Theorem carsgclctunlem2 34351

Description: Lemma for carsgclctun 34353. (Contributed by Thierry Arnoux, 25-May-2020.)

Hypotheses

Ref	Expression
carsgval.1	⊢ (𝜑 → 𝑂 ∈ 𝑉)
carsgval.2	⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))
carsgsiga.1	⊢ (𝜑 → (𝑀‘∅) = 0)
carsgsiga.2	⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))
carsgsiga.3	⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))
carsgclctunlem2.1	⊢ (𝜑 → Disj 𝑘 ∈ ℕ 𝐴)
carsgclctunlem2.2	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (toCaraSiga‘𝑀))
carsgclctunlem2.3	⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂)
carsgclctunlem2.4	⊢ (𝜑 → (𝑀‘𝐸) ≠ +∞)

Assertion

Ref	Expression
carsgclctunlem2	⊢ (𝜑 → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) ≤ (𝑀‘𝐸))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐸,𝑦   𝑥,𝑀,𝑦   𝑥,𝑂,𝑦   𝜑,𝑥,𝑦,𝑘   𝑘,𝐸   𝑘,𝑀   𝑘,𝑂   𝜑,𝑘

Allowed substitution hints:   𝐴(𝑘)   𝑉(𝑥,𝑦,𝑘)

