pmeasmono - Mathbox for Thierry Arnoux

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Theorem pmeasmono 34890

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 3988	. . . . . . 7 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → 𝐴 ⊆ (𝐵 ∖ 𝐴))
2		ssdifeq0 4441	. . . . . . 7 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅)
3	1, 2	sylib 221	. . . . . 6 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → 𝐴 = ∅)
4	3	fveq2d 6877	. . . . 5 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → (𝑃‘𝐴) = (𝑃‘∅))
5	4	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) = (𝑃‘∅))
6		caraext.2	. . . . 5 ⊢ (𝜑 → (𝑃‘∅) = 0)
7	6	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑃‘∅) = 0)
8	5, 7	eqtrd 2795	. . 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 7073	. . . . 5 ⊢ (𝜑 → (𝑃‘𝐵) ∈ (0[,]+∞))
12		elxrge0 13557	. . . . . 6 ⊢ ((𝑃‘𝐵) ∈ (0[,]+∞) ↔ ((𝑃‘𝐵) ∈ ℝ^* ∧ 0 ≤ (𝑃‘𝐵)))
13	12	simprbi 503	. . . . 5 ⊢ ((𝑃‘𝐵) ∈ (0[,]+∞) → 0 ≤ (𝑃‘𝐵))
14	11, 13	syl 18	. . . 4 ⊢ (𝜑 → 0 ≤ (𝑃‘𝐵))
15	14	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → 0 ≤ (𝑃‘𝐵))
16	8, 15	eqbrtrd 5126	. 2 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ≤ (𝑃‘𝐵))
17		iccssxr 13530	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
18	9	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → 𝑃:𝑅⟶(0[,]+∞))
19		pmeasmono.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑅)
20	19	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → 𝐴 ∈ 𝑅)
21	18, 20	ffvelcdmd 7073	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ∈ (0[,]+∞))
22	17, 21	sselid 3928	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ∈ ℝ^*)
23		pmeasmono.3	. . . . . 6 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑅)
24	23	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝐵 ∖ 𝐴) ∈ 𝑅)
25	18, 24	ffvelcdmd 7073	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞))
26		xrge0addge 33283	. . . 4 ⊢ (((𝑃‘𝐴) ∈ ℝ^* ∧ (𝑃‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞)) → (𝑃‘𝐴) ≤ ((𝑃‘𝐴) +_𝑒 (𝑃‘(𝐵 ∖ 𝐴))))
27	22, 25, 26	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ≤ ((𝑃‘𝐴) +_𝑒 (𝑃‘(𝐵 ∖ 𝐴))))
28		prct 33239	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑅 ∧ (𝐵 ∖ 𝐴) ∈ 𝑅) → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω)
29	19, 23, 28	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω)
30	29	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω)
31		prssi 4781	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑅 ∧ (𝐵 ∖ 𝐴) ∈ 𝑅) → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅)
32	19, 23, 31	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅)
33	32	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅)
34		disjdif 4425	. . . . . . 7 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
35		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → 𝐴 ≠ (𝐵 ∖ 𝐴))
36		id 23	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
37		id 23	. . . . . . . . 9 ⊢ (𝑦 = (𝐵 ∖ 𝐴) → 𝑦 = (𝐵 ∖ 𝐴))
38	36, 37	disjprg 5098	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑅 ∧ (𝐵 ∖ 𝐴) ∈ 𝑅 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦 ↔ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅))
39	20, 24, 35, 38	syl3anc 1398	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦 ↔ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅))
40	34, 39	mpbiri 261	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦)
