spanunsni - Hilbert Space Explorer

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Theorem spanunsni 31650

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 31298	. . . . . 6 ⊢ 𝐴 ∈ S_ℋ
3		spanunsn.2	. . . . . . 7 ⊢ 𝐵 ∈ ℋ
4		snssi 4729	. . . . . . 7 ⊢ (𝐵 ∈ ℋ → {𝐵} ⊆ ℋ)
5		spancl 31407	. . . . . . 7 ⊢ ({𝐵} ⊆ ℋ → (span‘{𝐵}) ∈ S_ℋ )
6	3, 4, 5	mp2b 10	. . . . . 6 ⊢ (span‘{𝐵}) ∈ S_ℋ
7	2, 6	shseli 31387	. . . . 5 ⊢ (𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵})) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ (span‘{𝐵})𝑥 = (𝑦 +_ℎ 𝑧))
8	3	elspansni 31629	. . . . . . . 8 ⊢ (𝑧 ∈ (span‘{𝐵}) ↔ ∃𝑤 ∈ ℂ 𝑧 = (𝑤 ·_ℎ 𝐵))
9	1, 3	pjclii 31492	. . . . . . . . . . . . . . . 16 ⊢ ((proj_ℎ‘𝐴)‘𝐵) ∈ 𝐴
10		shmulcl 31289	. . . . . . . . . . . . . . . 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 31288	. . . . . . . . . . . . . . 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 31378	. . . . . . . . . . . . . . . 16 ⊢ (⊥‘𝐴) ∈ C_ℋ
16	15, 3	pjhclii 31493	. . . . . . . . . . . . . . 15 ⊢ ((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ ℋ
17		spansnmul 31635	. . . . . . . . . . . . . . 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 31495	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 = (((proj_ℎ‘𝐴)‘𝐵) +_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵))
21	20	oveq2i 7378	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ·_ℎ 𝐵) = (𝑤 ·_ℎ (((proj_ℎ‘𝐴)‘𝐵) +_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))
22	1, 3	pjhclii 31493	. . . . . . . . . . . . . . . . . 18 ⊢ ((proj_ℎ‘𝐴)‘𝐵) ∈ ℋ
23		ax-hvdistr1 31079	. . . . . . . . . . . . . . . . . 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 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 7383	. . . . . . . . . . . . . 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 31303	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
29		hvmulcl 31084	. . . . . . . . . . . . . . . . 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 31084	. . . . . . . . . . . . . . . . 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 31073	. . . . . . . . . . . . . . . 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 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 7416	. . . . . . . . . . . . 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 4729	. . . . . . . . . . . . . 14 ⊢ (((proj_ℎ‘(⊥‘𝐴))‘𝐵) ∈ ℋ → {((proj_ℎ‘(⊥‘𝐴))‘𝐵)} ⊆ ℋ)
41		spancl 31407	. . . . . . . . . . . . . 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 31387	. . . . . . . . . . . 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 7375	. . . . . . . . . . . . 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 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 3136	. . . . . . . 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 3136	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (∃𝑧 ∈ (span‘{𝐵})𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))))
55	54	rexlimiv 3131	. . . . 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 31387	. . . . 5 ⊢ (𝑥 ∈ (𝐴 +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})𝑥 = (𝑦 +_ℎ 𝑧))
58	16	elspansni 31629	. . . . . . . 8 ⊢ (𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}) ↔ ∃𝑤 ∈ ℂ 𝑧 = (𝑤 ·_ℎ ((proj_ℎ‘(⊥‘𝐴))‘𝐵)))
59		negcl 11393	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ ℂ → -𝑤 ∈ ℂ)
60		shmulcl 31289	. . . . . . . . . . . . . . . . . 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 31288	. . . . . . . . . . . . . . . 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 31635	. . . . . . . . . . . . . . 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 31103	. . . . . . . . . . . . . . . . . . . 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 7383	. . . . . . . . . . . . . . . . . 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 31098	. . . . . . . . . . . . . . . . . . 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 31084	. . . . . . . . . . . . . . . . . . . 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 31072	. . . . . . . . . . . . . . . . . . 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 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 7382	. . . . . . . . . . . . . . 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 31083	. . . . . . . . . . . . . . . . 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 31094	. . . . . . . . . . . . . . . 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 31107	. . . . . . . . . . . . . . . 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 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 7383	. . . . . . . . . . . . . 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 7416	. . . . . . . . . . . . 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 31387	. . . . . . . . . . . 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 7375	. . . . . . . . . . . . 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 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 3136	. . . . . . . 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 3136	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → (∃𝑧 ∈ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)})𝑥 = (𝑦 +_ℎ 𝑧) → 𝑥 ∈ (𝐴 +_ℋ (span‘{𝐵}))))
108	107	rexlimiv 3131	. . . . 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 31302	. . . 4 ⊢ 𝐴 ⊆ ℋ
113	3, 4	ax-mp 5	. . . 4 ⊢ {𝐵} ⊆ ℋ
114	112, 113	spanuni 31615	. . 3 ⊢ (span‘(𝐴 ∪ {𝐵})) = ((span‘𝐴) +_ℋ (span‘{𝐵}))
115		spanid 31418	. . . . 5 ⊢ (𝐴 ∈ S_ℋ → (span‘𝐴) = 𝐴)
116	2, 115	ax-mp 5	. . . 4 ⊢ (span‘𝐴) = 𝐴
117	116	oveq1i 7377	. . 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 31615	. . 3 ⊢ (span‘(𝐴 ∪ {((proj_ℎ‘(⊥‘𝐴))‘𝐵)})) = ((span‘𝐴) +_ℋ (span‘{((proj_ℎ‘(⊥‘𝐴))‘𝐵)}))
121	116	oveq1i 7377	. . 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 1542 ∈ wcel 2114 ∃wrex 3061 ∪ cun 3887 ⊆ wss 3889 {csn 4567 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 1c1 11039 -cneg 11378 ℋchba 30990 +_ℎ cva 30991 ·_ℎ csm 30992 0_ℎc0v 30995 S_ℋ csh 30999 C_ℋ cch 31000 ⊥cort 31001 +_ℋ cph 31002 spancspn 31003 proj_ℎcpjh 31008

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cc 10357 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118 ax-hilex 31070 ax-hfvadd 31071 ax-hvcom 31072 ax-hvass 31073 ax-hv0cl 31074 ax-hvaddid 31075 ax-hfvmul 31076 ax-hvmulid 31077 ax-hvmulass 31078 ax-hvdistr1 31079 ax-hvdistr2 31080 ax-hvmul0 31081 ax-hfi 31150 ax-his1 31153 ax-his2 31154 ax-his3 31155 ax-his4 31156 ax-hcompl 31273

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 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 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 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 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-acn 9866 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-rlim 15451 df-sum 15649 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-cn 23192 df-cnp 23193 df-lm 23194 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cfil 25222 df-cau 25223 df-cmet 25224 df-grpo 30564 df-gid 30565 df-ginv 30566 df-gdiv 30567 df-ablo 30616 df-vc 30630 df-nv 30663 df-va 30666 df-ba 30667 df-sm 30668 df-0v 30669 df-vs 30670 df-nmcv 30671 df-ims 30672 df-dip 30772 df-ssp 30793 df-ph 30884 df-cbn 30934 df-hnorm 31039 df-hba 31040 df-hvsub 31042 df-hlim 31043 df-hcau 31044 df-sh 31278 df-ch 31292 df-oc 31323 df-ch0 31324 df-shs 31379 df-span 31380 df-pjh 31466

This theorem is referenced by: spansnji 31717

W3C validator