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Theorem siilem2 30796

Description: Lemma for sii 30798. (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 7356	. . . 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 2740	. . 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 7365	. . . . 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 5102	. . 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 2816	. . . . . 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 7356	. . . . . . 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 2813	. . . . . 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 5104	. . . . . 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 2816	. . . . . 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 7356	. . . . . . 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 2813	. . . . . 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 5104	. . . . . 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 11107	. . . . . 6 ⊢ 0 ∈ ℂ
26	10	phnvi 30760	. . . . . . . . 9 ⊢ 𝑈 ∈ NrmCVec
27	7, 9	dipcl 30656	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ)
28	26, 11, 12, 27	mp3an 1463	. . . . . . . 8 ⊢ (𝐴𝑃𝐵) ∈ ℂ
29	28	mul02i 11305	. . . . . . 7 ⊢ (0 · (𝐴𝑃𝐵)) = 0
30		0re 11117	. . . . . . 7 ⊢ 0 ∈ ℝ
31	29, 30	eqeltri 2824	. . . . . 6 ⊢ (0 · (𝐴𝑃𝐵)) ∈ ℝ
32		0le0 12229	. . . . . . 7 ⊢ 0 ≤ 0
33	32, 29	breqtrri 5119	. . . . . 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 4542	. . . 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 30795	. 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 4535	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 1540 ∈ wcel 2109 ifcif 4476 class class class wbr 5092 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 ℝcr 11008 0cc0 11009 · cmul 11014 ≤ cle 11150 2c2 12183 ↑cexp 13968 √csqrt 15140 NrmCVeccnv 30528 BaseSetcba 30530 ·_𝑠OLD cns 30531 −_𝑣 cnsb 30533 norm_CVcnmcv 30534 ·_𝑖OLDcdip 30644 CPreHil_OLDccphlo 30756

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-icc 13255 df-fz 13411 df-fzo 13558 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 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-cn 23112 df-cnp 23113 df-t1 23199 df-haus 23200 df-tx 23447 df-hmeo 23640 df-xms 24206 df-ms 24207 df-tms 24208 df-grpo 30437 df-gid 30438 df-ginv 30439 df-gdiv 30440 df-ablo 30489 df-vc 30503 df-nv 30536 df-va 30539 df-ba 30540 df-sm 30541 df-0v 30542 df-vs 30543 df-nmcv 30544 df-ims 30545 df-dip 30645 df-ph 30757

This theorem is referenced by: siii 30797

W3C validator