pjclem1 - Hilbert Space Explorer

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Theorem pjclem1 32097

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 31492	. . . . 5 ⊢ (𝐺 𝐶_ℋ 𝐻 ↔ 𝐺 = ((𝐺 ∩ 𝐻) ∨_ℋ (𝐺 ∩ (⊥‘𝐻))))
4		fveq2 6840	. . . . 5 ⊢ (𝐺 = ((𝐺 ∩ 𝐻) ∨_ℋ (𝐺 ∩ (⊥‘𝐻))) → (proj_ℎ‘𝐺) = (proj_ℎ‘((𝐺 ∩ 𝐻) ∨_ℋ (𝐺 ∩ (⊥‘𝐻)))))
5	3, 4	sylbi 217	. . . 4 ⊢ (𝐺 𝐶_ℋ 𝐻 → (proj_ℎ‘𝐺) = (proj_ℎ‘((𝐺 ∩ 𝐻) ∨_ℋ (𝐺 ∩ (⊥‘𝐻)))))
6		inss2 4197	. . . . . . . 8 ⊢ (𝐺 ∩ 𝐻) ⊆ 𝐻
7	1	choccli 31209	. . . . . . . . . 10 ⊢ (⊥‘𝐺) ∈ C_ℋ
8	2, 7	chub2i 31372	. . . . . . . . 9 ⊢ 𝐻 ⊆ ((⊥‘𝐺) ∨_ℋ 𝐻)
9	1, 2	chdmm3i 31381	. . . . . . . . 9 ⊢ (⊥‘(𝐺 ∩ (⊥‘𝐻))) = ((⊥‘𝐺) ∨_ℋ 𝐻)
10	8, 9	sseqtrri 3993	. . . . . . . 8 ⊢ 𝐻 ⊆ (⊥‘(𝐺 ∩ (⊥‘𝐻)))
11	6, 10	sstri 3953	. . . . . . 7 ⊢ (𝐺 ∩ 𝐻) ⊆ (⊥‘(𝐺 ∩ (⊥‘𝐻)))
12	1, 2	chincli 31362	. . . . . . . 8 ⊢ (𝐺 ∩ 𝐻) ∈ C_ℋ
13	2	choccli 31209	. . . . . . . . 9 ⊢ (⊥‘𝐻) ∈ C_ℋ
14	1, 13	chincli 31362	. . . . . . . 8 ⊢ (𝐺 ∩ (⊥‘𝐻)) ∈ C_ℋ
15	12, 14	pjscji 32072	. . . . . . 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 2742	. . . . 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 5812	. . . . . 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 31606	. . . . . . . . . 10 ⊢ (proj_ℎ‘(𝐺 ∩ 𝐻)): ℋ⟶ ℋ
20	14	pjfi 31606	. . . . . . . . . 10 ⊢ (proj_ℎ‘(𝐺 ∩ (⊥‘𝐻))): ℋ⟶ ℋ
21	2, 19, 20	pjsdii 32057	. . . . . . . . 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 32065	. . . . . . . . . . 11 ⊢ ((𝐺 ∩ 𝐻) ⊆ 𝐻 ↔ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(𝐺 ∩ 𝐻))) = (proj_ℎ‘(𝐺 ∩ 𝐻)))
23	6, 22	mpbi 230	. . . . . . . . . 10 ⊢ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(𝐺 ∩ 𝐻))) = (proj_ℎ‘(𝐺 ∩ 𝐻))
24	2, 14	pjorthcoi 32071	. . . . . . . . . . 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 7381	. . . . . . . . 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 31688	. . . . . . . . 9 ⊢ ((proj_ℎ‘(𝐺 ∩ 𝐻)) +_op 0_hop ) = (proj_ℎ‘(𝐺 ∩ 𝐻))
28	21, 26, 27	3eqtri 2756	. . . . . . . 8 ⊢ ((proj_ℎ‘𝐻) ∘ ((proj_ℎ‘(𝐺 ∩ 𝐻)) +_op (proj_ℎ‘(𝐺 ∩ (⊥‘𝐻))))) = (proj_ℎ‘(𝐺 ∩ 𝐻))
29	28	eqeq2i 2742	. . . . . . 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 5812	. . . . . . . 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 4196	. . . . . . . . 9 ⊢ (𝐺 ∩ 𝐻) ⊆ 𝐺
32	12, 1	pjss1coi 32065	. . . . . . . . 9 ⊢ ((𝐺 ∩ 𝐻) ⊆ 𝐺 ↔ ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘(𝐺 ∩ 𝐻))) = (proj_ℎ‘(𝐺 ∩ 𝐻)))
33	31, 32	mpbi 230	. . . . . . . 8 ⊢ ((proj_ℎ‘𝐺) ∘ (proj_ℎ‘(𝐺 ∩ 𝐻))) = (proj_ℎ‘(𝐺 ∩ 𝐻))
34	30, 33	eqtrdi 2780	. . . . . . 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 217	. . . . . 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 217	. . . 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 31496	. . . . 5 ⊢ (𝐺 𝐶_ℋ 𝐻 ↔ (⊥‘𝐺) 𝐶_ℋ 𝐻)
40	7, 2	cmbri 31492	. . . . 5 ⊢ ((⊥‘𝐺) 𝐶_ℋ 𝐻 ↔ (⊥‘𝐺) = (((⊥‘𝐺) ∩ 𝐻) ∨_ℋ ((⊥‘𝐺) ∩ (⊥‘𝐻))))
41	39, 40	bitri 275	. . . 4 ⊢ (𝐺 𝐶_ℋ 𝐻 ↔ (⊥‘𝐺) = (((⊥‘𝐺) ∩ 𝐻) ∨_ℋ ((⊥‘𝐺) ∩ (⊥‘𝐻))))
42		fveq2 6840	. . . . 5 ⊢ ((⊥‘𝐺) = (((⊥‘𝐺) ∩ 𝐻) ∨_ℋ ((⊥‘𝐺) ∩ (⊥‘𝐻))) → (proj_ℎ‘(⊥‘𝐺)) = (proj_ℎ‘(((⊥‘𝐺) ∩ 𝐻) ∨_ℋ ((⊥‘𝐺) ∩ (⊥‘𝐻)))))
43		inss2 4197	. . . . . . . . 9 ⊢ ((⊥‘𝐺) ∩ 𝐻) ⊆ 𝐻
44	2, 1	chub2i 31372	. . . . . . . . . 10 ⊢ 𝐻 ⊆ (𝐺 ∨_ℋ 𝐻)
45	1, 2	chdmm4i 31382	. . . . . . . . . 10 ⊢ (⊥‘((⊥‘𝐺) ∩ (⊥‘𝐻))) = (𝐺 ∨_ℋ 𝐻)
46	44, 45	sseqtrri 3993	. . . . . . . . 9 ⊢ 𝐻 ⊆ (⊥‘((⊥‘𝐺) ∩ (⊥‘𝐻)))
47	43, 46	sstri 3953	. . . . . . . 8 ⊢ ((⊥‘𝐺) ∩ 𝐻) ⊆ (⊥‘((⊥‘𝐺) ∩ (⊥‘𝐻)))
48	7, 2	chincli 31362	. . . . . . . . 9 ⊢ ((⊥‘𝐺) ∩ 𝐻) ∈ C_ℋ
49	7, 13	chincli 31362	. . . . . . . . 9 ⊢ ((⊥‘𝐺) ∩ (⊥‘𝐻)) ∈ C_ℋ
50	48, 49	pjscji 32072	. . . . . . . 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 2742	. . . . . 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 5812	. . . . . . 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 31606	. . . . . . . . . . 11 ⊢ (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)): ℋ⟶ ℋ
