spanunsni - Hilbert Space Explorer

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Theorem spanunsni 31654

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 31302	. . . . . 6 ⊢ 𝐴 ∈ S_ℋ
3		spanunsn.2	. . . . . . 7 ⊢ 𝐵 ∈ ℋ
4		snssi 4764	. . . . . . 7 ⊢ (𝐵 ∈ ℋ → {𝐵} ⊆ ℋ)
5		spancl 31411	. . . . . . 7 ⊢ ({𝐵} ⊆ ℋ → (span‘{𝐵}) ∈ S_ℋ )
6	3, 4, 5	mp2b 10	. . . . . 6 ⊢ (span‘{𝐵}) ∈ S_ℋ
7	2, 6	shseli 31391	. . . . 5 ⊢ (𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ (span‘{𝐵})𝑥 = (𝑦 +_ℎ 𝑧))
8	3	elspansni 31633	. . . . . . . 8 ⊢ (𝑧 ∈ (span‘{𝐵}) ↔ ∃𝑤 ∈ ℂ 𝑧 = (𝑤 ·_ℎ 𝐵))
9	1, 3	pjclii 31496	. . . . . . . . . . . . . . . 16 ⊢ ((proj_ℎ‘𝐴)‘𝐵) ∈ 𝐴
10		shmulcl 31293	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ S_ℋ ∧ 𝑤 ∈ ℂ ∧ ((proj_ℎ‘𝐴)‘𝐵) ∈ 𝐴) → (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ 𝐴)
11	2, 9, 10	mp3an13 1454	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℂ → (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ 𝐴)
12		shaddcl 31292	. . . . . . . . . . . . . . 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 1165	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ S_ℋ ∧ 𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) ∈ 𝐴)
14	2, 13	mp3an1 1450	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) ∈ 𝐴)
15	1	choccli 31382	. . . . . . . . . . . . . . . 16 ⊢ (⊥‘𝐴) ∈ C_ℋ
16	15, 3	pjhclii 31497	. . . . . . . . . . . . . . 15 ⊢ ((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ ℋ
17		spansnmul 31639	. . . . . . . . . . . . . . 15 ⊢ ((((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ ℋ ∧ 𝑤 ∈ ℂ) → (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
18	16, 17	mpan 690	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℂ → (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
19	18	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
20	1, 3	pjpji 31499	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 = (((proj_ℎ‘𝐴)‘𝐵) +_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))
21	20	oveq2i 7369	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ·_ℎ 𝐵) = (𝑤 ·_ℎ (((proj_ℎ‘𝐴)‘𝐵) +_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))
22	1, 3	pjhclii 31497	. . . . . . . . . . . . . . . . . 18 ⊢ ((proj_ℎ‘𝐴)‘𝐵) ∈ ℋ
23		ax-hvdistr1 31083	. . . . . . . . . . . . . . . . . 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 1455	. . . . . . . . . . . . . . . . 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	eqtrid 2783	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℂ → (𝑤 ·_ℎ 𝐵) = ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))))
26	25	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑤 ·_ℎ 𝐵) = ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))))
27	26	oveq2d 7374	. . . . . . . . . . . . . 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 31307	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
29		hvmulcl 31088	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℂ ∧ ((proj_ℎ‘𝐴)‘𝐵) ∈ ℋ) → (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ)
30	22, 29	mpan2 691	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℂ → (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ)
31		hvmulcl 31088	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℂ ∧ ((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ ℋ) → (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ ℋ)
32	16, 31	mpan2 691	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℂ → (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ ℋ)
33	30, 32	jca 511	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℂ → ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ ∧ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ ℋ))
34		ax-hvass 31077	. . . . . . . . . . . . . . . 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 1120	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . 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 7407	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → ∃𝑣 ∈ 𝐴 ∃𝑢 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})(𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)) = (𝑣 +_ℎ 𝑢))
40		snssi 4764	. . . . . . . . . . . . . 14 ⊢ (((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ ℋ → {((proj_ℎ‘(⊥‘𝐴))‘𝐵)} ⊆ ℋ)
41		spancl 31411	. . . . . . . . . . . . . 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 31391	. . . . . . . . . . . 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 234	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)) ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))
45		oveq2 7366	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑤 ·_ℎ 𝐵) → (𝑦 +_ℎ 𝑧) = (𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)))
46	45	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑤 ·_ℎ 𝐵) → (𝑥 = (𝑦 +_ℎ 𝑧) ↔ 𝑥 = (𝑦 +_ℎ (𝑤 ·_ℎ 𝐵))))
47	46	biimpa 476	. . . . . . . . . . 11 ⊢ ((𝑧 = (𝑤 ·_ℎ 𝐵) ∧ 𝑥 = (𝑦 +_ℎ 𝑧)) → 𝑥 = (𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)))
48		eleq1 2824	. . . . . . . . . . . 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 479	. . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) ∧ (𝑧 = (𝑤 ·_ℎ 𝐵) ∧ 𝑥 = (𝑦 +_ℎ 𝑧))) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))
51	50	exp43 436	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → (𝑤 ∈ ℂ → (𝑧 = (𝑤 ·_ℎ 𝐵) → (𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))))))
52	51	rexlimdv 3135	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (∃𝑤 ∈ ℂ 𝑧 = (𝑤 ·_ℎ 𝐵) → (𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))))
53	8, 52	biimtrid 242	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (𝑧 ∈ (span‘{𝐵}) → (𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))))
54	53	rexlimdv 3135	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (∃𝑧 ∈ (span‘{𝐵})𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))))
55	54	rexlimiv 3130	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ (span‘{𝐵})𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))
56	7, 55	sylbi 217	. . . 4 ⊢ (𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))
57	2, 42	shseli 31391	. . . . 5 ⊢ (𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})𝑥 = (𝑦 +_ℎ 𝑧))
58	16	elspansni 31633	. . . . . . . 8 ⊢ (𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}) ↔ ∃𝑤 ∈ ℂ 𝑧 = (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))
59		negcl 11380	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ℂ → -𝑤 ∈ ℂ)
