carsgclctunlem2 - Mathbox for Thierry Arnoux

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Theorem carsgclctunlem2 33306

Description: Lemma for carsgclctun 33308. (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 5073	. . . . 5 ⊢ ∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴) = (𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)
2	1	fveq2i 6891	. . . 4 ⊢ (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) = (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴))
3		iccssxr 13403	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
4		carsgval.2	. . . . . 6 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))
5		nnex 12214	. . . . . . . 8 ⊢ ℕ ∈ V
6	5	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℕ ∈ V)
7		carsgclctunlem2.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂)
8	7	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐸 ∈ 𝒫 𝑂)
9	8	elpwincl1 31750	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∩ 𝐴) ∈ 𝒫 𝑂)
10	6, 9	elpwiuncl 31752	. . . . . 6 ⊢ (𝜑 → ∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴) ∈ 𝒫 𝑂)
11	4, 10	ffvelcdmd 7084	. . . . 5 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
12	3, 11	sselid 3979	. . . 4 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ∈ ℝ^*)
13	2, 12	eqeltrrid 2838	. . 3 ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ^*)
14	4, 7	ffvelcdmd 7084	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐸) ∈ (0[,]+∞))
15	3, 14	sselid 3979	. . . 4 ⊢ (𝜑 → (𝑀‘𝐸) ∈ ℝ^*)
16	7	elpwdifcl 31751	. . . . . . 7 ⊢ (𝜑 → (𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴) ∈ 𝒫 𝑂)
17	4, 16	ffvelcdmd 7084	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ (0[,]+∞))
18	3, 17	sselid 3979	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ^*)
19	18	xnegcld 13275	. . . 4 ⊢ (𝜑 → -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ^*)
20	15, 19	xaddcld 13276	. . 3 ⊢ (𝜑 → ((𝑀‘𝐸) +_𝑒 -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) ∈ ℝ^*)
21	4	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
22	21, 9	ffvelcdmd 7084	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
23	22	ralrimiva 3146	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
24		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑘ℕ
25	24	esumcl 33016	. . . . . . 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 3979	. . . . 5 ⊢ (𝜑 → Σ^𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)) ∈ ℝ^)
28	9	ralrimiva 3146	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐸 ∩ 𝐴) ∈ 𝒫 𝑂)
29		dfiun3g 5961	. . . . . . . . 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 6892	. . . . . . 7 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) = (𝑀‘∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))))
32		nnct 13942	. . . . . . . . . 10 ⊢ ℕ ≼ ω
33		mptct 10529	. . . . . . . . . 10 ⊢ (ℕ ≼ ω → (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ≼ ω)
34		rnct 10516	. . . . . . . . . 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 2732	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) = (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))
38	37	rnmptss 7118	. . . . . . . . 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 7219	. . . . . . . . . 10 ⊢ (ℕ ∈ V → (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ∈ V)
41		rnexg 7891	. . . . . . . . . 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 5150	. . . . . . . . . . . 12 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → (𝑥 ≼ ω ↔ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ≼ ω))
44		sseq1 4006	. . . . . . . . . . . 12 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → (𝑥 ⊆ 𝒫 𝑂 ↔ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ⊆ 𝒫 𝑂))
45	43, 44	3anbi23d 1439	. . . . . . . . . . 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 4918	. . . . . . . . . . . . 13 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → ∪ 𝑥 = ∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)))
47	46	fveq2d 6892	. . . . . . . . . . . 12 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → (𝑀‘∪ 𝑥) = (𝑀‘∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))))
48		esumeq1 33020	. . . . . . . . . . . 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 5160	. . . . . . . . . . 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 3556	. . . . . . . . 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 1463	. . . . . . 7 ⊢ (𝜑 → (𝑀‘∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))) ≤ Σ^*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))(𝑀‘𝑦))
55	31, 54	eqbrtrd 5169	. . . . . 6 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ≤ Σ^*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))(𝑀‘𝑦))
