ⊢ (𝐺 𝐶_ℋ 𝐻 ↔ (⊥‘𝐺) 𝐶_ℋ 𝐻)
40	7, 2	cmbri 30830	. . . . 5 ⊢ ((⊥‘𝐺) 𝐶_ℋ 𝐻 ↔ (⊥‘𝐺) = (((⊥‘𝐺) ∩ 𝐻) ∨_ℋ ((⊥‘𝐺) ∩ (⊥‘𝐻))))
41	39, 40	bitri 274	. . . 4 ⊢ (𝐺 𝐶_ℋ 𝐻 ↔ (⊥‘𝐺) = (((⊥‘𝐺) ∩ 𝐻) ∨_ℋ ((⊥‘𝐺) ∩ (⊥‘𝐻))))
42		fveq2 6888	. . . . 5 ⊢ ((⊥‘𝐺) = (((⊥‘𝐺) ∩ 𝐻) ∨_ℋ ((⊥‘𝐺) ∩ (⊥‘𝐻))) → (proj_ℎ‘(⊥‘𝐺)) = (proj_ℎ‘(((⊥‘𝐺) ∩ 𝐻) ∨_ℋ ((⊥‘𝐺) ∩ (⊥‘𝐻)))))
43		inss2 4228	. . . . . . . . 9 ⊢ ((⊥‘𝐺) ∩ 𝐻) ⊆ 𝐻
44	2, 1	chub2i 30710	. . . . . . . . . 10 ⊢ 𝐻 ⊆ (𝐺 ∨_ℋ 𝐻)
45	1, 2	chdmm4i 30720	. . . . . . . . . 10 ⊢ (⊥‘((⊥‘𝐺) ∩ (⊥‘𝐻))) = (𝐺 ∨_ℋ 𝐻)
46	44, 45	sseqtrri 4018	. . . . . . . . 9 ⊢ 𝐻 ⊆ (⊥‘((⊥‘𝐺) ∩ (⊥‘𝐻)))
47	43, 46	sstri 3990	. . . . . . . 8 ⊢ ((⊥‘𝐺) ∩ 𝐻) ⊆ (⊥‘((⊥‘𝐺) ∩ (⊥‘𝐻)))
48	7, 2	chincli 30700	. . . . . . . . 9 ⊢ ((⊥‘𝐺) ∩ 𝐻) ∈ C_ℋ
49	7, 13	chincli 30700	. . . . . . . . 9 ⊢ ((⊥‘𝐺) ∩ (⊥‘𝐻)) ∈ C_ℋ
50	48, 49	pjscji 31410	. . . . . . . 8 ⊢ (((⊥‘𝐺) ∩ 𝐻) ⊆ (⊥‘((⊥‘𝐺) ∩ (⊥‘𝐻))) → (proj_ℎ‘(((⊥‘𝐺) ∩ 𝐻) ∨_ℋ ((⊥‘𝐺) ∩ (⊥‘𝐻)))) = ((proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)) +_op (proj_ℎ‘((⊥‘𝐺) ∩ (⊥‘𝐻)))))
51	47, 50	ax-mp 5	. . . . . . 7 ⊢ (proj_ℎ‘(((⊥‘𝐺) ∩ 𝐻) ∨_ℋ ((⊥‘𝐺) ∩ (⊥‘𝐻)))) = ((proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)) +_op (proj_ℎ‘((⊥‘𝐺) ∩ (⊥‘𝐻))))
52	51	eqeq2i 2745	. . . . . 6 ⊢ ((proj_ℎ‘(⊥‘𝐺)) = (proj_ℎ‘(((⊥‘𝐺) ∩ 𝐻) ∨_ℋ ((⊥‘𝐺) ∩ (⊥‘𝐻)))) ↔ (proj_ℎ‘(⊥‘𝐺)) = ((proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)) +_op (proj_ℎ‘((⊥‘𝐺) ∩ (⊥‘𝐻)))))
53		coeq2 5856	. . . . . . 7 ⊢ ((proj_ℎ‘(⊥‘𝐺)) = ((proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)) +_op (proj_ℎ‘((⊥‘𝐺) ∩ (⊥‘𝐻)))) → ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺))) = ((proj_ℎ‘𝐻) ∘ ((proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)) +_op (proj_ℎ‘((⊥‘𝐺) ∩ (⊥‘𝐻))))))
54	48	pjfi 30944	. . . . . . . . . . 11 ⊢ (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)): ℋ⟶ ℋ
55	49	pjfi 30944	. . . . . . . . . . 11 ⊢ (proj_ℎ‘((⊥‘𝐺) ∩ (⊥‘𝐻))): ℋ⟶ ℋ
56	2, 54, 55	pjsdii 31395	. . . . . . . . . 10 ⊢ ((proj_ℎ‘𝐻) ∘ ((proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)) +_op (proj_ℎ‘((⊥‘𝐺) ∩ (⊥‘𝐻))))) = (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻))) +_op ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘((⊥‘𝐺) ∩ (⊥‘𝐻)))))
57	48, 2	pjss1coi 31403	. . . . . . . . . . . 12 ⊢ (((⊥‘𝐺) ∩ 𝐻) ⊆ 𝐻 ↔ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻))) = (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)))
58	43, 57	mpbi 229	. . . . . . . . . . 11 ⊢ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻))) = (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻))
59	2, 49	pjorthcoi 31409	. . . . . . . . . . . 12 ⊢ (𝐻 ⊆ (⊥‘((⊥‘𝐺) ∩ (⊥‘𝐻))) → ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘((⊥‘𝐺) ∩ (⊥‘𝐻)))) = 0_hop )
60	46, 59	ax-mp 5	. . . . . . . . . . 11 ⊢ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘((⊥‘𝐺) ∩ (⊥‘𝐻)))) = 0_hop
61	58, 60	oveq12i 7417	. . . . . . . . . 10 ⊢ (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻))) +_op ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘((⊥‘𝐺) ∩ (⊥‘𝐻))))) = ((proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)) +_op 0_hop )
62	54	hoaddridi 31026	. . . . . . . . . 10 ⊢ ((proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)) +_op 0_hop ) = (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻))
