spanunsni - Hilbert Space Explorer

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Theorem spanunsni 29358

Description: The span of the union of a closed subspace with a singleton equals the span of its union with an orthogonal singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
spanunsn.1	⊢ 𝐴 ∈ C_ℋ
spanunsn.2	⊢ 𝐵 ∈ ℋ

**Assertion**
Ref	Expression
spanunsni	⊢ (span‘(𝐴 ∪ {𝐵})) = (span‘(𝐴 ∪ {((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))

Proof of Theorem spanunsni Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		spanunsn.1	. . . . . . 7 ⊢ 𝐴 ∈ C_ℋ
2	1	chshii 29006	. . . . . 6 ⊢ 𝐴 ∈ S_ℋ
3		spanunsn.2	. . . . . . 7 ⊢ 𝐵 ∈ ℋ
4		snssi 4743	. . . . . . 7 ⊢ (𝐵 ∈ ℋ → {𝐵} ⊆ ℋ)
5		spancl 29115	. . . . . . 7 ⊢ ({𝐵} ⊆ ℋ → (span‘{𝐵}) ∈ S_ℋ )
6	3, 4, 5	mp2b 10	. . . . . 6 ⊢ (span‘{𝐵}) ∈ S_ℋ
7	2, 6	shseli 29095	. . . . 5 ⊢ (𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ (span‘{𝐵})𝑥 = (𝑦 +_ℎ 𝑧))
8	3	elspansni 29337	. . . . . . . 8 ⊢ (𝑧 ∈ (span‘{𝐵}) ↔ ∃𝑤 ∈ ℂ 𝑧 = (𝑤 ·_ℎ 𝐵))
9	1, 3	pjclii 29200	. . . . . . . . . . . . . . . 16 ⊢ ((proj_ℎ‘𝐴)‘𝐵) ∈ 𝐴
10		shmulcl 28997	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ S_ℋ ∧ 𝑤 ∈ ℂ ∧ ((proj_ℎ‘𝐴)‘𝐵) ∈ 𝐴) → (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ 𝐴)
11	2, 9, 10	mp3an13 1448	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℂ → (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ 𝐴)
12		shaddcl 28996	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ S_ℋ ∧ 𝑦 ∈ 𝐴 ∧ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ 𝐴) → (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) ∈ 𝐴)
13	11, 12	syl3an3 1161	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ S_ℋ ∧ 𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) ∈ 𝐴)
14	2, 13	mp3an1 1444	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) ∈ 𝐴)
15	1	choccli 29086	. . . . . . . . . . . . . . . 16 ⊢ (⊥‘𝐴) ∈ C_ℋ
16	15, 3	pjhclii 29201	. . . . . . . . . . . . . . 15 ⊢ ((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ ℋ
17		spansnmul 29343	. . . . . . . . . . . . . . 15 ⊢ ((((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ ℋ ∧ 𝑤 ∈ ℂ) → (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
18	16, 17	mpan 688	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℂ → (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
19	18	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
20	1, 3	pjpji 29203	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 = (((proj_ℎ‘𝐴)‘𝐵) +_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))
21	20	oveq2i 7169	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ·_ℎ 𝐵) = (𝑤 ·_ℎ (((proj_ℎ‘𝐴)‘𝐵) +_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))
22	1, 3	pjhclii 29201	. . . . . . . . . . . . . . . . . 18 ⊢ ((proj_ℎ‘𝐴)‘𝐵) ∈ ℋ
23		ax-hvdistr1 28787	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ ℂ ∧ ((proj_ℎ‘𝐴)‘𝐵) ∈ ℋ ∧ ((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ ℋ) → (𝑤 ·_ℎ (((proj_ℎ‘𝐴)‘𝐵) +_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) = ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))))
24	22, 16, 23	mp3an23 1449	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ℂ → (𝑤 ·_ℎ (((proj_ℎ‘𝐴)‘𝐵) +_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) = ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))))
25	21, 24	syl5eq 2870	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℂ → (𝑤 ·_ℎ 𝐵) = ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))))
26	25	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑤 ·_ℎ 𝐵) = ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))))
27	26	oveq2d 7174	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)) = (𝑦 +_ℎ ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))))
28	1	cheli 29011	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
29		hvmulcl 28792	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℂ ∧ ((proj_ℎ‘𝐴)‘𝐵) ∈ ℋ) → (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ)
30	22, 29	mpan2 689	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℂ → (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ)
31		hvmulcl 28792	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℂ ∧ ((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ ℋ) → (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ ℋ)
32	16, 31	mpan2 689	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℂ → (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ ℋ)
33	30, 32	jca 514	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℂ → ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ ∧ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ ℋ))
34		ax-hvass 28781	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℋ ∧ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ ∧ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ ℋ) → ((𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) = (𝑦 +_ℎ ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))))
35	34	3expb 1116	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℋ ∧ ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ ∧ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ ℋ)) → ((𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) = (𝑦 +_ℎ ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))))
36	28, 33, 35	syl2an 597	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → ((𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) = (𝑦 +_ℎ ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))))
37	27, 36	eqtr4d 2861	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)) = ((𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))))
38		rspceov 7205	. . . . . . . . . . . . 13 ⊢ (((𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) ∈ 𝐴 ∧ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}) ∧ (𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)) = ((𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))) → ∃𝑣 ∈ 𝐴 ∃𝑢 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})(𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)) = (𝑣 +_ℎ 𝑢))
39	14, 19, 37, 38	syl3anc 1367	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → ∃𝑣 ∈ 𝐴 ∃𝑢 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})(𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)) = (𝑣 +_ℎ 𝑢))
40		snssi 4743	. . . . . . . . . . . . . 14 ⊢ (((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ ℋ → {((proj_ℎ‘(⊥‘𝐴))‘𝐵)} ⊆ ℋ)
41		spancl 29115	. . . . . . . . . . . . . 14 ⊢ ({((proj_ℎ‘(⊥‘𝐴))‘𝐵)} ⊆ ℋ → (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}) ∈ S_ℋ )
42	16, 40, 41	mp2b 10	. . . . . . . . . . . . 13 ⊢ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}) ∈ S_ℋ