55	49	pjfi 31606	. . . . . . . . . . 11 ⊢ (proj_ℎ‘((⊥‘𝐺) ∩ (⊥‘𝐻))): ℋ⟶ ℋ
56	2, 54, 55	pjsdii 32057	. . . . . . . . . 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 32065	. . . . . . . . . . . 12 ⊢ (((⊥‘𝐺) ∩ 𝐻) ⊆ 𝐻 ↔ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻))) = (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)))
58	43, 57	mpbi 230	. . . . . . . . . . 11 ⊢ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻))) = (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻))
59	2, 49	pjorthcoi 32071	. . . . . . . . . . . 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 7381	. . . . . . . . . 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 31688	. . . . . . . . . 10 ⊢ ((proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)) +_op 0_hop ) = (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻))
63	56, 61, 62	3eqtri 2756	. . . . . . . . 9 ⊢ ((proj_ℎ‘𝐻) ∘ ((proj_ℎ‘((⊥‘𝐺) ∩ 𝐻)) +_op (proj_ℎ‘((⊥‘𝐺) ∩ (⊥‘𝐻))))) = (proj_ℎ‘((⊥‘𝐺) ∩ 𝐻))
64	63	eqeq2i 2742	. . . . . . . 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 5812	. . . . . . . . 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 31371	. . . . . . . . . . 11 ⊢ 𝐺 ⊆ (𝐺 ∨_ℋ (⊥‘𝐻))
67	1, 2	chdmm2i 31380	. . . . . . . . . . 11 ⊢ (⊥‘((⊥‘𝐺) ∩ 𝐻)) = (𝐺 ∨_ℋ (⊥‘𝐻))
68	66, 67	sseqtrri 3993	. . . . . . . . . 10 ⊢ 𝐺 ⊆ (⊥‘((⊥‘𝐺) ∩ 𝐻))
69	1, 48	pjorthcoi 32071	. . . . . . . . . 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 2780	. . . . . . . 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 217	. . . . . . 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 217	. . . . 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 217	. . 3 ⊢ (𝐺 𝐶_ℋ 𝐻 → ((proj_ℎ‘𝐺) ∘ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺)))) = 0_hop )
77	38, 76	oveq12d 7387	. 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 31651	. . . . . . 7 ⊢ I_op = (proj_ℎ‘ ℋ)
79	78	coeq2i 5814	. . . . . 6 ⊢ ((proj_ℎ‘𝐻) ∘ I_op ) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘ ℋ))
80	2	pjfi 31606	. . . . . . 7 ⊢ (proj_ℎ‘𝐻): ℋ⟶ ℋ
81	80	hoid1i 31691	. . . . . 6 ⊢ ((proj_ℎ‘𝐻) ∘ I_op ) = (proj_ℎ‘𝐻)
82	79, 81	eqtr3i 2754	. . . . 5 ⊢ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘ ℋ)) = (proj_ℎ‘𝐻)
83	1	pjtoi 32081	. . . . . . 7 ⊢ ((proj_ℎ‘𝐺) +_op (proj_ℎ‘(⊥‘𝐺))) = (proj_ℎ‘ ℋ)
84	83	coeq2i 5814	. . . . . 6 ⊢ ((proj_ℎ‘𝐻) ∘ ((proj_ℎ‘𝐺) +_op (proj_ℎ‘(⊥‘𝐺)))) = ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘ ℋ))
85	1	pjfi 31606	. . . . . . 7 ⊢ (proj_ℎ‘𝐺): ℋ⟶ ℋ
86	7	pjfi 31606	. . . . . . 7 ⊢ (proj_ℎ‘(⊥‘𝐺)): ℋ⟶ ℋ
87	2, 85, 86	pjsdii 32057	. . . . . 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 2754	. . . . 5 ⊢ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘ ℋ)) = (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) +_op ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺))))
89	82, 88	eqtr3i 2754	. . . 4 ⊢ (proj_ℎ‘𝐻) = (((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)) +_op ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺))))
90	89	coeq2i 5814	. . 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 31668	. . . 4 ⊢ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘𝐺)): ℋ⟶ ℋ
92	80, 86	hocofi 31668	. . . 4 ⊢ ((proj_ℎ‘𝐻) ∘ (proj_ℎ‘(⊥‘𝐺))): ℋ⟶ ℋ
93	1, 91, 92	pjsdii 32057	. . 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 2753	. 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 2787	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 1540 ∈ wcel 2109 ∩ cin 3910 ⊆ wss 3911 class class class wbr 5102 ∘ ccom 5635 ‘cfv 6499 (class class class)co 7369 ℋchba 30821 C_ℋ cch 30831 ⊥cort 30832 ∨_ℋ chj 30835 𝐶_ℋ ccm 30838 proj_ℎcpjh 30839 +_op chos 30840 0_hop ch0o 30845 I_op chio 30846

