Theorem carsgclctunlem2 34303

Description: Lemma for carsgclctun 34305. (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 5020	. . . . 5 ⊢ ∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴) = (𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)
2	1	fveq2i 6825	. . . 4 ⊢ (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) = (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴))
3		iccssxr 13333	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ*
4		carsgval.2	. . . . . 6 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))
5		nnex 12134	. . . . . . . 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 32474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∩ 𝐴) ∈ 𝒫 𝑂)
10	6, 9	elpwiuncl 32476	. . . . . 6 ⊢ (𝜑 → ∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴) ∈ 𝒫 𝑂)
11	4, 10	ffvelcdmd 7019	. . . . 5 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
12	3, 11	sselid 3933	. . . 4 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ∈ ℝ*)
13	2, 12	eqeltrrid 2833	. . 3 ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
14	4, 7	ffvelcdmd 7019	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐸) ∈ (0[,]+∞))
15	3, 14	sselid 3933	. . . 4 ⊢ (𝜑 → (𝑀‘𝐸) ∈ ℝ*)
16	7	elpwdifcl 32475	. . . . . . 7 ⊢ (𝜑 → (𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴) ∈ 𝒫 𝑂)
17	4, 16	ffvelcdmd 7019	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ (0[,]+∞))
18	3, 17	sselid 3933	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
19	18	xnegcld 13202	. . . 4 ⊢ (𝜑 → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
20	15, 19	xaddcld 13203	. . 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 7019	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
23	22	ralrimiva 3121	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
24		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑘ℕ
25	24	esumcl 34013	. . . . . . 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 3933	. . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)) ∈ ℝ*)
28	9	ralrimiva 3121	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐸 ∩ 𝐴) ∈ 𝒫 𝑂)
29		dfiun3g 5909	. . . . . . . . 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 6826	. . . . . . 7 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) = (𝑀‘∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))))
32		nnct 13888	. . . . . . . . . 10 ⊢ ℕ ≼ ω
33		mptct 10432	. . . . . . . . . 10 ⊢ (ℕ ≼ ω → (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ≼ ω)
34		rnct 10419	. . . . . . . . . 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 2729	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) = (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))
38	37	rnmptss 7057	. . . . . . . . 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 7157	. . . . . . . . . 10 ⊢ (ℕ ∈ V → (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ∈ V)
41		rnexg 7835	. . . . . . . . . 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 5095	. . . . . . . . . . . 12 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → (𝑥 ≼ ω ↔ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ≼ ω))
44		sseq1 3961	. . . . . . . . . . . 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 4869	. . . . . . . . . . . . 13 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → ∪ 𝑥 = ∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)))
47	46	fveq2d 6826	. . . . . . . . . . . 12 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → (𝑀‘∪ 𝑥) = (𝑀‘∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))))
48		esumeq1 34017	. . . . . . . . . . . 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 5105	. . . . . . . . . . 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 3509	. . . . . . . . 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 5114	. . . . . 6 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))(𝑀‘𝑦))
56		fveq2 6822	. . . . . . 7 ⊢ (𝑦 = (𝐸 ∩ 𝐴) → (𝑀‘𝑦) = (𝑀‘(𝐸 ∩ 𝐴)))
57		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐸 ∩ 𝐴) = ∅) → (𝐸 ∩ 𝐴) = ∅)
58	57	fveq2d 6826	. . . . . . . 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 2764	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐸 ∩ 𝐴) = ∅) → (𝑀‘(𝐸 ∩ 𝐴)) = 0)
62		carsgclctunlem2.1	. . . . . . . . 9 ⊢ (𝜑 → Disj 𝑘 ∈ ℕ 𝐴)
63		disjin 32535	. . . . . . . . 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 3048	. . . . . . . . 9 ⊢ ∀𝑘 ∈ ℕ (𝐴 ∩ 𝐸) = (𝐸 ∩ 𝐴)
67		disjeq2 5063	. . . . . . . . 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 34051	. . . . . 6 ⊢ (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))(𝑀‘𝑦) = Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)))
71	55, 70	breqtrd 5118	. . . . 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 34298	. . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀‘𝐸))
