pmeasmono - Mathbox for Thierry Arnoux

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Theorem pmeasmono 34462

Description: This theorem's hypotheses define a pre-measure. A pre-measure is monotone. (Contributed by Thierry Arnoux, 19-Jul-2020.)

**Hypotheses**
Ref	Expression
caraext.1	⊢ (𝜑 → 𝑃:𝑅⟶(0[,]+∞))
caraext.2	⊢ (𝜑 → (𝑃‘∅) = 0)
caraext.3	⊢ ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑃‘∪ 𝑥) = Σ^*𝑦 ∈ 𝑥(𝑃‘𝑦))
pmeasmono.1	⊢ (𝜑 → 𝐴 ∈ 𝑅)
pmeasmono.2	⊢ (𝜑 → 𝐵 ∈ 𝑅)
pmeasmono.3	⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑅)
pmeasmono.4	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

**Assertion**
Ref	Expression
pmeasmono	⊢ (𝜑 → (𝑃‘𝐴) ≤ (𝑃‘𝐵))

Distinct variable groups: 𝑥,𝑃,𝑦 𝑥,𝑅,𝑦 𝜑,𝑥,𝑦 𝑥,𝐴,𝑦 𝑥,𝐵,𝑦

**Proof of Theorem pmeasmono**
Step	Hyp	Ref	Expression
1		eqimss 3993	. . . . . . 7 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → 𝐴 ⊆ (𝐵 ∖ 𝐴))
2		ssdifeq0 4440	. . . . . . 7 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅)
3	1, 2	sylib 218	. . . . . 6 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → 𝐴 = ∅)
4	3	fveq2d 6839	. . . . 5 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → (𝑃‘𝐴) = (𝑃‘∅))
5	4	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) = (𝑃‘∅))
6		caraext.2	. . . . 5 ⊢ (𝜑 → (𝑃‘∅) = 0)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑃‘∅) = 0)
8	5, 7	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) = 0)
9		caraext.1	. . . . . 6 ⊢ (𝜑 → 𝑃:𝑅⟶(0[,]+∞))
10		pmeasmono.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑅)
11	9, 10	ffvelcdmd 7032	. . . . 5 ⊢ (𝜑 → (𝑃‘𝐵) ∈ (0[,]+∞))
12		elxrge0 13377	. . . . . 6 ⊢ ((𝑃‘𝐵) ∈ (0[,]+∞) ↔ ((𝑃‘𝐵) ∈ ℝ^* ∧ 0 ≤ (𝑃‘𝐵)))
13	12	simprbi 496	. . . . 5 ⊢ ((𝑃‘𝐵) ∈ (0[,]+∞) → 0 ≤ (𝑃‘𝐵))
14	11, 13	syl 17	. . . 4 ⊢ (𝜑 → 0 ≤ (𝑃‘𝐵))
15	14	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → 0 ≤ (𝑃‘𝐵))
16	8, 15	eqbrtrd 5121	. 2 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ≤ (𝑃‘𝐵))
17		iccssxr 13350	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
18	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → 𝑃:𝑅⟶(0[,]+∞))
19		pmeasmono.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑅)
20	19	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → 𝐴 ∈ 𝑅)
21	18, 20	ffvelcdmd 7032	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ∈ (0[,]+∞))
22	17, 21	sselid 3932	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ∈ ℝ^*)
23		pmeasmono.3	. . . . . 6 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑅)
24	23	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝐵 ∖ 𝐴) ∈ 𝑅)
25	18, 24	ffvelcdmd 7032	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞))
26		xrge0addge 32819	. . . 4 ⊢ (((𝑃‘𝐴) ∈ ℝ^* ∧ (𝑃‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞)) → (𝑃‘𝐴) ≤ ((𝑃‘𝐴) +_𝑒 (𝑃‘(𝐵 ∖ 𝐴))))
27	22, 25, 26	syl2anc 585	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ≤ ((𝑃‘𝐴) +_𝑒 (𝑃‘(𝐵 ∖ 𝐴))))
28		prct 32773	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑅 ∧ (𝐵 ∖ 𝐴) ∈ 𝑅) → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω)
29	19, 23, 28	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω)
30	29	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω)
31		prssi 4778	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑅 ∧ (𝐵 ∖ 𝐴) ∈ 𝑅) → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅)
32	19, 23, 31	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅)
33	32	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅)
34		disjdif 4425	. . . . . . 7 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
35		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → 𝐴 ≠ (𝐵 ∖ 𝐴))
36		id 22	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
37		id 22	. . . . . . . . 9 ⊢ (𝑦 = (𝐵 ∖ 𝐴) → 𝑦 = (𝐵 ∖ 𝐴))
38	36, 37	disjprg 5095	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑅 ∧ (𝐵 ∖ 𝐴) ∈ 𝑅 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦 ↔ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅))
39	20, 24, 35, 38	syl3anc 1374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦 ↔ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅))
40	34, 39	mpbiri 258	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦)
41	30, 33, 40	3jca 1129	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → ({𝐴, (𝐵 ∖ 𝐴)} ≼ ω ∧ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅 ∧ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦))
42		prex 5383	. . . . . . 7 ⊢ {𝐴, (𝐵 ∖ 𝐴)} ∈ V
43		biidd 262	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝜑 ↔ 𝜑))
44		breq1 5102	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝑥 ≼ ω ↔ {𝐴, (𝐵 ∖ 𝐴)} ≼ ω))
45		sseq1 3960	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝑥 ⊆ 𝑅 ↔ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅))