63	56, 61, 62	3eqtri 2764	. . . . . . . . 9 ⊢ ((proj_ℎ‘𝐻) ∘ ((proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)) +_op (proj_ℎ‘((⊥‘𝐺) ∩ (⊥‘𝐻))))) = (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻))
64	63	eqeq2i 2745	. . . . . . . 8 ⊢ (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺))) = ((proj_ℎ‘𝐻) ∘ ((proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)) +_op (proj_ℎ‘((⊥‘𝐺) ∩ (⊥‘𝐻))))) ↔ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺))) = (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)))
65		coeq2 5856	. . . . . . . . 9 ⊢ (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺))) = (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)) → ((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺)))) = ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻))))
66	1, 13	chub1i 30709	. . . . . . . . . . 11 ⊢ 𝐺 ⊆ (𝐺 ∨_ℋ (⊥‘𝐻))
67	1, 2	chdmm2i 30718	. . . . . . . . . . 11 ⊢ (⊥‘((⊥‘𝐺) ∩ 𝐻)) = (𝐺 ∨_ℋ (⊥‘𝐻))
68	66, 67	sseqtrri 4018	. . . . . . . . . 10 ⊢ 𝐺 ⊆ (⊥‘((⊥‘𝐺) ∩ 𝐻))
69	1, 48	pjorthcoi 31409	. . . . . . . . . 10 ⊢ (𝐺 ⊆ (⊥‘((⊥‘𝐺) ∩ 𝐻)) → ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻))) = 0_hop )
70	68, 69	ax-mp 5	. . . . . . . . 9 ⊢ ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻))) = 0_hop
71	65, 70	eqtrdi 2788	. . . . . . . 8 ⊢ (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺))) = (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)) → ((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺)))) = 0_hop )
72	64, 71	sylbi 216	. . . . . . 7 ⊢ (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺))) = ((proj_ℎ‘𝐻) ∘ ((proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)) +_op (proj_ℎ‘((⊥‘𝐺) ∩ (⊥‘𝐻))))) → ((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺)))) = 0_hop )
73	53, 72	syl 17	. . . . . 6 ⊢ ((proj_ℎ‘(⊥‘𝐺)) = ((proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)) +_op (proj_ℎ‘((⊥‘𝐺) ∩ (⊥‘𝐻)))) → ((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺)))) = 0_hop )
74	52, 73	sylbi 216	. . . . 5 ⊢ ((proj_ℎ‘(⊥‘𝐺)) = (proj_ℎ‘(((⊥‘𝐺) ∩ 𝐻) ∨_ℋ ((⊥‘𝐺) ∩ (⊥‘𝐻)))) → ((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺)))) = 0_hop )
75	42, 74	syl 17	. . . 4 ⊢ ((⊥‘𝐺) = (((⊥‘𝐺) ∩ 𝐻) ∨_ℋ ((⊥‘𝐺) ∩ (⊥‘𝐻))) → ((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺)))) = 0_hop )
76	41, 75	sylbi 216	. . 3 ⊢ (𝐺 𝐶_ℋ 𝐻 → ((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺)))) = 0_hop )
77	38, 76	oveq12d 7423	. 2 ⊢ (𝐺 𝐶_ℋ 𝐻 → (((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))) +_op ((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺))))) = ((proj_ℎ‘(𝐺 ∩ 𝐻)) +_op 0_hop ))
78		df-iop 30989	. . . . . . 7 ⊢ I_op = (proj_ℎ‘ ℋ)
79	78	coeq2i 5858	. . . . . 6 ⊢ ((proj_ℎ‘𝐻) ∘ I_op ) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘ ℋ))
80	2	pjfi 30944	. . . . . . 7 ⊢ (proj_ℎ‘𝐻): ℋ⟶ ℋ
81	80	hoid1i 31029	. . . . . 6 ⊢ ((proj_ℎ‘𝐻) ∘ I_op ) = (proj_ℎ‘𝐻)
82	79, 81	eqtr3i 2762	. . . . 5 ⊢ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘ ℋ)) = (proj_ℎ‘𝐻)
83	1	pjtoi 31419	. . . . . . 7 ⊢ ((proj_ℎ‘𝐺) +_op (proj_ℎ‘(⊥‘𝐺))) = (proj_ℎ‘ ℋ)
84	83	coeq2i 5858	. . . . . 6 ⊢ ((proj_ℎ‘𝐻) ∘ ((proj_ℎ‘𝐺) +_op (proj_ℎ‘(⊥‘𝐺)))) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘ ℋ))
85	1	pjfi 30944	. . . . . . 7 ⊢ (proj_ℎ‘𝐺): ℋ⟶ ℋ
