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Theorem siilem2 31277

Description: Lemma for sii 31279. (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 7426	. . . 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 2776	. . 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 7435	. . . . 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 6889	. . . 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 5121	. . 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 347	. 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 2853	. . . . . 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 7426	. . . . . . 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 2850	. . . . . 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 5123	. . . . . 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 1464	. . . . 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 2853	. . . . . 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 7426	. . . . . . 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 2850	. . . . . 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 5123	. . . . . 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 1464	. . . . 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 11215	. . . . . 6 ⊢ 0 ∈ ℂ
26	10	phnvi 31241	. . . . . . . . 9 ⊢ 𝑈 ∈ NrmCVec
27	7, 9	dipcl 31137	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ)
28	26, 11, 12, 27	mp3an 1490	. . . . . . . 8 ⊢ (𝐴𝑃𝐵) ∈ ℂ
29	28	mul02i 11416	. . . . . . 7 ⊢ (0 · (𝐴𝑃𝐵)) = 0
30		0re 11227	. . . . . . 7 ⊢ 0 ∈ ℝ
31	29, 30	eqeltri 2861	. . . . . 6 ⊢ (0 · (𝐴𝑃𝐵)) ∈ ℝ
32		0le0 12359	. . . . . . 7 ⊢ 0 ≤ 0
33	32, 29	breqtrri 5140	. . . . . 6 ⊢ 0 ≤ (0 · (𝐴𝑃𝐵))
34	25, 31, 33	3pm3.2i 1358	. . . . 5 ⊢ (0 ∈ ℂ ∧ (0 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (0 · (𝐴𝑃𝐵)))
35	19, 24, 34	elimhyp 4555	. . . 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 1157	. . 3 ⊢ if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) ∈ ℂ
37	35	simp2i 1158	. . 3 ⊢ (if((𝐶 ∈ ℂ ∧ (𝐶 · (𝐴𝑃𝐵)) ∈ ℝ ∧ 0 ≤ (𝐶 · (𝐴𝑃𝐵))), 𝐶, 0) · (𝐴𝑃𝐵)) ∈ ℝ
38	35	simp3i 1159	. . 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 31276	. 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 4548	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

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ifcif 4489 class class class wbr 5111 ‘cfv 6540 (class class class)co 7419 ℂcc 11115 ℝcr 11116 0cc0 11117 · cmul 11122 ≤ cle 11261 2c2 12312 ↑cexp 14117 √csqrt 15310 NrmCVeccnv 31009 BaseSetcba 31011 ·_𝑠OLD cns 31012 −_𝑣 cnsb 31014 norm_CVcnmcv 31015 ·_𝑖OLDcdip 31125 CPreHil_OLDccphlo 31237

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-pre-sup 11195 ax-addf 11196 ax-mulf 11197

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-div 11889 df-nn 12251 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12522 df-z 12609 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13394 df-icc 13397 df-fz 13554 df-fzo 13702 df-seq 14058 df-exp 14118 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-clim 15565 df-sum 15764 df-struct 17231 df-sets 17248 df-slot 17266 df-ndx 17278 df-base 17294 df-ress 17315 df-plusg 17347 df-mulr 17348 df-starv 17349 df-sca 17350 df-vsca 17351 df-ip 17352 df-tset 17353 df-ple 17354 df-ds 17356 df-unif 17357 df-hom 17358 df-cco 17359 df-rest 17499 df-topn 17500 df-0g 17518 df-gsum 17519 df-topgen 17520 df-pt 17521 df-prds 17524 df-xrs 17580 df-qtop 17585 df-imas 17586 df-xps 17588 df-mre 17662 df-mrc 17663 df-acs 17665 df-mgm 18722 df-sgrp 18811 df-mnd 18827 df-submnd 18881 df-mulg 19180 df-cntz 19433 df-cmn 19898 df-psmet 21566 df-xmet 21567 df-met 21568 df-bl 21569 df-mopn 21570 df-cnfld 21575 df-top 23103 df-topon 23120 df-topsp 23142 df-bases 23155 df-cld 23228 df-ntr 23229 df-cls 23230 df-cn 23436 df-cnp 23437 df-t1 23523 df-haus 23524 df-tx 23772 df-hmeo 23965 df-xms 24530 df-ms 24531 df-tms 24532 df-grpo 30918 df-gid 30919 df-ginv 30920 df-gdiv 30921 df-ablo 30970 df-vc 30984 df-nv 31017 df-va 31020 df-ba 31021 df-sm 31022 df-0v 31023 df-vs 31024 df-nmcv 31025 df-ims 31026 df-dip 31126 df-ph 31238

This theorem is used by: siii 31278

W3C validator