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Theorem siilem2 30783

Description: Lemma for sii 30785. (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 7347	. . . 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 7356	. . . . 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 6820	. . . 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 5098	. . 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 7347	. . . . . . 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 5100	. . . . . 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 7347	. . . . . . 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 5100	. . . . . 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 11095	. . . . . 6 ⊢ 0 ∈ ℂ
26	10	phnvi 30747	. . . . . . . . 9 ⊢ 𝑈 ∈ NrmCVec
27	7, 9	dipcl 30643	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ)
28	26, 11, 12, 27	mp3an 1463	. . . . . . . 8 ⊢ (𝐴𝑃𝐵) ∈ ℂ
29	28	mul02i 11293	. . . . . . 7 ⊢ (0 · (𝐴𝑃𝐵)) = 0
30		0re 11105	. . . . . . 7 ⊢ 0 ∈ ℝ
31	29, 30	eqeltri 2824	. . . . . 6 ⊢ (0 · (𝐴𝑃𝐵)) ∈ ℝ
32		0le0 12217	. . . . . . 7 ⊢ 0 ≤ 0
33	32, 29	breqtrri 5115	. . . . . 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 30782	. 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 1540 ∈ wcel 2109 ifcif 4472 class class class wbr 5088 ‘cfv 6476 (class class class)co 7340 ℂcc 10995 ℝcr 10996 0cc0 10997 · cmul 11002 ≤ cle 11138 2c2 12171 ↑cexp 13956 √csqrt 15127 NrmCVeccnv 30515 BaseSetcba 30517 ·_𝑠OLD cns 30518 −_𝑣 cnsb 30520 norm_CVcnmcv 30521 ·_𝑖OLDcdip 30631 CPreHil_OLDccphlo 30743

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 5214 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5367 ax-un 7662 ax-inf2 9525 ax-cnex 11053 ax-resscn 11054 ax-1cn 11055 ax-icn 11056 ax-addcl 11057 ax-addrcl 11058 ax-mulcl 11059 ax-mulrcl 11060 ax-mulcom 11061 ax-addass 11062 ax-mulass 11063 ax-distr 11064 ax-i2m1 11065 ax-1ne0 11066 ax-1rid 11067 ax-rnegex 11068 ax-rrecex 11069 ax-cnre 11070 ax-pre-lttri 11071 ax-pre-lttrn 11072 ax-pre-ltadd 11073 ax-pre-mulgt0 11074 ax-pre-sup 11075 ax-addf 11076 ax-mulf 11077

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 3393 df-v 3435 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 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 4895 df-iun 4940 df-iin 4941 df-br 5089 df-opab 5151 df-mpt 5170 df-tr 5196 df-id 5508 df-eprel 5513 df-po 5521 df-so 5522 df-fr 5566 df-se 5567 df-we 5568 df-xp 5619 df-rel 5620 df-cnv 5621 df-co 5622 df-dm 5623 df-rn 5624 df-res 5625 df-ima 5626 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7297 df-ov 7343 df-oprab 7344 df-mpo 7345 df-of 7604 df-om 7791 df-1st 7915 df-2nd 7916 df-supp 8085 df-frecs 8205 df-wrecs 8236 df-recs 8285 df-rdg 8323 df-1o 8379 df-2o 8380 df-er 8616 df-map 8746 df-ixp 8816 df-en 8864 df-dom 8865 df-sdom 8866 df-fin 8867 df-fsupp 9240 df-fi 9289 df-sup 9320 df-inf 9321 df-oi 9390 df-card 9823 df-pnf 11139 df-mnf 11140 df-xr 11141 df-ltxr 11142 df-le 11143 df-sub 11337 df-neg 11338 df-div 11766 df-nn 12117 df-2 12179 df-3 12180 df-4 12181 df-5 12182 df-6 12183 df-7 12184 df-8 12185 df-9 12186 df-n0 12373 df-z 12460 df-dec 12580 df-uz 12724 df-q 12838 df-rp 12882 df-xneg 13002 df-xadd 13003 df-xmul 13004 df-ioo 13240 df-icc 13243 df-fz 13399 df-fzo 13546 df-seq 13897 df-exp 13957 df-hash 14226 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-clim 15382 df-sum 15581 df-struct 17045 df-sets 17062 df-slot 17080 df-ndx 17092 df-base 17108 df-ress 17129 df-plusg 17161 df-mulr 17162 df-starv 17163 df-sca 17164 df-vsca 17165 df-ip 17166 df-tset 17167 df-ple 17168 df-ds 17170 df-unif 17171 df-hom 17172 df-cco 17173 df-rest 17313 df-topn 17314 df-0g 17332 df-gsum 17333 df-topgen 17334 df-pt 17335 df-prds 17338 df-xrs 17393 df-qtop 17398 df-imas 17399 df-xps 17401 df-mre 17475 df-mrc 17476 df-acs 17478 df-mgm 18501 df-sgrp 18580 df-mnd 18596 df-submnd 18645 df-mulg 18934 df-cntz 19183 df-cmn 19648 df-psmet 21237 df-xmet 21238 df-met 21239 df-bl 21240 df-mopn 21241 df-cnfld 21246 df-top 22763 df-topon 22780 df-topsp 22802 df-bases 22815 df-cld 22888 df-ntr 22889 df-cls 22890 df-cn 23096 df-cnp 23097 df-t1 23183 df-haus 23184 df-tx 23431 df-hmeo 23624 df-xms 24189 df-ms 24190 df-tms 24191 df-grpo 30424 df-gid 30425 df-ginv 30426 df-gdiv 30427 df-ablo 30476 df-vc 30490 df-nv 30523 df-va 30526 df-ba 30527 df-sm 30528 df-0v 30529 df-vs 30530 df-nmcv 30531 df-ims 30532 df-dip 30632 df-ph 30744

This theorem is referenced by: siii 30784

W3C validator