41	30, 33, 40	3jca 1146	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → ({𝐴, (𝐵 ∖ 𝐴)} ≼ ω ∧ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅 ∧ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦))
42		prex 5395	. . . . . . 7 ⊢ {𝐴, (𝐵 ∖ 𝐴)} ∈ V
43		biidd 265	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝜑 ↔ 𝜑))
44		breq1 5105	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝑥 ≼ ω ↔ {𝐴, (𝐵 ∖ 𝐴)} ≼ ω))
45		sseq1 3955	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝑥 ⊆ 𝑅 ↔ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅))
46		disjeq1 5076	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (Disj 𝑦 ∈ 𝑥 𝑦 ↔ Disj 𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)}𝑦))
47	44, 45, 46	3anbi123d 1464	. . . . . . . . . 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 644	. . . . . . . . 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 4877	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → ∪ 𝑥 = ∪ {𝐴, (𝐵 ∖ 𝐴)})
50	49	fveq2d 6877	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝑃‘∪ 𝑥) = (𝑃‘∪ {𝐴, (𝐵 ∖ 𝐴)}))
51		esumeq1 34599	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → Σ^𝑦 ∈ 𝑥(𝑃‘𝑦) = Σ^𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑃‘𝑦))
52	50, 51	eqeq12d 2776	. . . . . . . . 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 347	. . . . . . . 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 3517	. . . . . . 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 728	. . . . 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 700	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = Σ^*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑃‘𝑦))
59		uniprg 4882	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑅 ∧ (𝐵 ∖ 𝐴) ∈ 𝑅) → ∪ {𝐴, (𝐵 ∖ 𝐴)} = (𝐴 ∪ (𝐵 ∖ 𝐴)))
60	19, 23, 59	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → ∪ {𝐴, (𝐵 ∖ 𝐴)} = (𝐴 ∪ (𝐵 ∖ 𝐴)))
61		pmeasmono.4	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
62		undif 4437	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
63	61, 62	sylib 221	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
64	60, 63	eqtrd 2795	. . . . . 6 ⊢ (𝜑 → ∪ {𝐴, (𝐵 ∖ 𝐴)} = 𝐵)
65	64	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → ∪ {𝐴, (𝐵 ∖ 𝐴)} = 𝐵)
66	65	fveq2d 6877	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = (𝑃‘𝐵))
67		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
68	67	fveq2d 6877	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) ∧ 𝑦 = 𝐴) → (𝑃‘𝑦) = (𝑃‘𝐴))
69		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) ∧ 𝑦 = (𝐵 ∖ 𝐴)) → 𝑦 = (𝐵 ∖ 𝐴))
70	69	fveq2d 6877	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) ∧ 𝑦 = (𝐵 ∖ 𝐴)) → (𝑃‘𝑦) = (𝑃‘(𝐵 ∖ 𝐴)))
71	68, 70, 20, 24, 21, 25, 35	esumpr 34631	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → Σ^*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑃‘𝑦) = ((𝑃‘𝐴) +_𝑒 (𝑃‘(𝐵 ∖ 𝐴))))
72	58, 66, 71	3eqtr3d 2803	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐵) = ((𝑃‘𝐴) +_𝑒 (𝑃‘(𝐵 ∖ 𝐴))))
73	27, 72	breqtrrd 5132	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ≤ (𝑃‘𝐵))
74	16, 73	pm2.61dane 3042	1 ⊢ (𝜑 → (𝑃‘𝐴) ≤ (𝑃‘𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 Vcvv 3450 ∖ cdif 3895 ∪ cun 3896 ∩ cin 3897 ⊆ wss 3898 ∅c0 4278 {cpr 4585 ∪ cuni 4866 Disj wdisj 5069 class class class wbr 5102 ⟶wf 6523 ‘cfv 6527 (class class class)co 7408 ωcom 7860 ≼ cdom 8949 0cc0 11171 +∞cpnf 11311 ℝ^*cxr 11313 ≤ cle 11315 +_𝑒 cxad 13208 [,]cicc 13448 Σ^*cesum 34592