Proof of Theorem carsgclctunlem2

Dummy variables 𝑒 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iunin2 5047	. . . . 5 ⊢ ∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴) = (𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)
2	1	fveq2i 6879	. . . 4 ⊢ (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) = (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴))
3		iccssxr 13447	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ*
4		carsgval.2	. . . . . 6 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))
5		nnex 12246	. . . . . . . 8 ⊢ ℕ ∈ V
6	5	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℕ ∈ V)
7		carsgclctunlem2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂)
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐸 ∈ 𝒫 𝑂)
9	8	elpwincl1 32506	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∩ 𝐴) ∈ 𝒫 𝑂)
10	6, 9	elpwiuncl 32508	. . . . . 6 ⊢ (𝜑 → ∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴) ∈ 𝒫 𝑂)
11	4, 10	ffvelcdmd 7075	. . . . 5 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
12	3, 11	sselid 3956	. . . 4 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ∈ ℝ*)
13	2, 12	eqeltrrid 2839	. . 3 ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
14	4, 7	ffvelcdmd 7075	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐸) ∈ (0[,]+∞))
15	3, 14	sselid 3956	. . . 4 ⊢ (𝜑 → (𝑀‘𝐸) ∈ ℝ*)
16	7	elpwdifcl 32507	. . . . . . 7 ⊢ (𝜑 → (𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴) ∈ 𝒫 𝑂)
17	4, 16	ffvelcdmd 7075	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ (0[,]+∞))
18	3, 17	sselid 3956	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
19	18	xnegcld 13316	. . . 4 ⊢ (𝜑 → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
20	15, 19	xaddcld 13317	. . 3 ⊢ (𝜑 → ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) ∈ ℝ*)
21	4	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
22	21, 9	ffvelcdmd 7075	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
23	22	ralrimiva 3132	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
24		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑘ℕ
25	24	esumcl 34061	. . . . . . 7 ⊢ ((ℕ ∈ V ∧ ∀𝑘 ∈ ℕ (𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞)) → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
26	6, 23, 25	syl2anc 584	. . . . . 6 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
27	3, 26	sselid 3956	. . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)) ∈ ℝ*)
28	9	ralrimiva 3132	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐸 ∩ 𝐴) ∈ 𝒫 𝑂)
29		dfiun3g 5947	. . . . . . . . 9 ⊢ (∀𝑘 ∈ ℕ (𝐸 ∩ 𝐴) ∈ 𝒫 𝑂 → ∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴) = ∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)))
30	28, 29	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴) = ∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)))
31	30	fveq2d 6880	. . . . . . 7 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) = (𝑀‘∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))))
32		nnct 13999	. . . . . . . . . 10 ⊢ ℕ ≼ ω
33		mptct 10552	. . . . . . . . . 10 ⊢ (ℕ ≼ ω → (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ≼ ω)
34		rnct 10539	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ≼ ω → ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ≼ ω)
35	32, 33, 34	mp2b 10	. . . . . . . . 9 ⊢ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ≼ ω
36	35	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ≼ ω)
37		eqid 2735	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) = (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))
38	37	rnmptss 7113	. . . . . . . . 9 ⊢ (∀𝑘 ∈ ℕ (𝐸 ∩ 𝐴) ∈ 𝒫 𝑂 → ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ⊆ 𝒫 𝑂)
39	28, 38	syl 17	. . . . . . . 8 ⊢ (𝜑 → ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ⊆ 𝒫 𝑂)
40		mptexg 7213	. . . . . . . . . 10 ⊢ (ℕ ∈ V → (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ∈ V)
41		rnexg 7898	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ∈ V → ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ∈ V)
42	5, 40, 41	mp2b 10	. . . . . . . . 9 ⊢ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ∈ V
43		breq1 5122	. . . . . . . . . . . 12 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → (𝑥 ≼ ω ↔ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ≼ ω))
44		sseq1 3984	. . . . . . . . . . . 12 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → (𝑥 ⊆ 𝒫 𝑂 ↔ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ⊆ 𝒫 𝑂))
45	43, 44	3anbi23d 1441	. . . . . . . . . . 11 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) ↔ (𝜑 ∧ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ≼ ω ∧ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ⊆ 𝒫 𝑂)))
46		unieq 4894	. . . . . . . . . . . . 13 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → ∪ 𝑥 = ∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)))
47	46	fveq2d 6880	. . . . . . . . . . . 12 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → (𝑀‘∪ 𝑥) = (𝑀‘∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))))
48		esumeq1 34065	. . . . . . . . . . . 12 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → Σ*𝑦 ∈ 𝑥(𝑀‘𝑦) = Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))(𝑀‘𝑦))
49	47, 48	breq12d 5132	. . . . . . . . . . 11 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → ((𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦) ↔ (𝑀‘∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))(𝑀‘𝑦)))
50	45, 49	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → (((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)) ↔ ((𝜑 ∧ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ≼ ω ∧ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ⊆ 𝒫 𝑂) → (𝑀‘∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))(𝑀‘𝑦))))
51		carsgsiga.2	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))
52	50, 51	vtoclg 3533	. . . . . . . . 9 ⊢ (ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ∈ V → ((𝜑 ∧ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ≼ ω ∧ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ⊆ 𝒫 𝑂) → (𝑀‘∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))(𝑀‘𝑦)))
53	42, 52	ax-mp 5	. . . . . . . 8 ⊢ ((𝜑 ∧ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ≼ ω ∧ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ⊆ 𝒫 𝑂) → (𝑀‘∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))(𝑀‘𝑦))