46		disjeq1 5073	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (Disj 𝑦 ∈ 𝑥 𝑦 ↔ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦))
47	44, 45, 46	3anbi123d 1439	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → ((𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦) ↔ ({𝐴, (𝐵 ∖ 𝐴)} ≼ ω ∧ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅 ∧ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦)))
48	43, 47	anbi12d 633	. . . . . . . . 9 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ↔ (𝜑 ∧ ({𝐴, (𝐵 ∖ 𝐴)} ≼ ω ∧ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅 ∧ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦))))
49		unieq 4875	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → ∪ 𝑥 = ∪ {𝐴, (𝐵 ∖ 𝐴)})
50	49	fveq2d 6839	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝑃‘∪ 𝑥) = (𝑃‘∪ {𝐴, (𝐵 ∖ 𝐴)}))
51		esumeq1 34172	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → Σ^𝑦 ∈ 𝑥(𝑃‘𝑦) = Σ^𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑃‘𝑦))
52	50, 51	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → ((𝑃‘∪ 𝑥) = Σ^𝑦 ∈ 𝑥(𝑃‘𝑦) ↔ (𝑃‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = Σ^𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑃‘𝑦)))
53	48, 52	imbi12d 344	. . . . . . . 8 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑃‘∪ 𝑥) = Σ^𝑦 ∈ 𝑥(𝑃‘𝑦)) ↔ ((𝜑 ∧ ({𝐴, (𝐵 ∖ 𝐴)} ≼ ω ∧ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅 ∧ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦)) → (𝑃‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = Σ^𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑃‘𝑦))))
54		caraext.3	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ≼ ω ∧ 𝑥 ⊆ 𝑅 ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑃‘∪ 𝑥) = Σ^*𝑦 ∈ 𝑥(𝑃‘𝑦))
55	53, 54	vtoclg 3512	. . . . . . 7 ⊢ ({𝐴, (𝐵 ∖ 𝐴)} ∈ V → ((𝜑 ∧ ({𝐴, (𝐵 ∖ 𝐴)} ≼ ω ∧ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅 ∧ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦)) → (𝑃‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = Σ^*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑃‘𝑦)))
56	42, 55	ax-mp 5	. . . . . 6 ⊢ ((𝜑 ∧ ({𝐴, (𝐵 ∖ 𝐴)} ≼ ω ∧ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅 ∧ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦)) → (𝑃‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = Σ^*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑃‘𝑦))
57	56	adantlr 716	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) ∧ ({𝐴, (𝐵 ∖ 𝐴)} ≼ ω ∧ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅 ∧ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦)) → (𝑃‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = Σ^*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑃‘𝑦))
58	41, 57	mpdan 688	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = Σ^*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑃‘𝑦))
59		uniprg 4880	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑅 ∧ (𝐵 ∖ 𝐴) ∈ 𝑅) → ∪ {𝐴, (𝐵 ∖ 𝐴)} = (𝐴 ∪ (𝐵 ∖ 𝐴)))
60	19, 23, 59	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → ∪ {𝐴, (𝐵 ∖ 𝐴)} = (𝐴 ∪ (𝐵 ∖ 𝐴)))
61		pmeasmono.4	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
62		undif 4435	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
63	61, 62	sylib 218	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
64	60, 63	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → ∪ {𝐴, (𝐵 ∖ 𝐴)} = 𝐵)
65	64	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → ∪ {𝐴, (𝐵 ∖ 𝐴)} = 𝐵)
66	65	fveq2d 6839	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = (𝑃‘𝐵))
67		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
68	67	fveq2d 6839	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) ∧ 𝑦 = 𝐴) → (𝑃‘𝑦) = (𝑃‘𝐴))
69		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) ∧ 𝑦 = (𝐵 ∖ 𝐴)) → 𝑦 = (𝐵 ∖ 𝐴))
70	69	fveq2d 6839	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) ∧ 𝑦 = (𝐵 ∖ 𝐴)) → (𝑃‘𝑦) = (𝑃‘(𝐵 ∖ 𝐴)))
71	68, 70, 20, 24, 21, 25, 35	esumpr 34204	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → Σ^*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑃‘𝑦) = ((𝑃‘𝐴) +_𝑒 (𝑃‘(𝐵 ∖ 𝐴))))
72	58, 66, 71	3eqtr3d 2780	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐵) = ((𝑃‘𝐴) +_𝑒 (𝑃‘(𝐵 ∖ 𝐴))))
73	27, 72	breqtrrd 5127	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ≤ (𝑃‘𝐵))
74	16, 73	pm2.61dane 3020	1 ⊢ (𝜑 → (𝑃‘𝐴) ≤ (𝑃‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 Vcvv 3441 ∖ cdif 3899 ∪ cun 3900 ∩ cin 3901 ⊆ wss 3902 ∅c0 4286 {cpr 4583 ∪ cuni 4864 Disj wdisj 5066 class class class wbr 5099 ⟶wf 6489 ‘cfv 6493 (class class class)co 7360 ωcom 7810 ≼ cdom 8885 0cc0 11030 +∞cpnf 11167 ℝ^*cxr 11169 ≤ cle 11171 +_𝑒 cxad 13028 [,]cicc 13268 Σ^*cesum 34165