76	14, 17, 75	xrge0subcld 32715	. . . . . 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 32474	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴) ∈ 𝒫 𝑂)
80	77, 79	ffvelcdmd 7019	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞))
81	3, 80	sselid 3933	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
82		xrge0neqmnf 13355	. . . . . . . . . . 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 32475	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴) ∈ 𝒫 𝑂)
85	77, 84	ffvelcdmd 7019	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞))
86	3, 85	sselid 3933	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
87		xrge0neqmnf 13355	. . . . . . . . . . 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 13202	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
90		xnegneg 13116	. . . . . . . . . . . . . . . . 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 13109	. . . . . . . . . . . . . . . . 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 13112	. . . . . . . . . . . . . . . 16 ⊢ -𝑒-∞ = +∞
96	94, 95	eqtrdi 2780	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → -𝑒-𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = +∞)
97	92, 96	eqtr3d 2766	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = +∞)
98	97	oveq2d 7365	. . . . . . . . . . . . 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 13458	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1...𝑛) ⊆ ℕ
101	100	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
102	101	sselda 3935	. . . . . . . . . . . . . . . . . . . . 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 3121	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀))
106		dfiun3g 5909	. . . . . . . . . . . . . . . . . . 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 13879	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1...𝑛) ∈ Fin
112		mptfi 9241	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1...𝑛) ∈ Fin → (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ Fin)
113		rnfi 9230	. . . . . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (1...𝑛) ↦ 𝐴) = (𝑘 ∈ (1...𝑛) ↦ 𝐴)
117	116	rnmptss 7057	. . . . . . . . . . . . . . . . . . . 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 34300	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ (toCaraSiga‘𝑀))
120	107, 119	eqeltrd 2828	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀))
121	108, 77	elcarsg 34289	. . . . . . . . . . . . . . . . 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 6826	. . . . . . . . . . . . . . . . . 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 6826	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = 𝐸 → (𝑀‘(𝑒 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)))
128	125, 127	oveq12d 7367	. . . . . . . . . . . . . . . . 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 6822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = 𝐸 → (𝑀‘𝑒) = (𝑀‘𝐸))
130	128, 129	eqeq12d 2745	. . . . . . . . . . . . . . . 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 3573	. . . . . . . . . . . . . . 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 13128	. . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . 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 2930	. . . . . . . . . . . 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 2932	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞)
143		xaddass 13151	. . . . . . . . . 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 13140	. . . . . . . . . . 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 7365	. . . . . . . . 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 13143	. . . . . . . . . 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 2768	. . . . . . . 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 7364	. . . . . . . 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 6826	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 ∩ ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))))
154		mptss 5993	. . . . . . . . . . . . 13 ⊢ ((1...𝑛) ⊆ ℕ → (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ (𝑘 ∈ ℕ ↦ 𝐴))
155		rnss 5881	. . . . . . . . . . . . 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 32534	. . . . . . . . . . . . 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 5065	. . . . . . . . . . 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 34301	. . . . . . . . 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 6826	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑀‘(𝐸 ∩ 𝑦)) = (𝑀‘(𝐸 ∩ 𝐴)))
166	111	elexi 3459	. . . . . . . . . . 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 4189	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∩ 𝐴) ⊆ 𝐴