54	36, 39, 53	mpd3an23 1465	. . . . . . 7 ⊢ (𝜑 → (𝑀‘∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))(𝑀‘𝑦))
55	31, 54	eqbrtrd 5141	. . . . . 6 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))(𝑀‘𝑦))
56		fveq2 6876	. . . . . . 7 ⊢ (𝑦 = (𝐸 ∩ 𝐴) → (𝑀‘𝑦) = (𝑀‘(𝐸 ∩ 𝐴)))
57		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐸 ∩ 𝐴) = ∅) → (𝐸 ∩ 𝐴) = ∅)
58	57	fveq2d 6880	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐸 ∩ 𝐴) = ∅) → (𝑀‘(𝐸 ∩ 𝐴)) = (𝑀‘∅))
59		carsgsiga.1	. . . . . . . . 9 ⊢ (𝜑 → (𝑀‘∅) = 0)
60	59	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐸 ∩ 𝐴) = ∅) → (𝑀‘∅) = 0)
61	58, 60	eqtrd 2770	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐸 ∩ 𝐴) = ∅) → (𝑀‘(𝐸 ∩ 𝐴)) = 0)
62		carsgclctunlem2.1	. . . . . . . . 9 ⊢ (𝜑 → Disj 𝑘 ∈ ℕ 𝐴)
63		disjin 32567	. . . . . . . . 9 ⊢ (Disj 𝑘 ∈ ℕ 𝐴 → Disj 𝑘 ∈ ℕ (𝐴 ∩ 𝐸))
64	62, 63	syl 17	. . . . . . . 8 ⊢ (𝜑 → Disj 𝑘 ∈ ℕ (𝐴 ∩ 𝐸))
65		incom 4184	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝐸) = (𝐸 ∩ 𝐴)
66	65	rgenw 3055	. . . . . . . . 9 ⊢ ∀𝑘 ∈ ℕ (𝐴 ∩ 𝐸) = (𝐸 ∩ 𝐴)
67		disjeq2 5090	. . . . . . . . 9 ⊢ (∀𝑘 ∈ ℕ (𝐴 ∩ 𝐸) = (𝐸 ∩ 𝐴) → (Disj 𝑘 ∈ ℕ (𝐴 ∩ 𝐸) ↔ Disj 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)))
68	66, 67	ax-mp 5	. . . . . . . 8 ⊢ (Disj 𝑘 ∈ ℕ (𝐴 ∩ 𝐸) ↔ Disj 𝑘 ∈ ℕ (𝐸 ∩ 𝐴))
69	64, 68	sylib 218	. . . . . . 7 ⊢ (𝜑 → Disj 𝑘 ∈ ℕ (𝐸 ∩ 𝐴))
70	56, 6, 22, 9, 61, 69	esumrnmpt2 34099	. . . . . 6 ⊢ (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))(𝑀‘𝑦) = Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)))
71	55, 70	breqtrd 5145	. . . . 5 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ≤ Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)))
72		carsgval.1	. . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ 𝑉)
73		difssd 4112	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴) ⊆ 𝐸)
74		carsgsiga.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))
75	72, 4, 73, 7, 74	carsgmon 34346	. . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀‘𝐸))
76	14, 17, 75	xrge0subcld 32740	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) ∈ (0[,]+∞))
77	4	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
78	7	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ 𝒫 𝑂)
79	78	elpwincl1 32506	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴) ∈ 𝒫 𝑂)
80	77, 79	ffvelcdmd 7075	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞))
81	3, 80	sselid 3956	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
82		xrge0neqmnf 13469	. . . . . . . . . . 11 ⊢ ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞)
83	80, 82	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞)
84	78	elpwdifcl 32507	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴) ∈ 𝒫 𝑂)
85	77, 84	ffvelcdmd 7075	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞))
86	3, 85	sselid 3956	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
87		xrge0neqmnf 13469	. . . . . . . . . . 11 ⊢ ((𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞)
88	85, 87	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞)
89	86	xnegcld 13316	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
90		xnegneg 13230	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ* → -𝑒-𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)))
91	86, 90	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -𝑒-𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)))
92	91	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → -𝑒-𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)))
93		xnegeq 13223	. . . . . . . . . . . . . . . . 17 ⊢ (-𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞ → -𝑒-𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -𝑒-∞)
94	93	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → -𝑒-𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -𝑒-∞)
95		xnegmnf 13226	. . . . . . . . . . . . . . . 16 ⊢ -𝑒-∞ = +∞
96	94, 95	eqtrdi 2786	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → -𝑒-𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = +∞)
97	92, 96	eqtr3d 2772	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = +∞)
98	97	oveq2d 7421	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) = ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 +∞))
99		simpll 766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
100		fz1ssnn 13572	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1...𝑛) ⊆ ℕ
101	100	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
102	101	sselda 3958	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
103		carsgclctunlem2.2	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (toCaraSiga‘𝑀))
104	99, 102, 103	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (toCaraSiga‘𝑀))
105	104	ralrimiva 3132	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀))
106		dfiun3g 5947	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀) → ∪ 𝑘 ∈ (1...𝑛)𝐴 = ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))
107	105, 106	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑘 ∈ (1...𝑛)𝐴 = ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))
108	72	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑂 ∈ 𝑉)
109	59	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘∅) = 0)
110	51	3adant1r 1178	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))
111		fzfi 13990	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1...𝑛) ∈ Fin
112		mptfi 9363	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1...𝑛) ∈ Fin → (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ Fin)
113		rnfi 9352	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ Fin → ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ Fin)
114	111, 112, 113	mp2b 10	. . . . . . . . . . . . . . . . . . . 20 ⊢ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ Fin
115	114	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ Fin)
116		eqid 2735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (1...𝑛) ↦ 𝐴) = (𝑘 ∈ (1...𝑛) ↦ 𝐴)
117	116	rnmptss 7113	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀) → ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ (toCaraSiga‘𝑀))
118	105, 117	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ (toCaraSiga‘𝑀))
