spanunsni - Hilbert Space Explorer

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Theorem spanunsni 31670

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 31318	. . . . . 6 ⊢ 𝐴 ∈ S_ℋ
3		spanunsn.2	. . . . . . 7 ⊢ 𝐵 ∈ ℋ
4		snssi 4752	. . . . . . 7 ⊢ (𝐵 ∈ ℋ → {𝐵} ⊆ ℋ)
5		spancl 31427	. . . . . . 7 ⊢ ({𝐵} ⊆ ℋ → (span‘{𝐵}) ∈ S_ℋ )
6	3, 4, 5	mp2b 10	. . . . . 6 ⊢ (span‘{𝐵}) ∈ S_ℋ
7	2, 6	shseli 31407	. . . . 5 ⊢ (𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ (span‘{𝐵})𝑥 = (𝑦 +_ℎ 𝑧))
8	3	elspansni 31649	. . . . . . . 8 ⊢ (𝑧 ∈ (span‘{𝐵}) ↔ ∃𝑤 ∈ ℂ 𝑧 = (𝑤 ·_ℎ 𝐵))
9	1, 3	pjclii 31512	. . . . . . . . . . . . . . . 16 ⊢ ((proj_ℎ‘𝐴)‘𝐵) ∈ 𝐴
10		shmulcl 31309	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ S_ℋ ∧ 𝑤 ∈ ℂ ∧ ((proj_ℎ‘𝐴)‘𝐵) ∈ 𝐴) → (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ 𝐴)
11	2, 9, 10	mp3an13 1455	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℂ → (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ 𝐴)
12		shaddcl 31308	. . . . . . . . . . . . . . 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 1166	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ S_ℋ ∧ 𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) ∈ 𝐴)
14	2, 13	mp3an1 1451	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) ∈ 𝐴)
15	1	choccli 31398	. . . . . . . . . . . . . . . 16 ⊢ (⊥‘𝐴) ∈ C_ℋ
16	15, 3	pjhclii 31513	. . . . . . . . . . . . . . 15 ⊢ ((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ ℋ
17		spansnmul 31655	. . . . . . . . . . . . . . 15 ⊢ ((((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ ℋ ∧ 𝑤 ∈ ℂ) → (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
18	16, 17	mpan 691	. . . . . . . . . . . . . 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 31515	. . . . . . . . . . . . . . . . . 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 31513	. . . . . . . . . . . . . . . . . 18 ⊢ ((proj_ℎ‘𝐴)‘𝐵) ∈ ℋ
23		ax-hvdistr1 31099	. . . . . . . . . . . . . . . . . 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 1456	. . . . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . . . 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 31323	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
29		hvmulcl 31104	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℂ ∧ ((proj_ℎ‘𝐴)‘𝐵) ∈ ℋ) → (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ)
30	22, 29	mpan2 692	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℂ → (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ)
31		hvmulcl 31104	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ ℂ ∧ ((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ ℋ) → (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ ℋ)
32	16, 31	mpan2 692	. . . . . . . . . . . . . . . 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 31093	. . . . . . . . . . . . . . . 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 1121	. . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → ∃𝑣 ∈ 𝐴 ∃𝑢 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})(𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)) = (𝑣 +_ℎ 𝑢))
40		snssi 4752	. . . . . . . . . . . . . 14 ⊢ (((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ ℋ → {((proj_ℎ‘(⊥‘𝐴))‘𝐵)} ⊆ ℋ)
41		spancl 31427	. . . . . . . . . . . . . 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 31407	. . . . . . . . . . . 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 2748	. . . . . . . . . . . 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 2825	. . . . . . . . . . . 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 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 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 3137	. . . . . . . 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 3137	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (∃𝑧 ∈ (span‘{𝐵})𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))))
55	54	rexlimiv 3132	. . . . 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 31407	. . . . 5 ⊢ (𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})𝑥 = (𝑦 +_ℎ 𝑧))
58	16	elspansni 31649	. . . . . . . 8 ⊢ (𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}) ↔ ∃𝑤 ∈ ℂ 𝑧 = (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))
59		negcl 11382	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ℂ → -𝑤 ∈ ℂ)
60		shmulcl 31309	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ S_ℋ ∧ -𝑤 ∈ ℂ ∧ ((proj_ℎ‘𝐴)‘𝐵) ∈ 𝐴) → (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ 𝐴)
61	2, 9, 60	mp3an13 1455	. . . . . . . . . . . . . . . . 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 31308	. . . . . . . . . . . . . . . 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 1165	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ S_ℋ ∧ 𝑤 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ 𝑦) ∈ 𝐴)
65	2, 64	mp3an1 1451	. . . . . . . . . . . . . 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 31655	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℋ ∧ 𝑤 ∈ ℂ) → (𝑤 ·_ℎ 𝐵) ∈ (span‘{𝐵}))
68	3, 67	mpan 691	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℂ → (𝑤 ·_ℎ 𝐵) ∈ (span‘{𝐵}))
69	68	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑤 ·_ℎ 𝐵) ∈ (span‘{𝐵}))
70		hvm1neg 31123	. . . . . . . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . . . . . . . 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 31118	. . . . . . . . . . . . . . . . . . 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 31104	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((-𝑤 ∈ ℂ ∧ ((proj_ℎ‘𝐴)‘𝐵) ∈ ℋ) → (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ)
76	59, 22, 75	sylancl 587	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ℂ → (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ)
77		ax-hvcom 31092	. . . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . . . 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 31103	. . . . . . . . . . . . . . . . 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		hvaddlid 31114	. . . . . . . . . . . . . . . 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 614	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 ∈ ℋ ∧ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ ℋ))
89		hvadd4 31127	. . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 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 31407	. . . . . . . . . . . 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 2748	. . . . . . . . . . . 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 2825	. . . . . . . . . . . 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 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 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 3137	. . . . . . . 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 3137	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (∃𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵}))))
108	107	rexlimiv 3132	. . . . 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 2734	. 2 ⊢ (𝐴 +_ℋ (span‘{𝐵})) = (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
112	1	chssii 31322	. . . 4 ⊢ 𝐴 ⊆ ℋ
113	3, 4	ax-mp 5	. . . 4 ⊢ {𝐵} ⊆ ℋ
114	112, 113	spanuni 31635	. . 3 ⊢ (span‘(𝐴 ∪ {𝐵})) = ((span‘𝐴) +_ℋ (span‘{𝐵}))
115		spanid 31438	. . . . 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 2760	. 2 ⊢ (span‘(𝐴 ∪ {𝐵})) = (𝐴 +_ℋ (span‘{𝐵}))
119	16, 40	ax-mp 5	. . . 4 ⊢ {((proj_ℎ‘(⊥‘𝐴))‘𝐵)} ⊆ ℋ
120	112, 119	spanuni 31635	. . 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 2760	. 2 ⊢ (span‘(𝐴 ∪ {((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) = (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
123	111, 118, 122	3eqtr4i 2770	1 ⊢ (span‘(𝐴 ∪ {𝐵})) = (span‘(𝐴 ∪ {((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∃wrex 3062 ∪ cun 3888 ⊆ wss 3890 {csn 4568 ‘cfv 6490 (class class class)co 7358 ℂcc 11025 1c1 11028 -cneg 11367 ℋchba 31010 +_ℎ cva 31011 ·_ℎ csm 31012 0_ℎc0v 31015 S_ℋ csh 31019 C_ℋ cch 31020 ⊥cort 31021 +_ℋ cph 31022 spancspn 31023 proj_ℎcpjh 31028