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cc 10364 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124 ax-hilex 30901 ax-hfvadd 30902 ax-hvcom 30903 ax-hvass 30904 ax-hv0cl 30905 ax-hvaddid 30906 ax-hfvmul 30907 ax-hvmulid 30908 ax-hvmulass 30909 ax-hvdistr1 30910 ax-hvdistr2 30911 ax-hvmul0 30912 ax-hfi 30981 ax-his1 30984 ax-his2 30985 ax-his3 30986 ax-his4 30987 ax-hcompl 31104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-omul 8416 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-acn 9871 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15430 df-rlim 15431 df-sum 15629 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19225 df-cmn 19688 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-cn 23090 df-cnp 23091 df-lm 23092 df-haus 23178 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-xms 24184 df-ms 24185 df-tms 24186 df-cfil 25131 df-cau 25132 df-cmet 25133 df-grpo 30395 df-gid 30396 df-ginv 30397 df-gdiv 30398 df-ablo 30447 df-vc 30461 df-nv 30494 df-va 30497 df-ba 30498 df-sm 30499 df-0v 30500 df-vs 30501 df-nmcv 30502 df-ims 30503 df-dip 30603 df-ssp 30624 df-ph 30715 df-cbn 30765 df-hnorm 30870 df-hba 30871 df-hvsub 30873 df-hlim 30874 df-hcau 30875 df-sh 31109 df-ch 31123 df-oc 31154 df-ch0 31155 df-shs 31210 df-chj 31212 df-pjh 31297 df-cm 31485 df-hosum 31632 df-h0op 31650 df-iop 31651

This theorem is referenced by: pjclem2 32098

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