56		fveq2 6888	. . . . . . 7 ⊢ (𝑦 = (𝐸 ∩ 𝐴) → (𝑀‘𝑦) = (𝑀‘(𝐸 ∩ 𝐴)))
57		simpr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐸 ∩ 𝐴) = ∅) → (𝐸 ∩ 𝐴) = ∅)
58	57	fveq2d 6892	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐸 ∩ 𝐴) = ∅) → (𝑀‘(𝐸 ∩ 𝐴)) = (𝑀‘∅))
59		carsgsiga.1	. . . . . . . . 9 ⊢ (𝜑 → (𝑀‘∅) = 0)
60	59	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐸 ∩ 𝐴) = ∅) → (𝑀‘∅) = 0)
61	58, 60	eqtrd 2772	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐸 ∩ 𝐴) = ∅) → (𝑀‘(𝐸 ∩ 𝐴)) = 0)
62		carsgclctunlem2.1	. . . . . . . . 9 ⊢ (𝜑 → Disj 𝑘 ∈ ℕ 𝐴)
63		disjin 31804	. . . . . . . . 9 ⊢ (Disj 𝑘 ∈ ℕ 𝐴 → Disj 𝑘 ∈ ℕ (𝐴 ∩ 𝐸))
64	62, 63	syl 17	. . . . . . . 8 ⊢ (𝜑 → Disj 𝑘 ∈ ℕ (𝐴 ∩ 𝐸))
65		incom 4200	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝐸) = (𝐸 ∩ 𝐴)
66	65	rgenw 3065	. . . . . . . . 9 ⊢ ∀𝑘 ∈ ℕ (𝐴 ∩ 𝐸) = (𝐸 ∩ 𝐴)
67		disjeq2 5116	. . . . . . . . 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 217	. . . . . . 7 ⊢ (𝜑 → Disj 𝑘 ∈ ℕ (𝐸 ∩ 𝐴))
70	56, 6, 22, 9, 61, 69	esumrnmpt2 33054	. . . . . 6 ⊢ (𝜑 → Σ^𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))(𝑀‘𝑦) = Σ^𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)))
71	55, 70	breqtrd 5173	. . . . 5 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ≤ Σ^*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)))
72		carsgval.1	. . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ 𝑉)
73		difssd 4131	. . . . . . . 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 33301	. . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀‘𝐸))
76	14, 17, 75	xrge0subcld 31963	. . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐸) +_𝑒 -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) ∈ (0[,]+∞))
77	4	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
78	7	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ 𝒫 𝑂)
79	78	elpwincl1 31750	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴) ∈ 𝒫 𝑂)
80	77, 79	ffvelcdmd 7084	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞))
81	3, 80	sselid 3979	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ^*)
82		xrge0neqmnf 13425	. . . . . . . . . . 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 31751	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴) ∈ 𝒫 𝑂)
85	77, 84	ffvelcdmd 7084	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞))
86	3, 85	sselid 3979	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ^*)
87		xrge0neqmnf 13425	. . . . . . . . . . 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 13275	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ^*)
90		xnegneg 13189	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 13182	. . . . . . . . . . . . . . . . 17 ⊢ (-_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞ → -_𝑒-_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -_𝑒-∞)
94	93	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → -_𝑒-_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -_𝑒-∞)
95		xnegmnf 13185	. . . . . . . . . . . . . . . 16 ⊢ -_𝑒-∞ = +∞
96	94, 95	eqtrdi 2788	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → -_𝑒-_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = +∞)
97	92, 96	eqtr3d 2774	. . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
100		fz1ssnn 13528	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1...𝑛) ⊆ ℕ
101	100	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
102	101	sselda 3981	. . . . . . . . . . . . . . . . . . . . 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 3146	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀))
106		dfiun3g 5961	. . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑂 ∈ 𝑉)
109	59	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘∅) = 0)
110	51	3adant1r 1177	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ^*𝑦 ∈ 𝑥(𝑀‘𝑦))
111		fzfi 13933	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1...𝑛) ∈ Fin
112		mptfi 9347	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1...𝑛) ∈ Fin → (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ Fin)
113		rnfi 9331	. . . . . . . . . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (1...𝑛) ↦ 𝐴) = (𝑘 ∈ (1...𝑛) ↦ 𝐴)
117	116	rnmptss 7118	. . . . . . . . . . . . . . . . . . . 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 33303	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ (toCaraSiga‘𝑀))
120	107, 119	eqeltrd 2833	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀))
121	108, 77	elcarsg 33292	. . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀‘𝑒))
124		ineq1 4204	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝐸 → (𝑒 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴))