119	108, 77, 109, 110, 115, 118	fiunelcarsg 34348	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ (toCaraSiga‘𝑀))
120	107, 119	eqeltrd 2834	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀))
121	108, 77	elcarsg 34337	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∪ 𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀) ↔ (∪ 𝑘 ∈ (1...𝑛)𝐴 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀‘𝑒))))
122	120, 121	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∪ 𝑘 ∈ (1...𝑛)𝐴 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀‘𝑒)))
123	122	simprd 495	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀‘𝑒))
124		ineq1 4188	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝐸 → (𝑒 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴))
125	124	fveq2d 6880	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = 𝐸 → (𝑀‘(𝑒 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)))
126		difeq1 4094	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝐸 → (𝑒 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))
127	126	fveq2d 6880	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = 𝐸 → (𝑀‘(𝑒 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)))
128	125, 127	oveq12d 7423	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = 𝐸 → ((𝑀‘(𝑒 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) = ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))))
129		fveq2 6876	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = 𝐸 → (𝑀‘𝑒) = (𝑀‘𝐸))
130	128, 129	eqeq12d 2751	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = 𝐸 → (((𝑀‘(𝑒 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀‘𝑒) ↔ ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀‘𝐸)))
131	130	rspcv 3597	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∈ 𝒫 𝑂 → (∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀‘𝑒) → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀‘𝐸)))
132	78, 123, 131	sylc 65	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀‘𝐸))
133	132	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀‘𝐸))
134		xaddpnf1 13242	. . . . . . . . . . . . . . 15 ⊢ (((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ* ∧ (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞) → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 +∞) = +∞)
135	81, 83, 134	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 +∞) = +∞)
136	135	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 +∞) = +∞)
137	98, 133, 136	3eqtr3d 2778	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀‘𝐸) = +∞)
138		carsgclctunlem2.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀‘𝐸) ≠ +∞)
139	138	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀‘𝐸) ≠ +∞)
140	139	neneqd 2937	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → ¬ (𝑀‘𝐸) = +∞)
141	137, 140	pm2.65da 816	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞)
142	141	neqned 2939	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞)
143		xaddass 13265	. . . . . . . . . 10 ⊢ ((((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ* ∧ (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞) ∧ ((𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ* ∧ (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞) ∧ (-𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ* ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞)) → (((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) = ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 ((𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)))))
144	81, 83, 86, 88, 89, 142, 143	syl222anc 1388	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) = ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 ((𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)))))
145		xnegid 13254	. . . . . . . . . . 11 ⊢ ((𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ* → ((𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) = 0)
146	86, 145	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) = 0)
147	146	oveq2d 7421	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 ((𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)))) = ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 0))
148		xaddrid 13257	. . . . . . . . . 10 ⊢ ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ* → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 0) = (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)))
149	81, 148	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 0) = (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)))
150	144, 147, 149	3eqtrd 2774	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)))
151	132	oveq1d 7420	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) = ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))))
152	107	ineq2d 4195	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 ∩ ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)))
153	152	fveq2d 6880	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 ∩ ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))))
154		mptss 6029	. . . . . . . . . . . . 13 ⊢ ((1...𝑛) ⊆ ℕ → (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ (𝑘 ∈ ℕ ↦ 𝐴))
155		rnss 5919	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ (𝑘 ∈ ℕ ↦ 𝐴) → ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ ran (𝑘 ∈ ℕ ↦ 𝐴))
156	100, 154, 155	mp2b 10	. . . . . . . . . . . 12 ⊢ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ ran (𝑘 ∈ ℕ ↦ 𝐴)
157	156	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ ran (𝑘 ∈ ℕ ↦ 𝐴))
158		disjrnmpt 32566	. . . . . . . . . . . . 13 ⊢ (Disj 𝑘 ∈ ℕ 𝐴 → Disj 𝑦 ∈ ran (𝑘 ∈ ℕ ↦ 𝐴)𝑦)
159	62, 158	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → Disj 𝑦 ∈ ran (𝑘 ∈ ℕ ↦ 𝐴)𝑦)
160	159	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Disj 𝑦 ∈ ran (𝑘 ∈ ℕ ↦ 𝐴)𝑦)
161		disjss1 5092	. . . . . . . . . . 11 ⊢ (ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ ran (𝑘 ∈ ℕ ↦ 𝐴) → (Disj 𝑦 ∈ ran (𝑘 ∈ ℕ ↦ 𝐴)𝑦 → Disj 𝑦 ∈ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)𝑦))
162	157, 160, 161	sylc 65	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Disj 𝑦 ∈ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)𝑦)
