Theorem carsgclctunlem2 34455

Description: Lemma for carsgclctun 34457. (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 5025	. . . . 5 ⊢ ∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴) = (𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)
2	1	fveq2i 6836	. . . 4 ⊢ (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) = (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴))
3		iccssxr 13348	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ*
4		carsgval.2	. . . . . 6 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))
5		nnex 12153	. . . . . . . 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 32580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∩ 𝐴) ∈ 𝒫 𝑂)
10	6, 9	elpwiuncl 32582	. . . . . 6 ⊢ (𝜑 → ∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴) ∈ 𝒫 𝑂)
11	4, 10	ffvelcdmd 7030	. . . . 5 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
12	3, 11	sselid 3930	. . . 4 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ∈ ℝ*)
13	2, 12	eqeltrrid 2840	. . 3 ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
14	4, 7	ffvelcdmd 7030	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐸) ∈ (0[,]+∞))
15	3, 14	sselid 3930	. . . 4 ⊢ (𝜑 → (𝑀‘𝐸) ∈ ℝ*)
16	7	elpwdifcl 32581	. . . . . . 7 ⊢ (𝜑 → (𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴) ∈ 𝒫 𝑂)
17	4, 16	ffvelcdmd 7030	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ (0[,]+∞))
18	3, 17	sselid 3930	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
19	18	xnegcld 13217	. . . 4 ⊢ (𝜑 → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
20	15, 19	xaddcld 13218	. . 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 7030	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
23	22	ralrimiva 3127	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
24		nfcv 2897	. . . . . . . 8 ⊢ Ⅎ𝑘ℕ
25	24	esumcl 34166	. . . . . . 7 ⊢ ((ℕ ∈ V ∧ ∀𝑘 ∈ ℕ (𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞)) → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
26	6, 23, 25	syl2anc 585	. . . . . 6 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
27	3, 26	sselid 3930	. . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)) ∈ ℝ*)
28	9	ralrimiva 3127	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐸 ∩ 𝐴) ∈ 𝒫 𝑂)
29		dfiun3g 5916	. . . . . . . . 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 6837	. . . . . . 7 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) = (𝑀‘∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))))
32		nnct 13906	. . . . . . . . . 10 ⊢ ℕ ≼ ω
33		mptct 10450	. . . . . . . . . 10 ⊢ (ℕ ≼ ω → (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ≼ ω)
34		rnct 10437	. . . . . . . . . 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 7068	. . . . . . . . 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 7167	. . . . . . . . . 10 ⊢ (ℕ ∈ V → (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ∈ V)
41		rnexg 7844	. . . . . . . . . 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 5100	. . . . . . . . . . . 12 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → (𝑥 ≼ ω ↔ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ≼ ω))
44		sseq1 3958	. . . . . . . . . . . 12 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → (𝑥 ⊆ 𝒫 𝑂 ↔ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ⊆ 𝒫 𝑂))
45	43, 44	3anbi23d 1442	. . . . . . . . . . 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 4873	. . . . . . . . . . . . 13 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → ∪ 𝑥 = ∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)))
47	46	fveq2d 6837	. . . . . . . . . . . 12 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → (𝑀‘∪ 𝑥) = (𝑀‘∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))))
48		esumeq1 34170	. . . . . . . . . . . 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 5110	. . . . . . . . . . 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 3510	. . . . . . . . 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 1466	. . . . . . 7 ⊢ (𝜑 → (𝑀‘∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))(𝑀‘𝑦))
55	31, 54	eqbrtrd 5119	. . . . . 6 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))(𝑀‘𝑦))
56		fveq2 6833	. . . . . . 7 ⊢ (𝑦 = (𝐸 ∩ 𝐴) → (𝑀‘𝑦) = (𝑀‘(𝐸 ∩ 𝐴)))
57		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐸 ∩ 𝐴) = ∅) → (𝐸 ∩ 𝐴) = ∅)
58	57	fveq2d 6837	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐸 ∩ 𝐴) = ∅) → (𝑀‘(𝐸 ∩ 𝐴)) = (𝑀‘∅))
59		carsgsiga.1	. . . . . . . . 9 ⊢ (𝜑 → (𝑀‘∅) = 0)
60	59	ad2antrr 727	. . . . . . . 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 32641	. . . . . . . . 9 ⊢ (Disj 𝑘 ∈ ℕ 𝐴 → Disj 𝑘 ∈ ℕ (𝐴 ∩ 𝐸))
64	62, 63	syl 17	. . . . . . . 8 ⊢ (𝜑 → Disj 𝑘 ∈ ℕ (𝐴 ∩ 𝐸))
65		incom 4160	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝐸) = (𝐸 ∩ 𝐴)
66	65	rgenw 3054	. . . . . . . . 9 ⊢ ∀𝑘 ∈ ℕ (𝐴 ∩ 𝐸) = (𝐸 ∩ 𝐴)
67		disjeq2 5068	. . . . . . . . 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 34204	. . . . . 6 ⊢ (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))(𝑀‘𝑦) = Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)))
71	55, 70	breqtrd 5123	. . . . 5 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ≤ Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)))
72		carsgval.1	. . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ 𝑉)
73		difssd 4088	. . . . . . . 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 34450	. . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀‘𝐸))