60		shmulcl 31293	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ S_ℋ ∧ -𝑤 ∈ ℂ ∧ ((proj_ℎ‘𝐴)‘𝐵) ∈ 𝐴) → (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ 𝐴)
61	2, 9, 60	mp3an13 1454	. . . . . . . . . . . . . . . . 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 31292	. . . . . . . . . . . . . . . 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 1164	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ S_ℋ ∧ 𝑤 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ 𝑦) ∈ 𝐴)
65	2, 64	mp3an1 1450	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ 𝑦) ∈ 𝐴)
66	65	ancoms 458	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ 𝑦) ∈ 𝐴)
67		spansnmul 31639	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℋ ∧ 𝑤 ∈ ℂ) → (𝑤 ·_ℎ 𝐵) ∈ (span‘{𝐵}))
68	3, 67	mpan 690	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℂ → (𝑤 ·_ℎ 𝐵) ∈ (span‘{𝐵}))
69	68	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑤 ·_ℎ 𝐵) ∈ (span‘{𝐵}))
70		hvm1neg 31107	. . . . . . . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ℂ → (-1 ·_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) = (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)))
72	71	oveq2d 7374	. . . . . . . . . . . . . . . . . 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 31102	. . . . . . . . . . . . . . . . . . 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 31088	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((-𝑤 ∈ ℂ ∧ ((proj_ℎ‘𝐴)‘𝐵) ∈ ℋ) → (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ)
76	59, 22, 75	sylancl 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ℂ → (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ)
77		ax-hvcom 31076	. . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ℂ → 0_ℎ = ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))))
80	79	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → 0_ℎ = ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))))
81	80	oveq1d 7373	. . . . . . . . . . . . . . 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 31087	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℋ ∧ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ ℋ) → (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) ∈ ℋ)
83	28, 32, 82	syl2an 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) ∈ ℋ)
84		hvaddlid 31098	. . . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ℂ → ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ ∧ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ))
87	86	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ ∧ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ))
88	28, 32	anim12i 613	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 ∈ ℋ ∧ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ ℋ))
89		hvadd4 31111	. . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . 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 7374	. . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . 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 7407	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 31391	. . . . . . . . . . . 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 234	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) ∈ (𝐴 +_ℋ (span‘{𝐵})))
98		oveq2 7366	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) → (𝑦 +_ℎ 𝑧) = (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))))
99	98	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) → (𝑥 = (𝑦 +_ℎ 𝑧) ↔ 𝑥 = (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))))
100	99	biimpa 476	. . . . . . . . . . 11 ⊢ ((𝑧 = (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∧ 𝑥 = (𝑦 +_ℎ 𝑧)) → 𝑥 = (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))))
101		eleq1 2824	. . . . . . . . . . . 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 479	. . . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) ∧ (𝑧 = (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∧ 𝑥 = (𝑦 +_ℎ 𝑧))) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})))
104	103	exp43 436	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → (𝑤 ∈ ℂ → (𝑧 = (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) → (𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵}))))))
105	104	rexlimdv 3135	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → (∃𝑤 ∈ ℂ 𝑧 = (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) → (𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})))))
106	58, 105	biimtrid 242	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}) → (𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})))))
107	106	rexlimdv 3135	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (∃𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵}))))
108	107	rexlimiv 3130	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})))
109	57, 108	sylbi 217	. . . 4 ⊢ (𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})))
110	56, 109	impbii 209	. . 3 ⊢ (𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})) ↔ 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))
111	110	eqriv 2733	. 2 ⊢ (𝐴 +_ℋ (span‘{𝐵})) = (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
112	1	chssii 31306	. . . 4 ⊢ 𝐴 ⊆ ℋ
113	3, 4	ax-mp 5	. . . 4 ⊢ {𝐵} ⊆ ℋ
114	112, 113	spanuni 31619	. . 3 ⊢ (span‘(𝐴 ∪ {𝐵})) = ((span‘𝐴) +_ℋ (span‘{𝐵}))
115		spanid 31422	. . . . 5 ⊢ (𝐴 ∈ S_ℋ → (span‘𝐴) = 𝐴)
116	2, 115	ax-mp 5	. . . 4 ⊢ (span‘𝐴) = 𝐴
117	116	oveq1i 7368	. . 3 ⊢ ((span‘𝐴) +_ℋ (span‘{𝐵})) = (𝐴 +_ℋ (span‘{𝐵}))
118	114, 117	eqtri 2759	. 2 ⊢ (span‘(𝐴 ∪ {𝐵})) = (𝐴 +_ℋ (span‘{𝐵}))
119	16, 40	ax-mp 5	. . . 4 ⊢ {((proj_ℎ‘(⊥‘𝐴))‘𝐵)} ⊆ ℋ
120	112, 119	spanuni 31619	. . 3 ⊢ (span‘(𝐴 ∪ {((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) = ((span‘𝐴) +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
121	116	oveq1i 7368	. . 3 ⊢ ((span‘𝐴) +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) = (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
122	120, 121	eqtri 2759	. 2 ⊢ (span‘(𝐴 ∪ {((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) = (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
123	111, 118, 122	3eqtr4i 2769	1 ⊢ (span‘(𝐴 ∪ {𝐵})) = (span‘(𝐴 ∪ {((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∃wrex 3060 ∪ cun 3899 ⊆ wss 3901 {csn 4580 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 1c1 11027 -cneg 11365 ℋchba 30994 +_ℎ cva 30995 ·_ℎ csm 30996 0_ℎc0v 30999 S_ℋ csh 31003 C_ℋ cch 31004 ⊥cort 31005 +_ℋ cph 31006 spancspn 31007 proj_ℎcpjh 31012

