pmeasmono - Mathbox for Thierry Arnoux

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Theorem pmeasmono 34291

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 3996	. . . . . . 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 6830	. . . . 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 2764	. . 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 7023	. . . . 5 ⊢ (𝜑 → (𝑃‘𝐵) ∈ (0[,]+∞))
12		elxrge0 13378	. . . . . 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 5117	. 2 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ≤ (𝑃‘𝐵))
17		iccssxr 13351	. . . . 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 7023	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ∈ (0[,]+∞))
22	17, 21	sselid 3935	. . . 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 7023	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞))
26		xrge0addge 32714	. . . 4 ⊢ (((𝑃‘𝐴) ∈ ℝ^* ∧ (𝑃‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞)) → (𝑃‘𝐴) ≤ ((𝑃‘𝐴) +_𝑒 (𝑃‘(𝐵 ∖ 𝐴))))
27	22, 25, 26	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ≤ ((𝑃‘𝐴) +_𝑒 (𝑃‘(𝐵 ∖ 𝐴))))
28		prct 32671	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑅 ∧ (𝐵 ∖ 𝐴) ∈ 𝑅) → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω)
29	19, 23, 28	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω)
30	29	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω)
31		prssi 4775	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑅 ∧ (𝐵 ∖ 𝐴) ∈ 𝑅) → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅)
32	19, 23, 31	syl2anc 584	. . . . . . 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 5091	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑅 ∧ (𝐵 ∖ 𝐴) ∈ 𝑅 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦 ↔ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅))
39	20, 24, 35, 38	syl3anc 1373	. . . . . . 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 1128	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → ({𝐴, (𝐵 ∖ 𝐴)} ≼ ω ∧ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅 ∧ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦))
42		prex 5379	. . . . . . 7 ⊢ {𝐴, (𝐵 ∖ 𝐴)} ∈ V
43		biidd 262	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝜑 ↔ 𝜑))
44		breq1 5098	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝑥 ≼ ω ↔ {𝐴, (𝐵 ∖ 𝐴)} ≼ ω))
45		sseq1 3963	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝑥 ⊆ 𝑅 ↔ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅))
46		disjeq1 5069	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (Disj 𝑦 ∈ 𝑥 𝑦 ↔ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦))
47	44, 45, 46	3anbi123d 1438	. . . . . . . . . 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 632	. . . . . . . . 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 4872	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → ∪ 𝑥 = ∪ {𝐴, (𝐵 ∖ 𝐴)})
50	49	fveq2d 6830	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝑃‘∪ 𝑥) = (𝑃‘∪ {𝐴, (𝐵 ∖ 𝐴)}))
51		esumeq1 34000	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → Σ^𝑦 ∈ 𝑥(𝑃‘𝑦) = Σ^𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑃‘𝑦))
52	50, 51	eqeq12d 2745	. . . . . . . . 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 3511	. . . . . . 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 715	. . . . 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 687	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = Σ^*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑃‘𝑦))
59		uniprg 4877	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑅 ∧ (𝐵 ∖ 𝐴) ∈ 𝑅) → ∪ {𝐴, (𝐵 ∖ 𝐴)} = (𝐴 ∪ (𝐵 ∖ 𝐴)))
60	19, 23, 59	syl2anc 584	. . . . . . 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 2764	. . . . . 6 ⊢ (𝜑 → ∪ {𝐴, (𝐵 ∖ 𝐴)} = 𝐵)
65	64	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → ∪ {𝐴, (𝐵 ∖ 𝐴)} = 𝐵)
66	65	fveq2d 6830	. . . 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 6830	. . . . 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 6830	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) ∧ 𝑦 = (𝐵 ∖ 𝐴)) → (𝑃‘𝑦) = (𝑃‘(𝐵 ∖ 𝐴)))
71	68, 70, 20, 24, 21, 25, 35	esumpr 34032	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → Σ^*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑃‘𝑦) = ((𝑃‘𝐴) +_𝑒 (𝑃‘(𝐵 ∖ 𝐴))))
72	58, 66, 71	3eqtr3d 2772	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐵) = ((𝑃‘𝐴) +_𝑒 (𝑃‘(𝐵 ∖ 𝐴))))
73	27, 72	breqtrrd 5123	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ≤ (𝑃‘𝐵))
74	16, 73	pm2.61dane 3012	1 ⊢ (𝜑 → (𝑃‘𝐴) ≤ (𝑃‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 Vcvv 3438 ∖ cdif 3902 ∪ cun 3903 ∩ cin 3904 ⊆ wss 3905 ∅c0 4286 {cpr 4581 ∪ cuni 4861 Disj wdisj 5062 class class class wbr 5095 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 ωcom 7806 ≼ cdom 8877 0cc0 11028 +∞cpnf 11165 ℝ^*cxr 11167 ≤ cle 11169 +_𝑒 cxad 13030 [,]cicc 13269 Σ^*cesum 33993

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 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107 ax-mulf 11108

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 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-disj 5063 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-dju 9816 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13270 df-ioc 13271 df-ico 13272 df-icc 13273 df-fz 13429 df-fzo 13576 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-fac 14199 df-bc 14228 df-hash 14256 df-shft 14992 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-limsup 15396 df-clim 15413 df-rlim 15414 df-sum 15612 df-ef 15992 df-sin 15994 df-cos 15995 df-pi 15997 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-ordt 17423 df-xrs 17424 df-qtop 17429 df-imas 17430 df-xps 17432 df-mre 17506 df-mrc 17507 df-acs 17509 df-ps 18490 df-tsr 18491 df-plusf 18531 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-mhm 18675 df-submnd 18676 df-grp 18833 df-minusg 18834 df-sbg 18835 df-mulg 18965 df-subg 19020 df-cntz 19214 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-ring 20138 df-cring 20139 df-subrng 20449 df-subrg 20473 df-abv 20712 df-lmod 20783 df-scaf 20784 df-sra 21095 df-rgmod 21096 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22797 df-topon 22814 df-topsp 22836 df-bases 22849 df-cld 22922 df-ntr 22923 df-cls 22924 df-nei 23001 df-lp 23039 df-perf 23040 df-cn 23130 df-cnp 23131 df-haus 23218 df-tx 23465 df-hmeo 23658 df-fil 23749 df-fm 23841 df-flim 23842 df-flf 23843 df-tmd 23975 df-tgp 23976 df-tsms 24030 df-trg 24063 df-xms 24224 df-ms 24225 df-tms 24226 df-nm 24486 df-ngp 24487 df-nrg 24489 df-nlm 24490 df-ii 24786 df-cncf 24787 df-limc 25783 df-dv 25784 df-log 26481 df-esum 33994

This theorem is referenced by: (None)

W3C validator