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-inf2 9551 ax-cc 10346 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106 ax-mulf 11107 ax-hilex 31090 ax-hfvadd 31091 ax-hvcom 31092 ax-hvass 31093 ax-hv0cl 31094 ax-hvaddid 31095 ax-hfvmul 31096 ax-hvmulid 31097 ax-hvmulass 31098 ax-hvdistr1 31099 ax-hvdistr2 31100 ax-hvmul0 31101 ax-hfi 31170 ax-his1 31173 ax-his2 31174 ax-his3 31175 ax-his4 31176 ax-hcompl 31293

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 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 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-oadd 8400 df-omul 8401 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-card 9852 df-acn 9855 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-q 12888 df-rp 12932 df-xneg 13052 df-xadd 13053 df-xmul 13054 df-ioo 13291 df-ico 13293 df-icc 13294 df-fz 13451 df-fzo 13598 df-fl 13740 df-seq 13953 df-exp 14013 df-hash 14282 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15439 df-rlim 15440 df-sum 15638 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-starv 17224 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-unif 17232 df-hom 17233 df-cco 17234 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18741 df-mulg 19033 df-cntz 19281 df-cmn 19746 df-psmet 21334 df-xmet 21335 df-met 21336 df-bl 21337 df-mopn 21338 df-fbas 21339 df-fg 21340 df-cnfld 21343 df-top 22868 df-topon 22885 df-topsp 22907 df-bases 22920 df-cld 22993 df-ntr 22994 df-cls 22995 df-nei 23072 df-cn 23201 df-cnp 23202 df-lm 23203 df-haus 23289 df-tx 23536 df-hmeo 23729 df-fil 23820 df-fm 23912 df-flim 23913 df-flf 23914 df-xms 24294 df-ms 24295 df-tms 24296 df-cfil 25231 df-cau 25232 df-cmet 25233 df-grpo 30584 df-gid 30585 df-ginv 30586 df-gdiv 30587 df-ablo 30636 df-vc 30650 df-nv 30683 df-va 30686 df-ba 30687 df-sm 30688 df-0v 30689 df-vs 30690 df-nmcv 30691 df-ims 30692 df-dip 30792 df-ssp 30813 df-ph 30904 df-cbn 30954 df-hnorm 31059 df-hba 31060 df-hvsub 31062 df-hlim 31063 df-hcau 31064 df-sh 31298 df-ch 31312 df-oc 31343 df-ch0 31344 df-shs 31399 df-span 31400 df-pjh 31486

This theorem is referenced by: spansnji 31737

W3C validator