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cc 10345 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105 ax-mulf 11106 ax-hilex 31074 ax-hfvadd 31075 ax-hvcom 31076 ax-hvass 31077 ax-hv0cl 31078 ax-hvaddid 31079 ax-hfvmul 31080 ax-hvmulid 31081 ax-hvmulass 31082 ax-hvdistr1 31083 ax-hvdistr2 31084 ax-hvmul0 31085 ax-hfi 31154 ax-his1 31157 ax-his2 31158 ax-his3 31159 ax-his4 31160 ax-hcompl 31277

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-omul 8402 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-acn 9854 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-rlim 15412 df-sum 15610 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-cn 23171 df-cnp 23172 df-lm 23173 df-haus 23259 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cfil 25211 df-cau 25212 df-cmet 25213 df-grpo 30568 df-gid 30569 df-ginv 30570 df-gdiv 30571 df-ablo 30620 df-vc 30634 df-nv 30667 df-va 30670 df-ba 30671 df-sm 30672 df-0v 30673 df-vs 30674 df-nmcv 30675 df-ims 30676 df-dip 30776 df-ssp 30797 df-ph 30888 df-cbn 30938 df-hnorm 31043 df-hba 31044 df-hvsub 31046 df-hlim 31047 df-hcau 31048 df-sh 31282 df-ch 31296 df-oc 31327 df-ch0 31328 df-shs 31383 df-span 31384 df-pjh 31470

This theorem is referenced by: spansnji 31721

W3C validator