170		sseq2 3962	. . . . . . . . . . . . . . 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 6826	. . . . . . . . . . 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 2764	. . . . . . . . . 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 5065	. . . . . . . . . . 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 34051	. . . . . . . . 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 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸 ∩ 𝐴)))
183	150, 151, 182	3eqtr3rd 2773	. . . . . . 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 3933	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
186	185	xnegcld 13202	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
187	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝐸) ∈ ℝ*)
188		iunss1 4956	. . . . . . . . . . . 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 1178	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))
192	108, 77, 190, 84, 191	carsgmon 34298	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)))
193		xleneg 13120	. . . . . . . . . 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 13156	. . . . . . . 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 5114	. . . . . 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 34073	. . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)) ≤ ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
200	12, 27, 20, 71, 199	xrletrd 13064	. . . 4 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ≤ ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
201	2, 200	eqbrtrrid 5128	. . 3 ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) ≤ ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
202		xleadd1a 13155	. . 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 32983	. . 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 5118	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 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3436   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  ∅c0 4284  𝒫 cpw 4551  ∪ cuni 4858  ∪ ciun 4941  Disj wdisj 5059   class class class wbr 5092   ↦ cmpt 5173  ran crn 5620  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  ωcom 7799   ≼ cdom 8870  Fincfn 8872  0cc0 11009  1c1 11010  +∞cpnf 11146  -∞cmnf 11147  ℝ^*cxr 11148   ≤ cle 11150  ℕcn 12128  -_𝑒cxne 13011   +_𝑒 cxad 13012  [,]cicc 13251  ...cfz 13410  Σ^*cesum 34010  toCaraSigaccarsg 34285

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537  ax-ac2 10357  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087  ax-addf 11088  ax-mulf 11089

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-disj 5060  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-supp 8094  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-pm 8756  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fsupp 9252  df-fi 9301  df-sup 9332  df-inf 9333  df-oi 9402  df-dju 9797  df-card 9835  df-acn 9838  df-ac 10010  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-q 12850  df-rp 12894  df-xneg 13014  df-xadd 13015  df-xmul 13016  df-ioo 13252  df-ioc 13253  df-ico 13254  df-icc 13255  df-fz 13411  df-fzo 13558  df-fl 13696  df-mod 13774  df-seq 13909  df-exp 13969  df-fac 14181  df-bc 14210  df-hash 14238  df-shft 14974  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-limsup 15378  df-clim 15395  df-rlim 15396  df-sum 15594  df-ef 15974  df-sin 15976  df-cos 15977  df-pi 15979  df-struct 17058  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-plusg 17174  df-mulr 17175  df-starv 17176  df-sca 17177  df-vsca 17178  df-ip 17179  df-tset 17180  df-ple 17181  df-ds 17183  df-unif 17184  df-hom 17185  df-cco 17186  df-rest 17326  df-topn 17327  df-0g 17345  df-gsum 17346  df-topgen 17347  df-pt 17348  df-prds 17351  df-ordt 17405  df-xrs 17406  df-qtop 17411  df-imas 17412  df-xps 17414  df-mre 17488  df-mrc 17489  df-acs 17491  df-ps 18472  df-tsr 18473  df-plusf 18513  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-mhm 18657  df-submnd 18658  df-grp 18815  df-minusg 18816  df-sbg 18817  df-mulg 18947  df-subg 19002  df-cntz 19196  df-cmn 19661  df-abl 19662  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-cring 20121  df-subrng 20431  df-subrg 20455  df-abv 20694  df-lmod 20765  df-scaf 20766  df-sra 21077  df-rgmod 21078  df-psmet 21253  df-xmet 21254  df-met 21255  df-bl 21256  df-mopn 21257  df-fbas 21258  df-fg 21259  df-cnfld 21262  df-top 22779  df-topon 22796  df-topsp 22818  df-bases 22831  df-cld 22904  df-ntr 22905  df-cls 22906  df-nei 22983  df-lp 23021  df-perf 23022  df-cn 23112  df-cnp 23113  df-haus 23200  df-tx 23447  df-hmeo 23640  df-fil 23731  df-fm 23823  df-flim 23824  df-flf 23825  df-tmd 23957  df-tgp 23958  df-tsms 24012  df-trg 24045  df-xms 24206  df-ms 24207  df-tms 24208  df-nm 24468  df-ngp 24469  df-nrg 24471  df-nlm 24472  df-ii 24768  df-cncf 24769  df-limc 25765  df-dv 25766  df-log 26463  df-esum 34011  df-carsg 34286

This theorem is referenced by:  carsgclctunlem3  34304