43	2, 42	shseli 29095	. . . . . . . . . . . 12 ⊢ ((𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)) ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) ↔ ∃𝑣 ∈ 𝐴 ∃𝑢 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})(𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)) = (𝑣 +_ℎ 𝑢))
44	39, 43	sylibr 236	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)) ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))
45		oveq2 7166	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑤 ·_ℎ 𝐵) → (𝑦 +_ℎ 𝑧) = (𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)))
46	45	eqeq2d 2834	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑤 ·_ℎ 𝐵) → (𝑥 = (𝑦 +_ℎ 𝑧) ↔ 𝑥 = (𝑦 +_ℎ (𝑤 ·_ℎ 𝐵))))
47	46	biimpa 479	. . . . . . . . . . 11 ⊢ ((𝑧 = (𝑤 ·_ℎ 𝐵) ∧ 𝑥 = (𝑦 +_ℎ 𝑧)) → 𝑥 = (𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)))
48		eleq1 2902	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)) → (𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) ↔ (𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)) ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))))
49	48	biimparc 482	. . . . . . . . . . 11 ⊢ (((𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)) ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) ∧ 𝑥 = (𝑦 +_ℎ (𝑤 ·_ℎ 𝐵))) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))
50	44, 47, 49	syl2an 597	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) ∧ (𝑧 = (𝑤 ·_ℎ 𝐵) ∧ 𝑥 = (𝑦 +_ℎ 𝑧))) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))
51	50	exp43 439	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → (𝑤 ∈ ℂ → (𝑧 = (𝑤 ·_ℎ 𝐵) → (𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))))))
52	51	rexlimdv 3285	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (∃𝑤 ∈ ℂ 𝑧 = (𝑤 ·_ℎ 𝐵) → (𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))))
53	8, 52	syl5bi 244	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (𝑧 ∈ (span‘{𝐵}) → (𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))))
54	53	rexlimdv 3285	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (∃𝑧 ∈ (span‘{𝐵})𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))))
55	54	rexlimiv 3282	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ (span‘{𝐵})𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))
56	7, 55	sylbi 219	. . . 4 ⊢ (𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))
57	2, 42	shseli 29095	. . . . 5 ⊢ (𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})𝑥 = (𝑦 +_ℎ 𝑧))
58	16	elspansni 29337	. . . . . . . 8 ⊢ (𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}) ↔ ∃𝑤 ∈ ℂ 𝑧 = (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))
59		negcl 10888	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ℂ → -𝑤 ∈ ℂ)
60		shmulcl 28997	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ S_ℋ ∧ -𝑤 ∈ ℂ ∧ ((proj_ℎ‘𝐴)‘𝐵) ∈ 𝐴) → (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ 𝐴)
61	2, 9, 60	mp3an13 1448	. . . . . . . . . . . . . . . . 17 ⊢ (-𝑤 ∈ ℂ → (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ 𝐴)
62	59, 61	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℂ → (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ 𝐴)
63		shaddcl 28996	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ S_ℋ ∧ (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ 𝑦) ∈ 𝐴)
64	62, 63	syl3an2 1160	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ S_ℋ ∧ 𝑤 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ 𝑦) ∈ 𝐴)
65	2, 64	mp3an1 1444	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ 𝑦) ∈ 𝐴)
66	65	ancoms 461	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ 𝑦) ∈ 𝐴)
67		spansnmul 29343	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℋ ∧ 𝑤 ∈ ℂ) → (𝑤 ·_ℎ 𝐵) ∈ (span‘{𝐵}))
68	3, 67	mpan 688	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℂ → (𝑤 ·_ℎ 𝐵) ∈ (span‘{𝐵}))
69	68	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑤 ·_ℎ 𝐵) ∈ (span‘{𝐵}))
70		hvm1neg 28811	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ∈ ℂ ∧ ((proj_ℎ‘𝐴)‘𝐵) ∈ ℋ) → (-1 ·_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) = (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)))
71	22, 70	mpan2 689	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ℂ → (-1 ·_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) = (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)))
72	71	oveq2d 7174	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ℂ → ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (-1 ·_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)))) = ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))))
73		hvnegid 28806	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ → ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (-1 ·_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)))) = 0_ℎ)
74	30, 73	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ℂ → ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (-1 ·_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)))) = 0_ℎ)
75		hvmulcl 28792	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((-𝑤 ∈ ℂ ∧ ((proj_ℎ‘𝐴)‘𝐵) ∈ ℋ) → (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ)
76	59, 22, 75	sylancl 588	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ℂ → (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ)
77		ax-hvcom 28780	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ ∧ (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ) → ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) = ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))))
78	30, 76, 77	syl2anc 586	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ ℂ → ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) = ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))))
79	72, 74, 78	3eqtr3d 2866	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ℂ → 0_ℎ = ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))))
80	79	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → 0_ℎ = ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))))
81	80	oveq1d 7173	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (0_ℎ +_ℎ (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))) = (((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) +_ℎ (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))))
82		hvaddcl 28791	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℋ ∧ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ ℋ) → (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) ∈ ℋ)
83	28, 32, 82	syl2an 597	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) ∈ ℋ)
