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Theorem siilem2 30832

Description: Lemma for sii 30834. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
siii.1	⊢ 𝑋 = (BaseSet‘𝑈)
siii.6	⊢ 𝑁 = (norm_CV‘𝑈)
siii.7	⊢ 𝑃 = (·_𝑖OLD‘𝑈)
siii.9	⊢ 𝑈 ∈ CPreHil_OLD
siii.a	⊢ 𝐴 ∈ 𝑋
siii.b	⊢ 𝐵 ∈ 𝑋
siii2.3	⊢ 𝑀 = ( −_𝑣 ‘𝑈)
siii2.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)

**Assertion**
Ref	Expression
siilem2	⊢ ((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))) → ((𝐵𝑃𝐴) = (𝐶 · ((𝑁‘𝐵)↑2)) → (√‘((𝐴𝑃𝐵) · (𝐶 · ((𝑁‘𝐵)↑2)))) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵))))

**Proof of Theorem siilem2**
Step	Hyp	Ref	Expression
1		oveq1 7353	. . . 4 ⊢ (𝐶 = if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) → (𝐶 · ((𝑁‘𝐵)↑2)) = (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · ((𝑁‘𝐵)↑2)))
2	1	eqeq2d 2742	. . 3 ⊢ (𝐶 = if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) → ((𝐵𝑃𝐴) = (𝐶 · ((𝑁‘𝐵)↑2)) ↔ (𝐵𝑃𝐴) = (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · ((𝑁‘𝐵)↑2))))
3	1	oveq2d 7362	. . . . 5 ⊢ (𝐶 = if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) → ((𝐴𝑃𝐵) · (𝐶 · ((𝑁‘𝐵)↑2))) = ((𝐴𝑃𝐵) · (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · ((𝑁‘𝐵)↑2))))
4	3	fveq2d 6826	. . . 4 ⊢ (𝐶 = if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) → (√‘((𝐴𝑃𝐵) · (𝐶 · ((𝑁‘𝐵)↑2)))) = (√‘((𝐴𝑃𝐵) · (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · ((𝑁‘𝐵)↑2)))))
5	4	breq1d 5099	. . 3 ⊢ (𝐶 = if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) → ((√‘((𝐴𝑃𝐵) · (𝐶 · ((𝑁‘𝐵)↑2)))) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵)) ↔ (√‘((𝐴𝑃𝐵) · (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · ((𝑁‘𝐵)↑2)))) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵))))
6	2, 5	imbi12d 344	. 2 ⊢ (𝐶 = if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) → (((𝐵𝑃𝐴) = (𝐶 · ((𝑁‘𝐵)↑2)) → (√‘((𝐴𝑃𝐵) · (𝐶 · ((𝑁‘𝐵)↑2)))) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵))) ↔ ((𝐵𝑃𝐴) = (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · ((𝑁‘𝐵)↑2)) → (√‘((𝐴𝑃𝐵) · (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · ((𝑁‘𝐵)↑2)))) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵)))))
7		siii.1	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
8		siii.6	. . 3 ⊢ 𝑁 = (norm_CV‘𝑈)
9		siii.7	. . 3 ⊢ 𝑃 = (·_𝑖OLD‘𝑈)
10		siii.9	. . 3 ⊢ 𝑈 ∈ CPreHil_OLD
11		siii.a	. . 3 ⊢ 𝐴 ∈ 𝑋
12		siii.b	. . 3 ⊢ 𝐵 ∈ 𝑋
13		siii2.3	. . 3 ⊢ 𝑀 = ( −_𝑣 ‘𝑈)
14		siii2.4	. . 3 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
15		eleq1 2819	. . . . . 6 ⊢ (𝐶 = if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) → (𝐶 ∈ ℂ ↔ if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) ∈ ℂ))
16		oveq1 7353	. . . . . . 7 ⊢ (𝐶 = if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) → (𝐶 · (𝐴𝑃𝐵)) = (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · (𝐴𝑃𝐵)))
17	16	eleq1d 2816	. . . . . 6 ⊢ (𝐶 = if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) → ((𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ↔ (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · (𝐴𝑃𝐵)) ∈ ℝ))
18	16	breq2d 5101	. . . . . 6 ⊢ (𝐶 = if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) → (0 ≤ (𝐶 · (𝐴𝑃𝐵)) ↔ 0 ≤ (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · (𝐴𝑃𝐵))))
19	15, 17, 18	3anbi123d 1438	. . . . 5 ⊢ (𝐶 = if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) → ((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))) ↔ (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) ∈ ℂ ∧ (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · (𝐴𝑃𝐵)))))