163	108, 77, 109, 110, 115, 118, 162, 78	carsgclctunlem1 34349	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))) = Σ*𝑦 ∈ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)(𝑀‘(𝐸 ∩ 𝑦)))
164		ineq2 4189	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝐸 ∩ 𝑦) = (𝐸 ∩ 𝐴))
165	164	fveq2d 6880	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑀‘(𝐸 ∩ 𝑦)) = (𝑀‘(𝐸 ∩ 𝐴)))
166	111	elexi 3482	. . . . . . . . . . 11 ⊢ (1...𝑛) ∈ V
167	166	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ V)
168	99, 102, 22	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
169		inss2 4213	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∩ 𝐴) ⊆ 𝐴
170		sseq2 3985	. . . . . . . . . . . . . . 15 ⊢ (𝐴 = ∅ → ((𝐸 ∩ 𝐴) ⊆ 𝐴 ↔ (𝐸 ∩ 𝐴) ⊆ ∅))
171	169, 170	mpbii 233	. . . . . . . . . . . . . 14 ⊢ (𝐴 = ∅ → (𝐸 ∩ 𝐴) ⊆ ∅)
172		ss0 4377	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∩ 𝐴) ⊆ ∅ → (𝐸 ∩ 𝐴) = ∅)
173	171, 172	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ → (𝐸 ∩ 𝐴) = ∅)
174	173	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝐸 ∩ 𝐴) = ∅)
175	174	fveq2d 6880	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘(𝐸 ∩ 𝐴)) = (𝑀‘∅))
176	109	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘∅) = 0)
177	175, 176	eqtrd 2770	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘(𝐸 ∩ 𝐴)) = 0)
178	62	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Disj 𝑘 ∈ ℕ 𝐴)
179		disjss1 5092	. . . . . . . . . . 11 ⊢ ((1...𝑛) ⊆ ℕ → (Disj 𝑘 ∈ ℕ 𝐴 → Disj 𝑘 ∈ (1...𝑛)𝐴))
180	101, 178, 179	sylc 65	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Disj 𝑘 ∈ (1...𝑛)𝐴)
181	165, 167, 168, 104, 177, 180	esumrnmpt2 34099	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ*𝑦 ∈ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)(𝑀‘(𝐸 ∩ 𝑦)) = Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸 ∩ 𝐴)))
182	153, 163, 181	3eqtrd 2774	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸 ∩ 𝐴)))
183	150, 151, 182	3eqtr3rd 2779	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸 ∩ 𝐴)) = ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))))
184	17	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ (0[,]+∞))
185	3, 184	sselid 3956	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
186	185	xnegcld 13316	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
187	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝐸) ∈ ℝ*)
188		iunss1 4982	. . . . . . . . . . . 12 ⊢ ((1...𝑛) ⊆ ℕ → ∪ 𝑘 ∈ (1...𝑛)𝐴 ⊆ ∪ 𝑘 ∈ ℕ 𝐴)
189	100, 188	mp1i 13	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑘 ∈ (1...𝑛)𝐴 ⊆ ∪ 𝑘 ∈ ℕ 𝐴)
190	189	sscond 4121	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴) ⊆ (𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))
191	74	3adant1r 1178	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))
192	108, 77, 190, 84, 191	carsgmon 34346	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)))
193		xleneg 13234	. . . . . . . . . 10 ⊢ (((𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ* ∧ (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*) → ((𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ↔ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≤ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
194	193	biimpa 476	. . . . . . . . 9 ⊢ ((((𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ* ∧ (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*) ∧ (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≤ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)))
195	185, 86, 192, 194	syl21anc 837	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≤ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)))
196		xleadd2a 13270	. . . . . . . 8 ⊢ (((-𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ* ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ* ∧ (𝑀‘𝐸) ∈ ℝ*) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≤ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) → ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) ≤ ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
197	89, 186, 187, 195, 196	syl31anc 1375	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) ≤ ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
198	183, 197	eqbrtrd 5141	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸 ∩ 𝐴)) ≤ ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
199	76, 22, 198	esumgect 34121	. . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)) ≤ ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
200	12, 27, 20, 71, 199	xrletrd 13178	. . . 4 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ≤ ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
201	2, 200	eqbrtrrid 5155	. . 3 ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) ≤ ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
202		xleadd1a 13269	. . 3 ⊢ ((((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ* ∧ ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) ∈ ℝ* ∧ (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*) ∧ (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) ≤ ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)))) → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) ≤ (((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
203	13, 20, 18, 201, 202	syl31anc 1375	. 2 ⊢ (𝜑 → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) ≤ (((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
204		xrge0npcan 33015	. . 3 ⊢ (((𝑀‘𝐸) ∈ (0[,]+∞) ∧ (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ (0[,]+∞) ∧ (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀‘𝐸)) → (((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) = (𝑀‘𝐸))
205	14, 17, 75, 204	syl3anc 1373	. 2 ⊢ (𝜑 → (((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) = (𝑀‘𝐸))
206	203, 205	breqtrd 5145	1 ⊢ (𝜑 → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) ≤ (𝑀‘𝐸))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  Vcvv 3459   ∖ cdif 3923   ∩ cin 3925   ⊆ wss 3926  ∅c0 4308  𝒫 cpw 4575  ∪ cuni 4883  ∪ ciun 4967  Disj wdisj 5086   class class class wbr 5119   ↦ cmpt 5201  ran crn 5655  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  ωcom 7861   ≼ cdom 8957  Fincfn 8959  0cc0 11129  1c1 11130  +∞cpnf 11266  -∞cmnf 11267  ℝ^*cxr 11268   ≤ cle 11270  ℕcn 12240  -_𝑒cxne 13125   +_𝑒 cxad 13126  [,]cicc 13365  ...cfz 13524  Σ^*cesum 34058  toCaraSigaccarsg 34333