84		hvaddid2 28802	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) ∈ ℋ → (0_ℎ +_ℎ (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))) = (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))))
85	83, 84	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (0_ℎ +_ℎ (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))) = (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))))
86	76, 30	jca 514	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ℂ → ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ ∧ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ))
87	86	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ ∧ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ))
88	28, 32	anim12i 614	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 ∈ ℋ ∧ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ ℋ))
89		hvadd4 28815	. . . . . . . . . . . . . . . 16 ⊢ ((((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ ∧ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ) ∧ (𝑦 ∈ ℋ ∧ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ ℋ)) → (((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) +_ℎ (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))) = (((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ 𝑦) +_ℎ ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))))
90	87, 88, 89	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) +_ℎ (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))) = (((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ 𝑦) +_ℎ ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))))
91	81, 85, 90	3eqtr3d 2866	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) = (((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ 𝑦) +_ℎ ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))))
92	26	oveq2d 7174	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ 𝑦) +_ℎ (𝑤 ·_ℎ 𝐵)) = (((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ 𝑦) +_ℎ ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))))
93	91, 92	eqtr4d 2861	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) = (((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ 𝑦) +_ℎ (𝑤 ·_ℎ 𝐵)))
94		rspceov 7205	. . . . . . . . . . . . 13 ⊢ ((((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ 𝑦) ∈ 𝐴 ∧ (𝑤 ·_ℎ 𝐵) ∈ (span‘{𝐵}) ∧ (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) = (((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ 𝑦) +_ℎ (𝑤 ·_ℎ 𝐵))) → ∃𝑣 ∈ 𝐴 ∃𝑢 ∈ (span‘{𝐵})(𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) = (𝑣 +_ℎ 𝑢))
95	66, 69, 93, 94	syl3anc 1367	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → ∃𝑣 ∈ 𝐴 ∃𝑢 ∈ (span‘{𝐵})(𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) = (𝑣 +_ℎ 𝑢))
96	2, 6	shseli 29095	. . . . . . . . . . . 12 ⊢ ((𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) ∈ (𝐴 +_ℋ (span‘{𝐵})) ↔ ∃𝑣 ∈ 𝐴 ∃𝑢 ∈ (span‘{𝐵})(𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) = (𝑣 +_ℎ 𝑢))
97	95, 96	sylibr 236	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) ∈ (𝐴 +_ℋ (span‘{𝐵})))
98		oveq2 7166	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) → (𝑦 +_ℎ 𝑧) = (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))))
99	98	eqeq2d 2834	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) → (𝑥 = (𝑦 +_ℎ 𝑧) ↔ 𝑥 = (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))))
100	99	biimpa 479	. . . . . . . . . . 11 ⊢ ((𝑧 = (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∧ 𝑥 = (𝑦 +_ℎ 𝑧)) → 𝑥 = (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))))
101		eleq1 2902	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) → (𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})) ↔ (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) ∈ (𝐴 +_ℋ (span‘{𝐵}))))
102	101	biimparc 482	. . . . . . . . . . 11 ⊢ (((𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) ∈ (𝐴 +_ℋ (span‘{𝐵})) ∧ 𝑥 = (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})))
103	97, 100, 102	syl2an 597	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) ∧ (𝑧 = (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∧ 𝑥 = (𝑦 +_ℎ 𝑧))) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})))
104	103	exp43 439	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → (𝑤 ∈ ℂ → (𝑧 = (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) → (𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵}))))))
105	104	rexlimdv 3285	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (∃𝑤 ∈ ℂ 𝑧 = (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) → (𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})))))
106	58, 105	syl5bi 244	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}) → (𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})))))
107	106	rexlimdv 3285	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (∃𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵}))))
108	107	rexlimiv 3282	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})))
109	57, 108	sylbi 219	. . . 4 ⊢ (𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})))
110	56, 109	impbii 211	. . 3 ⊢ (𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})) ↔ 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))
111	110	eqriv 2820	. 2 ⊢ (𝐴 +_ℋ (span‘{𝐵})) = (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
112	1	chssii 29010	. . . 4 ⊢ 𝐴 ⊆ ℋ
113	3, 4	ax-mp 5	. . . 4 ⊢ {𝐵} ⊆ ℋ
114	112, 113	spanuni 29323	. . 3 ⊢ (span‘(𝐴 ∪ {𝐵})) = ((span‘𝐴) +_ℋ (span‘{𝐵}))
115		spanid 29126	. . . . 5 ⊢ (𝐴 ∈ S_ℋ → (span‘𝐴) = 𝐴)
116	2, 115	ax-mp 5	. . . 4 ⊢ (span‘𝐴) = 𝐴
117	116	oveq1i 7168	. . 3 ⊢ ((span‘𝐴) +_ℋ (span‘{𝐵})) = (𝐴 +_ℋ (span‘{𝐵}))
118	114, 117	eqtri 2846	. 2 ⊢ (span‘(𝐴 ∪ {𝐵})) = (𝐴 +_ℋ (span‘{𝐵}))
119	16, 40	ax-mp 5	. . . 4 ⊢ {((proj_ℎ‘(⊥‘𝐴))‘𝐵)} ⊆ ℋ
120	112, 119	spanuni 29323	. . 3 ⊢ (span‘(𝐴 ∪ {((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) = ((span‘𝐴) +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
121	116	oveq1i 7168	. . 3 ⊢ ((span‘𝐴) +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) = (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
122	120, 121	eqtri 2846	. 2 ⊢ (span‘(𝐴 ∪ {((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) = (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
123	111, 118, 122	3eqtr4i 2856	1 ⊢ (span‘(𝐴 ∪ {𝐵})) = (span‘(𝐴 ∪ {((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 ∪ cun 3936 ⊆ wss 3938 {csn 4569 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 1c1 10540 -cneg 10873 ℋchba 28698 +_ℎ cva 28699 ·_ℎ csm 28700 0_ℎc0v 28703 S_ℋ csh 28707 C_ℋ cch 28708 ⊥cort 28709 +_ℋ cph 28710 spancspn 28711 proj_ℎcpjh 28716