20		eleq1 2819	. . . . . 6 ⊢ (0 = if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) → (0 ∈ ℂ ↔ if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) ∈ ℂ))
21		oveq1 7353	. . . . . . 7 ⊢ (0 = if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) → (0 · (𝐴𝑃𝐵)) = (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · (𝐴𝑃𝐵)))
22	21	eleq1d 2816	. . . . . 6 ⊢ (0 = if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) → ((0 · (𝐴𝑃𝐵)) ∈ ℝ ↔ (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · (𝐴𝑃𝐵)) ∈ ℝ))
23	21	breq2d 5101	. . . . . 6 ⊢ (0 = if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) → (0 ≤ (0 · (𝐴𝑃𝐵)) ↔ 0 ≤ (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · (𝐴𝑃𝐵))))
24	20, 22, 23	3anbi123d 1438	. . . . 5 ⊢ (0 = if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) → ((0 ∈ ℂ ∧ (0 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (0 · (𝐴𝑃𝐵))) ↔ (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) ∈ ℂ ∧ (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · (𝐴𝑃𝐵)))))
25		0cn 11104	. . . . . 6 ⊢ 0 ∈ ℂ
26	10	phnvi 30796	. . . . . . . . 9 ⊢ 𝑈 ∈ NrmCVec
27	7, 9	dipcl 30692	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ)
28	26, 11, 12, 27	mp3an 1463	. . . . . . . 8 ⊢ (𝐴𝑃𝐵) ∈ ℂ
29	28	mul02i 11302	. . . . . . 7 ⊢ (0 · (𝐴𝑃𝐵)) = 0
30		0re 11114	. . . . . . 7 ⊢ 0 ∈ ℝ
31	29, 30	eqeltri 2827	. . . . . 6 ⊢ (0 · (𝐴𝑃𝐵)) ∈ ℝ
32		0le0 12226	. . . . . . 7 ⊢ 0 ≤ 0
33	32, 29	breqtrri 5116	. . . . . 6 ⊢ 0 ≤ (0 · (𝐴𝑃𝐵))
34	25, 31, 33	3pm3.2i 1340	. . . . 5 ⊢ (0 ∈ ℂ ∧ (0 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (0 · (𝐴𝑃𝐵)))
35	19, 24, 34	elimhyp 4538	. . . 4 ⊢ (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) ∈ ℂ ∧ (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · (𝐴𝑃𝐵)))
36	35	simp1i 1139	. . 3 ⊢ if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) ∈ ℂ
37	35	simp2i 1140	. . 3 ⊢ (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · (𝐴𝑃𝐵)) ∈ ℝ
38	35	simp3i 1141	. . 3 ⊢ 0 ≤ (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · (𝐴𝑃𝐵))
39	7, 8, 9, 10, 11, 12, 13, 14, 36, 37, 38	siilem1 30831	. 2 ⊢ ((𝐵𝑃𝐴) = (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · ((𝑁‘𝐵)↑2)) → (√‘((𝐴𝑃𝐵) · (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · ((𝑁‘𝐵)↑2)))) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵)))
40	6, 39	dedth 4531	1 ⊢ ((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))) → ((𝐵𝑃𝐴) = (𝐶 · ((𝑁‘𝐵)↑2)) → (√‘((𝐴𝑃𝐵) · (𝐶 · ((𝑁‘𝐵)↑2)))) ≤ ((𝑁‘𝐴) · (𝑁‘𝐵))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ifcif 4472 class class class wbr 5089 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 ℝcr 11005 0cc0 11006 · cmul 11011 ≤ cle 11147 2c2 12180 ↑cexp 13968 √csqrt 15140 NrmCVeccnv 30564 BaseSetcba 30566 ·_𝑠OLD cns 30567 −_𝑣 cnsb 30569 norm_CVcnmcv 30570 ·_𝑖OLDcdip 30680 CPreHil_OLDccphlo 30792

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085 ax-mulf 11086

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-icc 13252 df-fz 13408 df-fzo 13555 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936 df-cn 23142 df-cnp 23143 df-t1 23229 df-haus 23230 df-tx 23477 df-hmeo 23670 df-xms 24235 df-ms 24236 df-tms 24237 df-grpo 30473 df-gid 30474 df-ginv 30475 df-gdiv 30476 df-ablo 30525 df-vc 30539 df-nv 30572 df-va 30575 df-ba 30576 df-sm 30577 df-0v 30578 df-vs 30579 df-nmcv 30580 df-ims 30581 df-dip 30681 df-ph 30793

This theorem is referenced by: siii 30833

W3C validator