spanunsni - Hilbert Space Explorer

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Theorem spanunsni 30521

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 30169	. . . . . 6 ⊢ 𝐴 ∈ S_ℋ
3		spanunsn.2	. . . . . . 7 ⊢ 𝐵 ∈ ℋ
4		snssi 4768	. . . . . . 7 ⊢ (𝐵 ∈ ℋ → {𝐵} ⊆ ℋ)
5		spancl 30278	. . . . . . 7 ⊢ ({𝐵} ⊆ ℋ → (span‘{𝐵}) ∈ S_ℋ )
6	3, 4, 5	mp2b 10	. . . . . 6 ⊢ (span‘{𝐵}) ∈ S_ℋ
7	2, 6	shseli 30258	. . . . 5 ⊢ (𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ (span‘{𝐵})𝑥 = (𝑦 +_ℎ 𝑧))
8	3	elspansni 30500	. . . . . . . 8 ⊢ (𝑧 ∈ (span‘{𝐵}) ↔ ∃𝑤 ∈ ℂ 𝑧 = (𝑤 ·_ℎ 𝐵))
9	1, 3	pjclii 30363	. . . . . . . . . . . . . . . 16 ⊢ ((proj_ℎ‘𝐴)‘𝐵) ∈ 𝐴
10		shmulcl 30160	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ S_ℋ ∧ 𝑤 ∈ ℂ ∧ ((proj_ℎ‘𝐴)‘𝐵) ∈ 𝐴) → (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ 𝐴)
11	2, 9, 10	mp3an13 1452	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℂ → (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ 𝐴)
12		shaddcl 30159	. . . . . . . . . . . . . . 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 1448	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))) ∈ 𝐴)
15	1	choccli 30249	. . . . . . . . . . . . . . . 16 ⊢ (⊥‘𝐴) ∈ C_ℋ
16	15, 3	pjhclii 30364	. . . . . . . . . . . . . . 15 ⊢ ((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ ℋ
17		spansnmul 30506	. . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
20	1, 3	pjpji 30366	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 = (((proj_ℎ‘𝐴)‘𝐵) +_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))
21	20	oveq2i 7368	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ·_ℎ 𝐵) = (𝑤 ·_ℎ (((proj_ℎ‘𝐴)‘𝐵) +_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))
22	1, 3	pjhclii 30364	. . . . . . . . . . . . . . . . . 18 ⊢ ((proj_ℎ‘𝐴)‘𝐵) ∈ ℋ
23		ax-hvdistr1 29950	. . . . . . . . . . . . . . . . . 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 1453	. . . . . . . . . . . . . . . . 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 2788	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ℂ → (𝑤 ·_ℎ 𝐵) = ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))))
26	25	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑤 ·_ℎ 𝐵) = ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))))
27	26	oveq2d 7373	. . . . . . . . . . . . . 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 30174	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
29		hvmulcl 29955	. . . . . . . . . . . . . . . . 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 29955	. . . . . . . . . . . . . . . . 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 512	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℂ → ((𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ ∧ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) ∈ ℋ))
34		ax-hvass 29944	. . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . 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 7404	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → ∃𝑣 ∈ 𝐴 ∃𝑢 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})(𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)) = (𝑣 +_ℎ 𝑢))
40		snssi 4768	. . . . . . . . . . . . . 14 ⊢ (((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ ℋ → {((proj_ℎ‘(⊥‘𝐴))‘𝐵)} ⊆ ℋ)
41		spancl 30278	. . . . . . . . . . . . . 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 30258	. . . . . . . . . . . 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 233	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 +_ℎ (𝑤 ·_ℎ 𝐵)) ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))
45		oveq2 7365	. . . . . . . . . . . . 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 477	. . . . . . . . . . 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 480	. . . . . . . . . . 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 437	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → (𝑤 ∈ ℂ → (𝑧 = (𝑤 ·_ℎ 𝐵) → (𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))))))
52	51	rexlimdv 3150	. . . . . . . 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 241	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (𝑧 ∈ (span‘{𝐵}) → (𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))))
54	53	rexlimdv 3150	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (∃𝑧 ∈ (span‘{𝐵})𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))))
55	54	rexlimiv 3145	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ (span‘{𝐵})𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))
56	7, 55	sylbi 216	. . . 4 ⊢ (𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})))
57	2, 42	shseli 30258	. . . . 5 ⊢ (𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})𝑥 = (𝑦 +_ℎ 𝑧))
58	16	elspansni 30500	. . . . . . . 8 ⊢ (𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}) ↔ ∃𝑤 ∈ ℂ 𝑧 = (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))
59		negcl 11401	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ℂ → -𝑤 ∈ ℂ)
60		shmulcl 30160	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ S_ℋ ∧ -𝑤 ∈ ℂ ∧ ((proj_ℎ‘𝐴)‘𝐵) ∈ 𝐴) → (-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ 𝐴)
61	2, 9, 60	mp3an13 1452	. . . . . . . . . . . . . . . . 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 30159	. . . . . . . . . . . . . . . 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 1448	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ ℂ ∧ 𝑦 ∈ 𝐴) → ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ 𝑦) ∈ 𝐴)
66	65	ancoms 459	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ 𝑦) ∈ 𝐴)
67		spansnmul 30506	. . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑤 ·_ℎ 𝐵) ∈ (span‘{𝐵}))
70		hvm1neg 29974	. . . . . . . . . . . . . . . . . . . 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 7373	. . . . . . . . . . . . . . . . . 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 29969	. . . . . . . . . . . . . . . . . . 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 29955	. . . . . . . . . . . . . . . . . . . 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 29943	. . . . . . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ℂ → 0_ℎ = ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))))
80	79	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → 0_ℎ = ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵))))
81	80	oveq1d 7372	. . . . . . . . . . . . . . 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 29954	. . . . . . . . . . . . . . . . 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		hvaddid2 29965	. . . . . . . . . . . . . . . 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 512	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ℂ → ((-𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ ∧ (𝑤 ·_ℎ ((proj_ℎ‘𝐴)‘𝐵)) ∈ ℋ))
87	86	adantl 482	. . . . . . . . . . . . . . . 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 29978	. . . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . 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 7373	. . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . 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 7404	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 30258	. . . . . . . . . . . 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 233	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑤 ∈ ℂ) → (𝑦 +_ℎ (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))) ∈ (𝐴 +_ℋ (span‘{𝐵})))
98		oveq2 7365	. . . . . . . . . . . . 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 477	. . . . . . . . . . 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 480	. . . . . . . . . . 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 437	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → (𝑤 ∈ ℂ → (𝑧 = (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)) → (𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵}))))))
105	104	rexlimdv 3150	. . . . . . . 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 241	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}) → (𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})))))
107	106	rexlimdv 3150	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (∃𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵}))))
108	107	rexlimiv 3145	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})))
109	57, 108	sylbi 216	. . . 4 ⊢ (𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})))
110	56, 109	impbii 208	. . 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 30173	. . . 4 ⊢ 𝐴 ⊆ ℋ
113	3, 4	ax-mp 5	. . . 4 ⊢ {𝐵} ⊆ ℋ
114	112, 113	spanuni 30486	. . 3 ⊢ (span‘(𝐴 ∪ {𝐵})) = ((span‘𝐴) +_ℋ (span‘{𝐵}))
115		spanid 30289	. . . . 5 ⊢ (𝐴 ∈ S_ℋ → (span‘𝐴) = 𝐴)
116	2, 115	ax-mp 5	. . . 4 ⊢ (span‘𝐴) = 𝐴
117	116	oveq1i 7367	. . 3 ⊢ ((span‘𝐴) +_ℋ (span‘{𝐵})) = (𝐴 +_ℋ (span‘{𝐵}))
118	114, 117	eqtri 2764	. 2 ⊢ (span‘(𝐴 ∪ {𝐵})) = (𝐴 +_ℋ (span‘{𝐵}))
119	16, 40	ax-mp 5	. . . 4 ⊢ {((proj_ℎ‘(⊥‘𝐴))‘𝐵)} ⊆ ℋ
120	112, 119	spanuni 30486	. . 3 ⊢ (span‘(𝐴 ∪ {((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) = ((span‘𝐴) +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
121	116	oveq1i 7367	. . 3 ⊢ ((span‘𝐴) +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) = (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
122	120, 121	eqtri 2764	. 2 ⊢ (span‘(𝐴 ∪ {((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) = (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
123	111, 118, 122	3eqtr4i 2774	1 ⊢ (span‘(𝐴 ∪ {𝐵})) = (span‘(𝐴 ∪ {((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3073 ∪ cun 3908 ⊆ wss 3910 {csn 4586 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 1c1 11052 -cneg 11386 ℋchba 29861 +_ℎ cva 29862 ·_ℎ csm 29863 0_ℎc0v 29866 S_ℋ csh 29870 C_ℋ cch 29871 ⊥cort 29872 +_ℋ cph 29873 spancspn 29874 proj_ℎcpjh 29879