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cc 9859 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 ax-hilex 28778 ax-hfvadd 28779 ax-hvcom 28780 ax-hvass 28781 ax-hv0cl 28782 ax-hvaddid 28783 ax-hfvmul 28784 ax-hvmulid 28785 ax-hvmulass 28786 ax-hvdistr1 28787 ax-hvdistr2 28788 ax-hvmul0 28789 ax-hfi 28858 ax-his1 28861 ax-his2 28862 ax-his3 28863 ax-his4 28864 ax-hcompl 28981

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-rlim 14848 df-sum 15045 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-cn 21837 df-cnp 21838 df-lm 21839 df-haus 21925 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cfil 23860 df-cau 23861 df-cmet 23862 df-grpo 28272 df-gid 28273 df-ginv 28274 df-gdiv 28275 df-ablo 28324 df-vc 28338 df-nv 28371 df-va 28374 df-ba 28375 df-sm 28376 df-0v 28377 df-vs 28378 df-nmcv 28379 df-ims 28380 df-dip 28480 df-ssp 28501 df-ph 28592 df-cbn 28642 df-hnorm 28747 df-hba 28748 df-hvsub 28750 df-hlim 28751 df-hcau 28752 df-sh 28986 df-ch 29000 df-oc 29031 df-ch0 29032 df-shs 29087 df-span 29088 df-pjh 29174

This theorem is referenced by: spansnji 29425

W3C validator