pmeasmono - Mathbox for Thierry Arnoux

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Theorem pmeasmono 32291

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 3977	. . . . . . 7 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → 𝐴 ⊆ (𝐵 ∖ 𝐴))
2		ssdifeq0 4417	. . . . . . 7 ⊢ (𝐴 ⊆ (𝐵 ∖ 𝐴) ↔ 𝐴 = ∅)
3	1, 2	sylib 217	. . . . . 6 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → 𝐴 = ∅)
4	3	fveq2d 6778	. . . . 5 ⊢ (𝐴 = (𝐵 ∖ 𝐴) → (𝑃‘𝐴) = (𝑃‘∅))
5	4	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) = (𝑃‘∅))
6		caraext.2	. . . . 5 ⊢ (𝜑 → (𝑃‘∅) = 0)
7	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑃‘∅) = 0)
8	5, 7	eqtrd 2778	. . 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	ffvelrnd 6962	. . . . 5 ⊢ (𝜑 → (𝑃‘𝐵) ∈ (0[,]+∞))
12		elxrge0 13189	. . . . . 6 ⊢ ((𝑃‘𝐵) ∈ (0[,]+∞) ↔ ((𝑃‘𝐵) ∈ ℝ^* ∧ 0 ≤ (𝑃‘𝐵)))
13	12	simprbi 497	. . . . 5 ⊢ ((𝑃‘𝐵) ∈ (0[,]+∞) → 0 ≤ (𝑃‘𝐵))
14	11, 13	syl 17	. . . 4 ⊢ (𝜑 → 0 ≤ (𝑃‘𝐵))
15	14	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → 0 ≤ (𝑃‘𝐵))
16	8, 15	eqbrtrd 5096	. 2 ⊢ ((𝜑 ∧ 𝐴 = (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ≤ (𝑃‘𝐵))
17		iccssxr 13162	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
18	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → 𝑃:𝑅⟶(0[,]+∞))
19		pmeasmono.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑅)
20	19	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → 𝐴 ∈ 𝑅)
21	18, 20	ffvelrnd 6962	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ∈ (0[,]+∞))
22	17, 21	sselid 3919	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ∈ ℝ^*)
23		pmeasmono.3	. . . . . 6 ⊢ (𝜑 → (𝐵 ∖ 𝐴) ∈ 𝑅)
24	23	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝐵 ∖ 𝐴) ∈ 𝑅)
25	18, 24	ffvelrnd 6962	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘(𝐵 ∖ 𝐴)) ∈ (0[,]+∞))
26		xrge0addge 31080	. . . 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 31049	. . . . . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → {𝐴, (𝐵 ∖ 𝐴)} ≼ ω)
31		prssi 4754	. . . . . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅)
34		disjdif 4405	. . . . . . 7 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
35		simpr 485	. . . . . . . 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 5070	. . . . . . . 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 257	. . . . . 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 5355	. . . . . . 7 ⊢ {𝐴, (𝐵 ∖ 𝐴)} ∈ V
43		biidd 261	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝜑 ↔ 𝜑))
44		breq1 5077	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝑥 ≼ ω ↔ {𝐴, (𝐵 ∖ 𝐴)} ≼ ω))
45		sseq1 3946	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝑥 ⊆ 𝑅 ↔ {𝐴, (𝐵 ∖ 𝐴)} ⊆ 𝑅))
46		disjeq1 5046	. . . . . . . . . . 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 631	. . . . . . . . 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 4850	. . . . . . . . . . 11 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → ∪ 𝑥 = ∪ {𝐴, (𝐵 ∖ 𝐴)})
50	49	fveq2d 6778	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → (𝑃‘∪ 𝑥) = (𝑃‘∪ {𝐴, (𝐵 ∖ 𝐴)}))
51		esumeq1 32002	. . . . . . . . . 10 ⊢ (𝑥 = {𝐴, (𝐵 ∖ 𝐴)} → Σ^𝑦 ∈ 𝑥(𝑃‘𝑦) = Σ^𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑃‘𝑦))
52	50, 51	eqeq12d 2754	. . . . . . . . 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 345	. . . . . . . 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 3505	. . . . . . 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 712	. . . . 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 684	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = Σ^*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑃‘𝑦))
59		uniprg 4856	. . . . . . . 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 4415	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
63	61, 62	sylib 217	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∪ (𝐵 ∖ 𝐴)) = 𝐵)
64	60, 63	eqtrd 2778	. . . . . 6 ⊢ (𝜑 → ∪ {𝐴, (𝐵 ∖ 𝐴)} = 𝐵)
65	64	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → ∪ {𝐴, (𝐵 ∖ 𝐴)} = 𝐵)
66	65	fveq2d 6778	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘∪ {𝐴, (𝐵 ∖ 𝐴)}) = (𝑃‘𝐵))
67		simpr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
68	67	fveq2d 6778	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) ∧ 𝑦 = 𝐴) → (𝑃‘𝑦) = (𝑃‘𝐴))
69		simpr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) ∧ 𝑦 = (𝐵 ∖ 𝐴)) → 𝑦 = (𝐵 ∖ 𝐴))
70	69	fveq2d 6778	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) ∧ 𝑦 = (𝐵 ∖ 𝐴)) → (𝑃‘𝑦) = (𝑃‘(𝐵 ∖ 𝐴)))
71	68, 70, 20, 24, 21, 25, 35	esumpr 32034	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → Σ^*𝑦 ∈ {𝐴, (𝐵 ∖ 𝐴)} (𝑃‘𝑦) = ((𝑃‘𝐴) +_𝑒 (𝑃‘(𝐵 ∖ 𝐴))))
72	58, 66, 71	3eqtr3d 2786	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐵) = ((𝑃‘𝐴) +_𝑒 (𝑃‘(𝐵 ∖ 𝐴))))
73	27, 72	breqtrrd 5102	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ (𝐵 ∖ 𝐴)) → (𝑃‘𝐴) ≤ (𝑃‘𝐵))
74	16, 73	pm2.61dane 3032	1 ⊢ (𝜑 → (𝑃‘𝐴) ≤ (𝑃‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 Vcvv 3432 ∖ cdif 3884 ∪ cun 3885 ∩ cin 3886 ⊆ wss 3887 ∅c0 4256 {cpr 4563 ∪ cuni 4839 Disj wdisj 5039 class class class wbr 5074 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ωcom 7712 ≼ cdom 8731 0cc0 10871 +∞cpnf 11006 ℝ^*cxr 11008 ≤ cle 11010 +_𝑒 cxad 12846 [,]cicc 13082 Σ^*cesum 31995

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-pi 15782 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-ordt 17212 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-ps 18284 df-tsr 18285 df-plusf 18325 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-cntz 18923 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-subrg 20022 df-abv 20077 df-lmod 20125 df-scaf 20126 df-sra 20434 df-rgmod 20435 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-tmd 23223 df-tgp 23224 df-tsms 23278 df-trg 23311 df-xms 23473 df-ms 23474 df-tms 23475 df-nm 23738 df-ngp 23739 df-nrg 23741 df-nlm 23742 df-ii 24040 df-cncf 24041 df-limc 25030 df-dv 25031 df-log 25712 df-esum 31996

This theorem is referenced by: (None)

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