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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cc 10371 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 ax-hilex 29941 ax-hfvadd 29942 ax-hvcom 29943 ax-hvass 29944 ax-hv0cl 29945 ax-hvaddid 29946 ax-hfvmul 29947 ax-hvmulid 29948 ax-hvmulass 29949 ax-hvdistr1 29950 ax-hvdistr2 29951 ax-hvmul0 29952 ax-hfi 30021 ax-his1 30024 ax-his2 30025 ax-his3 30026 ax-his4 30027 ax-hcompl 30144

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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-omul 8417 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-acn 9878 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-rlim 15371 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-cn 22578 df-cnp 22579 df-lm 22580 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cfil 24619 df-cau 24620 df-cmet 24621 df-grpo 29435 df-gid 29436 df-ginv 29437 df-gdiv 29438 df-ablo 29487 df-vc 29501 df-nv 29534 df-va 29537 df-ba 29538 df-sm 29539 df-0v 29540 df-vs 29541 df-nmcv 29542 df-ims 29543 df-dip 29643 df-ssp 29664 df-ph 29755 df-cbn 29805 df-hnorm 29910 df-hba 29911 df-hvsub 29913 df-hlim 29914 df-hcau 29915 df-sh 30149 df-ch 30163 df-oc 30194 df-ch0 30195 df-shs 30250 df-span 30251 df-pjh 30337

This theorem is referenced by: spansnji 30588

W3C validator