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-inf2 9620 ax-cnex 11227 ax-resscn 11228 ax-1cn 11229 ax-icn 11230 ax-addcl 11231 ax-addrcl 11232 ax-mulcl 11233 ax-mulrcl 11234 ax-mulcom 11235 ax-addass 11236 ax-mulass 11237 ax-distr 11238 ax-i2m1 11239 ax-1ne0 11240 ax-1rid 11241 ax-rnegex 11242 ax-rrecex 11243 ax-cnre 11244 ax-pre-lttri 11245 ax-pre-lttrn 11246 ax-pre-ltadd 11247 ax-pre-mulgt0 11248 ax-pre-sup 11249 ax-addf 11250 ax-mulf 11251

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-disj 5070 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-isom 6536 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8454 df-2o 8455 df-er 8695 df-map 8827 df-pm 8828 df-ixp 8904 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-dju 9953 df-card 9991 df-pnf 11316 df-mnf 11317 df-xr 11318 df-ltxr 11319 df-le 11320 df-sub 11514 df-neg 11515 df-div 11943 df-nn 12305 df-2 12374 df-3 12375 df-4 12376 df-5 12377 df-6 12378 df-7 12379 df-8 12380 df-9 12381 df-n0 12576 df-z 12663 df-dec 12784 df-uz 12935 df-q 13045 df-rp 13090 df-xneg 13210 df-xadd 13211 df-xmul 13212 df-ioo 13449 df-ioc 13450 df-ico 13451 df-icc 13452 df-fz 13609 df-fzo 13757 df-fl 13900 df-mod 13978 df-seq 14113 df-exp 14173 df-fac 14385 df-bc 14414 df-hash 14442 df-shft 15187 df-cj 15233 df-re 15234 df-im 15235 df-sqrt 15369 df-abs 15370 df-limsup 15605 df-clim 15622 df-rlim 15623 df-sum 15821 df-ef 16200 df-sin 16202 df-cos 16203 df-pi 16205 df-struct 17286 df-sets 17303 df-slot 17321 df-ndx 17333 df-base 17349 df-ress 17370 df-plusg 17402 df-mulr 17403 df-starv 17404 df-sca 17405 df-vsca 17406 df-ip 17407 df-tset 17408 df-ple 17409 df-ds 17411 df-unif 17412 df-hom 17413 df-cco 17414 df-rest 17554 df-topn 17555 df-0g 17573 df-gsum 17574 df-topgen 17575 df-pt 17576 df-prds 17579 df-ordt 17634 df-xrs 17635 df-qtop 17640 df-imas 17641 df-xps 17643 df-mre 17717 df-mrc 17718 df-acs 17720 df-ps 18701 df-tsr 18702 df-plusf 18776 df-mgm 18777 df-sgrp 18869 df-mnd 18885 df-mhm 18939 df-submnd 18940 df-grp 19108 df-minusg 19109 df-sbg 19110 df-mulg 19239 df-subg 19294 df-cntz 19492 df-cmn 19957 df-abl 19958 df-mgp 20322 df-rng 20336 df-ur 20369 df-ring 20422 df-cring 20423 df-subrng 20759 df-subrg 20783 df-abv 21027 df-lmod 21098 df-scaf 21099 df-sra 21409 df-rgmod 21410 df-psmet 21631 df-xmet 21632 df-met 21633 df-bl 21634 df-mopn 21635 df-fbas 21636 df-fg 21637 df-cnfld 21640 df-top 23173 df-topon 23190 df-topsp 23212 df-bases 23225 df-cld 23298 df-ntr 23299 df-cls 23300 df-nei 23377 df-lp 23415 df-perf 23416 df-cn 23506 df-cnp 23507 df-haus 23594 df-tx 23842 df-hmeo 24035 df-fil 24126 df-fm 24218 df-flim 24219 df-flf 24220 df-tmd 24352 df-tgp 24353 df-tsms 24407 df-trg 24440 df-xms 24600 df-ms 24601 df-tms 24602 df-nm 24862 df-ngp 24863 df-nrg 24865 df-nlm 24866 df-ii 25159 df-cncf 25160 df-limc 26147 df-dv 26148 df-log 26847 df-esum 34593

This theorem is used by: (None)

W3C validator