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 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109 ax-mulf 11110

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-disj 5067 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9817 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-z 12493 df-dec 12612 df-uz 12756 df-q 12866 df-rp 12910 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-ioo 13269 df-ioc 13270 df-ico 13271 df-icc 13272 df-fz 13428 df-fzo 13575 df-fl 13716 df-mod 13794 df-seq 13929 df-exp 13989 df-fac 14201 df-bc 14230 df-hash 14258 df-shft 14994 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-limsup 15398 df-clim 15415 df-rlim 15416 df-sum 15614 df-ef 15994 df-sin 15996 df-cos 15997 df-pi 15999 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-rest 17346 df-topn 17347 df-0g 17365 df-gsum 17366 df-topgen 17367 df-pt 17368 df-prds 17371 df-ordt 17426 df-xrs 17427 df-qtop 17432 df-imas 17433 df-xps 17435 df-mre 17509 df-mrc 17510 df-acs 17512 df-ps 18493 df-tsr 18494 df-plusf 18568 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18712 df-submnd 18713 df-grp 18870 df-minusg 18871 df-sbg 18872 df-mulg 19002 df-subg 19057 df-cntz 19250 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-subrng 20483 df-subrg 20507 df-abv 20746 df-lmod 20817 df-scaf 20818 df-sra 21129 df-rgmod 21130 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22842 df-topon 22859 df-topsp 22881 df-bases 22894 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cn 23175 df-cnp 23176 df-haus 23263 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-tmd 24020 df-tgp 24021 df-tsms 24075 df-trg 24108 df-xms 24268 df-ms 24269 df-tms 24270 df-nm 24530 df-ngp 24531 df-nrg 24533 df-nlm 24534 df-ii 24830 df-cncf 24831 df-limc 25827 df-dv 25828 df-log 26525 df-esum 34166

This theorem is referenced by: (None)

W3C validator