76	14, 17, 75	xrge0subcld 32822	. . . . . 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 32580	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴) ∈ 𝒫 𝑂)
80	77, 79	ffvelcdmd 7030	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞))
81	3, 80	sselid 3930	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
82		xrge0neqmnf 13370	. . . . . . . . . . 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 32581	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴) ∈ 𝒫 𝑂)
85	77, 84	ffvelcdmd 7030	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞))
86	3, 85	sselid 3930	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
87		xrge0neqmnf 13370	. . . . . . . . . . 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 13217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
90		xnegneg 13131	. . . . . . . . . . . . . . . . 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 13124	. . . . . . . . . . . . . . . . 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 13127	. . . . . . . . . . . . . . . 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 7374	. . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
100		fz1ssnn 13473	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1...𝑛) ⊆ ℕ
101	100	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
102	101	sselda 3932	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
103		carsgclctunlem2.2	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (toCaraSiga‘𝑀))
104	99, 102, 103	syl2anc 585	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (toCaraSiga‘𝑀))
105	104	ralrimiva 3127	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀))
106		dfiun3g 5916	. . . . . . . . . . . . . . . . . . 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 1179	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))
111		fzfi 13897	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1...𝑛) ∈ Fin
112		mptfi 9253	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1...𝑛) ∈ Fin → (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ Fin)
113		rnfi 9242	. . . . . . . . . . . . . . . . . . . . 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 7068	. . . . . . . . . . . . . . . . . . . 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 34452	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ (toCaraSiga‘𝑀))
120	107, 119	eqeltrd 2835	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀))
121	108, 77	elcarsg 34441	. . . . . . . . . . . . . . . . 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 4164	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝐸 → (𝑒 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴))
125	124	fveq2d 6837	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = 𝐸 → (𝑀‘(𝑒 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)))
126		difeq1 4070	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝐸 → (𝑒 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))
127	126	fveq2d 6837	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = 𝐸 → (𝑀‘(𝑒 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)))
128	125, 127	oveq12d 7376	. . . . . . . . . . . . . . . . 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 6833	. . . . . . . . . . . . . . . . 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 3571	. . . . . . . . . . . . . . 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 13143	. . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . 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 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀‘𝐸) ≠ +∞)
140	139	neneqd 2936	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → ¬ (𝑀‘𝐸) = +∞)
141	137, 140	pm2.65da 817	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞)
142	141	neqned 2938	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞)
143		xaddass 13166	. . . . . . . . . 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 1389	. . . . . . . . 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 13155	. . . . . . . . . . 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 7374	. . . . . . . . 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 13158	. . . . . . . . . 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 7373	. . . . . . . 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 4171	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 ∩ ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)))
153	152	fveq2d 6837	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 ∩ ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))))
154		mptss 6000	. . . . . . . . . . . . 13 ⊢ ((1...𝑛) ⊆ ℕ → (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ (𝑘 ∈ ℕ ↦ 𝐴))
155		rnss 5887	. . . . . . . . . . . . 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 32640	. . . . . . . . . . . . 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 5070	. . . . . . . . . . 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 34453	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))) = Σ*𝑦 ∈ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)(𝑀‘(𝐸 ∩ 𝑦)))
164		ineq2 4165	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝐸 ∩ 𝑦) = (𝐸 ∩ 𝐴))
165	164	fveq2d 6837	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑀‘(𝐸 ∩ 𝑦)) = (𝑀‘(𝐸 ∩ 𝐴)))
166	111	elexi 3462	. . . . . . . . . . 11 ⊢ (1...𝑛) ∈ V