86	7	pjfi 30944	. . . . . . 7 ⊢ (proj_ℎ‘(⊥‘𝐺)): ℋ⟶ ℋ
87	2, 85, 86	pjsdii 31395	. . . . . 6 ⊢ ((proj_ℎ‘𝐻) ∘ ((proj_ℎ‘𝐺) +_op (proj_ℎ‘(⊥‘𝐺)))) = (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) +_op ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺))))
88	84, 87	eqtr3i 2762	. . . . 5 ⊢ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘ ℋ)) = (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) +_op ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺))))
89	82, 88	eqtr3i 2762	. . . 4 ⊢ (proj_ℎ‘𝐻) = (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) +_op ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺))))
90	89	coeq2i 5858	. . 3 ⊢ ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = ((proj_ℎ‘𝐺) ∘ (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) +_op ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺)))))
91	80, 85	hocofi 31006	. . . 4 ⊢ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)): ℋ⟶ ℋ
92	80, 86	hocofi 31006	. . . 4 ⊢ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺))): ℋ⟶ ℋ
93	1, 91, 92	pjsdii 31395	. . 3 ⊢ ((proj_ℎ‘𝐺) ∘ (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) +_op ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺))))) = (((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))) +_op ((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺)))))
94	90, 93	eqtr2i 2761	. 2 ⊢ (((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺))) +_op ((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺))))) = ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻))
95	77, 94, 27	3eqtr3g 2795	1 ⊢ (𝐺 𝐶_ℋ 𝐻 → ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘𝐻)) = (proj_ℎ‘(𝐺 ∩ 𝐻)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∩ cin 3946 ⊆ wss 3947 class class class wbr 5147 ∘ ccom 5679 ‘cfv 6540 (class class class)co 7405 ℋchba 30159 C_ℋ cch 30169 ⊥cort 30170 ∨_ℋ chj 30173 𝐶_ℋ ccm 30176 proj_ℎcpjh 30177 +_op chos 30178 0_hop ch0o 30183 I_op chio 30184

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cc 10426 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 ax-hilex 30239 ax-hfvadd 30240 ax-hvcom 30241 ax-hvass 30242 ax-hv0cl 30243 ax-hvaddid 30244 ax-hfvmul 30245 ax-hvmulid 30246 ax-hvmulass 30247 ax-hvdistr1 30248 ax-hvdistr2 30249 ax-hvmul0 30250 ax-hfi 30319 ax-his1 30322 ax-his2 30323 ax-his3 30324 ax-his4 30325 ax-hcompl 30442

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-acn 9933 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429 df-sum 15629 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-cn 22722 df-cnp 22723 df-lm 22724 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cfil 24763 df-cau 24764 df-cmet 24765 df-grpo 29733 df-gid 29734 df-ginv 29735 df-gdiv 29736 df-ablo 29785 df-vc 29799 df-nv 29832 df-va 29835 df-ba 29836 df-sm 29837 df-0v 29838 df-vs 29839 df-nmcv 29840 df-ims 29841 df-dip 29941 df-ssp 29962 df-ph 30053 df-cbn 30103 df-hnorm 30208 df-hba 30209 df-hvsub 30211 df-hlim 30212 df-hcau 30213 df-sh 30447 df-ch 30461 df-oc 30492 df-ch0 30493 df-shs 30548 df-chj 30550 df-pjh 30635 df-cm 30823 df-hosum 30970 df-h0op 30988 df-iop 30989

This theorem is referenced by: pjclem2 31436

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