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-inf2 9655  ax-ac2 10477  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-pre-sup 11207  ax-addf 11208  ax-mulf 11209

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-disj 5087  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7671  df-om 7862  df-1st 7988  df-2nd 7989  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8719  df-map 8842  df-pm 8843  df-ixp 8912  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fsupp 9374  df-fi 9423  df-sup 9454  df-inf 9455  df-oi 9524  df-dju 9915  df-card 9953  df-acn 9956  df-ac 10130  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-div 11895  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-z 12589  df-dec 12709  df-uz 12853  df-q 12965  df-rp 13009  df-xneg 13128  df-xadd 13129  df-xmul 13130  df-ioo 13366  df-ioc 13367  df-ico 13368  df-icc 13369  df-fz 13525  df-fzo 13672  df-fl 13809  df-mod 13887  df-seq 14020  df-exp 14080  df-fac 14292  df-bc 14321  df-hash 14349  df-shft 15086  df-cj 15118  df-re 15119  df-im 15120  df-sqrt 15254  df-abs 15255  df-limsup 15487  df-clim 15504  df-rlim 15505  df-sum 15703  df-ef 16083  df-sin 16085  df-cos 16086  df-pi 16088  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-mulr 17285  df-starv 17286  df-sca 17287  df-vsca 17288  df-ip 17289  df-tset 17290  df-ple 17291  df-ds 17293  df-unif 17294  df-hom 17295  df-cco 17296  df-rest 17436  df-topn 17437  df-0g 17455  df-gsum 17456  df-topgen 17457  df-pt 17458  df-prds 17461  df-ordt 17515  df-xrs 17516  df-qtop 17521  df-imas 17522  df-xps 17524  df-mre 17598  df-mrc 17599  df-acs 17601  df-ps 18576  df-tsr 18577  df-plusf 18617  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-mhm 18761  df-submnd 18762  df-grp 18919  df-minusg 18920  df-sbg 18921  df-mulg 19051  df-subg 19106  df-cntz 19300  df-cmn 19763  df-abl 19764  df-mgp 20101  df-rng 20113  df-ur 20142  df-ring 20195  df-cring 20196  df-subrng 20506  df-subrg 20530  df-abv 20769  df-lmod 20819  df-scaf 20820  df-sra 21131  df-rgmod 21132  df-psmet 21307  df-xmet 21308  df-met 21309  df-bl 21310  df-mopn 21311  df-fbas 21312  df-fg 21313  df-cnfld 21316  df-top 22832  df-topon 22849  df-topsp 22871  df-bases 22884  df-cld 22957  df-ntr 22958  df-cls 22959  df-nei 23036  df-lp 23074  df-perf 23075  df-cn 23165  df-cnp 23166  df-haus 23253  df-tx 23500  df-hmeo 23693  df-fil 23784  df-fm 23876  df-flim 23877  df-flf 23878  df-tmd 24010  df-tgp 24011  df-tsms 24065  df-trg 24098  df-xms 24259  df-ms 24260  df-tms 24261  df-nm 24521  df-ngp 24522  df-nrg 24524  df-nlm 24525  df-ii 24821  df-cncf 24822  df-limc 25819  df-dv 25820  df-log 26517  df-esum 34059  df-carsg 34334

This theorem is referenced by:  carsgclctunlem3  34352