167	166	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ V)
168	99, 102, 22	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
169		inss2 4189	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∩ 𝐴) ⊆ 𝐴
170		sseq2 3959	. . . . . . . . . . . . . . 15 ⊢ (𝐴 = ∅ → ((𝐸 ∩ 𝐴) ⊆ 𝐴 ↔ (𝐸 ∩ 𝐴) ⊆ ∅))
171	169, 170	mpbii 233	. . . . . . . . . . . . . 14 ⊢ (𝐴 = ∅ → (𝐸 ∩ 𝐴) ⊆ ∅)
172		ss0 4353	. . . . . . . . . . . . . 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 6837	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘(𝐸 ∩ 𝐴)) = (𝑀‘∅))
176	109	ad2antrr 727	. . . . . . . . . . 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 5070	. . . . . . . . . . 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 34204	. . . . . . . . 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 3930	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
186	185	xnegcld 13217	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
187	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝐸) ∈ ℝ*)
188		iunss1 4960	. . . . . . . . . . . 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 4097	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴) ⊆ (𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))
191	74	3adant1r 1179	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))
192	108, 77, 190, 84, 191	carsgmon 34450	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)))
193		xleneg 13135	. . . . . . . . . 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 838	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≤ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)))
196		xleadd2a 13171	. . . . . . . 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 1376	. . . . . . 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 5119	. . . . . 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 34226	. . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)) ≤ ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
200	12, 27, 20, 71, 199	xrletrd 13078	. . . 4 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ≤ ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
201	2, 200	eqbrtrrid 5133	. . 3 ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) ≤ ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
202		xleadd1a 13170	. . 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 1376	. 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 33081	. . 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 1374	. 2 ⊢ (𝜑 → (((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) = (𝑀‘𝐸))
206	203, 205	breqtrd 5123	1 ⊢ (𝜑 → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) ≤ (𝑀‘𝐸))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2931  ∀wral 3050  Vcvv 3439   ∖ cdif 3897   ∩ cin 3899   ⊆ wss 3900  ∅c0 4284  𝒫 cpw 4553  ∪ cuni 4862  ∪ ciun 4945  Disj wdisj 5064   class class class wbr 5097   ↦ cmpt 5178  ran crn 5624  ⟶wf 6487  ‘cfv 6491  (class class class)co 7358  ωcom 7808   ≼ cdom 8883  Fincfn 8885  0cc0 11028  1c1 11029  +∞cpnf 11165  -∞cmnf 11166  ℝ^*cxr 11167   ≤ cle 11169  ℕcn 12147  -_𝑒cxne 13025   +_𝑒 cxad 13026  [,]cicc 13266  ...cfz 13425  Σ^*cesum 34163  toCaraSigaccarsg 34437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-inf2 9552  ax-ac2 10375  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 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-iin 4948  df-disj 5065  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  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 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-isom 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8767  df-pm 8768  df-ixp 8838  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-fsupp 9267  df-fi 9316  df-sup 9347  df-inf 9348  df-oi 9417  df-dju 9815  df-card 9853  df-acn 9856  df-ac 10028  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-div 11797  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-n0 12404  df-z 12491  df-dec 12610  df-uz 12754  df-q 12864  df-rp 12908  df-xneg 13028  df-xadd 13029  df-xmul 13030  df-ioo 13267  df-ioc 13268  df-ico 13269  df-icc 13270  df-fz 13426  df-fzo 13573  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 17424  df-xrs 17425  df-qtop 17430  df-imas 17431  df-xps 17433  df-mre 17507  df-mrc 17508  df-acs 17510  df-ps 18491  df-tsr 18492  df-plusf 18566  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18710  df-submnd 18711  df-grp 18868  df-minusg 18869  df-sbg 18870  df-mulg 19000  df-subg 19055  df-cntz 19248  df-cmn 19713  df-abl 19714  df-mgp 20078  df-rng 20090  df-ur 20119  df-ring 20172  df-cring 20173  df-subrng 20481  df-subrg 20505  df-abv 20744  df-lmod 20815  df-scaf 20816  df-sra 21127  df-rgmod 21128  df-psmet 21303  df-xmet 21304  df-met 21305  df-bl 21306  df-mopn 21307  df-fbas 21308  df-fg 21309  df-cnfld 21312  df-top 22840  df-topon 22857  df-topsp 22879  df-bases 22892  df-cld 22965  df-ntr 22966  df-cls 22967  df-nei 23044  df-lp 23082  df-perf 23083  df-cn 23173  df-cnp 23174  df-haus 23261  df-tx 23508  df-hmeo 23701  df-fil 23792  df-fm 23884  df-flim 23885  df-flf 23886  df-tmd 24018  df-tgp 24019  df-tsms 24073  df-trg 24106  df-xms 24266  df-ms 24267  df-tms 24268  df-nm 24528  df-ngp 24529  df-nrg 24531  df-nlm 24532  df-ii 24828  df-cncf 24829  df-limc 25825  df-dv 25826  df-log 26523  df-esum 34164  df-carsg 34438

This theorem is referenced by:  carsgclctunlem3  34456