pmeasmono - Mathbox for Thierry Arnoux

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Theorem pmeasmono 34306

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 4054	. . . . . . 7 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → 𝐴 ⊆ (𝐵 ∖ 𝐴))
2		ssdifeq0 4493	. . . . . . 7 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅)
3	1, 2	sylib 218	. . . . . 6 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → 𝐴 = ∅)
4	3	fveq2d 6911	. . . . 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 2775	. . 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 7105	. . . . 5 ⊢ (𝜑 → (𝑃‘𝐵) ∈ (0[,]+∞))
12		elxrge0 13494	. . . . . 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 5170	. 2 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ≤ (𝑃‘𝐵))
17		iccssxr 13467	. . . . 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 7105	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ∈ (0[,]+∞))
22	17, 21	sselid 3993	. . . 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 7105	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞))
26		xrge0addge 32768	. . . 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 32732	. . . . . . . 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 4826	. . . . . . . 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 4478	. . . . . . 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 5144	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑅 ∧ (𝐵 ∖ 𝐴) ∈ 𝑅 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦 ↔ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅))
39	20, 24, 35, 38	syl3anc 1370	. . . . . . 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 1127	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → ({𝐴, (𝐵 ∖ 𝐴)} ≼ ω ∧ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅 ∧ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦))
42		prex 5443	. . . . . . 7 ⊢ {𝐴, (𝐵 ∖ 𝐴)} ∈ V
43		biidd 262	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝜑 ↔ 𝜑))
44		breq1 5151	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝑥 ≼ ω ↔ {𝐴, (𝐵 ∖ 𝐴)} ≼ ω))
45		sseq1 4021	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝑥 ⊆ 𝑅 ↔ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅))
46		disjeq1 5122	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (Disj 𝑦 ∈ 𝑥 𝑦 ↔ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦))
47	44, 45, 46	3anbi123d 1435	. . . . . . . . . 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 4923	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → ∪ 𝑥 = ∪ {𝐴, (𝐵 ∖ 𝐴)})
50	49	fveq2d 6911	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝑃‘∪ 𝑥) = (𝑃‘∪ {𝐴, (𝐵 ∖ 𝐴)}))
51		esumeq1 34015	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → Σ^𝑦 ∈ 𝑥(𝑃‘𝑦) = Σ^𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑃‘𝑦))
52	50, 51	eqeq12d 2751	. . . . . . . . 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 3554	. . . . . . 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 4928	. . . . . . . 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 4488	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
63	61, 62	sylib 218	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
64	60, 63	eqtrd 2775	. . . . . 6 ⊢ (𝜑 → ∪ {𝐴, (𝐵 ∖ 𝐴)} = 𝐵)
65	64	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → ∪ {𝐴, (𝐵 ∖ 𝐴)} = 𝐵)
66	65	fveq2d 6911	. . . 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 6911	. . . . 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 6911	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) ∧ 𝑦 = (𝐵 ∖ 𝐴)) → (𝑃‘𝑦) = (𝑃‘(𝐵 ∖ 𝐴)))
71	68, 70, 20, 24, 21, 25, 35	esumpr 34047	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → Σ^*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑃‘𝑦) = ((𝑃‘𝐴) +_𝑒 (𝑃‘(𝐵 ∖ 𝐴))))
72	58, 66, 71	3eqtr3d 2783	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐵) = ((𝑃‘𝐴) +_𝑒 (𝑃‘(𝐵 ∖ 𝐴))))
73	27, 72	breqtrrd 5176	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ≤ (𝑃‘𝐵))
74	16, 73	pm2.61dane 3027	1 ⊢ (𝜑 → (𝑃‘𝐴) ≤ (𝑃‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∖ cdif 3960 ∪ cun 3961 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 {cpr 4633 ∪ cuni 4912 Disj wdisj 5115 class class class wbr 5148 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ωcom 7887 ≼ cdom 8982 0cc0 11153 +∞cpnf 11290 ℝ^*cxr 11292 ≤ cle 11294 +_𝑒 cxad 13150 [,]cicc 13387 Σ^*cesum 34008

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-ordt 17548 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-ps 18624 df-tsr 18625 df-plusf 18665 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-subrng 20563 df-subrg 20587 df-abv 20827 df-lmod 20877 df-scaf 20878 df-sra 21190 df-rgmod 21191 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-tmd 24096 df-tgp 24097 df-tsms 24151 df-trg 24184 df-xms 24346 df-ms 24347 df-tms 24348 df-nm 24611 df-ngp 24612 df-nrg 24614 df-nlm 24615 df-ii 24917 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-esum 34009

This theorem is referenced by: (None)

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