125	124	fveq2d 6892	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = 𝐸 → (𝑀‘(𝑒 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)))
126		difeq1 4114	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝐸 → (𝑒 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))
127	126	fveq2d 6892	. . . . . . . . . . . . . . . . . 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 6888	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = 𝐸 → (𝑀‘𝑒) = (𝑀‘𝐸))
130	128, 129	eqeq12d 2748	. . . . . . . . . . . . . . . 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 3608	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 13201	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +_𝑒 +∞) = +∞)
137	98, 133, 136	3eqtr3d 2780	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀‘𝐸) = +∞)
138		carsgclctunlem2.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀‘𝐸) ≠ +∞)
139	138	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀‘𝐸) ≠ +∞)
140	139	neneqd 2945	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → ¬ (𝑀‘𝐸) = +∞)
141	137, 140	pm2.65da 815	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞)
142	141	neqned 2947	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞)
143		xaddass 13224	. . . . . . . . . 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 1386	. . . . . . . . 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 13213	. . . . . . . . . . 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 13216	. . . . . . . . . 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 2776	. . . . . . . 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 4211	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 ∩ ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)))
153	152	fveq2d 6892	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 ∩ ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))))
154		mptss 6040	. . . . . . . . . . . . 13 ⊢ ((1...𝑛) ⊆ ℕ → (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ (𝑘 ∈ ℕ ↦ 𝐴))
155		rnss 5936	. . . . . . . . . . . . 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 31803	. . . . . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Disj 𝑦 ∈ ran (𝑘 ∈ ℕ ↦ 𝐴)𝑦)
161		disjss1 5118	. . . . . . . . . . 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 33304	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))) = Σ^*𝑦 ∈ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)(𝑀‘(𝐸 ∩ 𝑦)))
164		ineq2 4205	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝐸 ∩ 𝑦) = (𝐸 ∩ 𝐴))
165	164	fveq2d 6892	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑀‘(𝐸 ∩ 𝑦)) = (𝑀‘(𝐸 ∩ 𝐴)))
166	111	elexi 3493	. . . . . . . . . . 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 4228	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∩ 𝐴) ⊆ 𝐴
170		sseq2 4007	. . . . . . . . . . . . . . 15 ⊢ (𝐴 = ∅ → ((𝐸 ∩ 𝐴) ⊆ 𝐴 ↔ (𝐸 ∩ 𝐴) ⊆ ∅))
171	169, 170	mpbii 232	. . . . . . . . . . . . . 14 ⊢ (𝐴 = ∅ → (𝐸 ∩ 𝐴) ⊆ ∅)
172		ss0 4397	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∩ 𝐴) ⊆ ∅ → (𝐸 ∩ 𝐴) = ∅)
173	171, 172	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ → (𝐸 ∩ 𝐴) = ∅)
174	173	adantl 482	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝐸 ∩ 𝐴) = ∅)
175	174	fveq2d 6892	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘(𝐸 ∩ 𝐴)) = (𝑀‘∅))
176	109	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘∅) = 0)
177	175, 176	eqtrd 2772	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘(𝐸 ∩ 𝐴)) = 0)
178	62	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Disj 𝑘 ∈ ℕ 𝐴)
179		disjss1 5118	. . . . . . . . . . 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 33054	. . . . . . . . 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 2776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = Σ^*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸 ∩ 𝐴)))
183	150, 151, 182	3eqtr3rd 2781	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Σ^*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸 ∩ 𝐴)) = ((𝑀‘𝐸) +_𝑒 -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))))
184	17	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ (0[,]+∞))
185	3, 184	sselid 3979	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ^*)
186	185	xnegcld 13275	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ^*)
187	15	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝐸) ∈ ℝ^*)
188		iunss1 5010	. . . . . . . . . . . 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 4140	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴) ⊆ (𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))
191	74	3adant1r 1177	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))
192	108, 77, 190, 84, 191	carsgmon 33301	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)))
193		xleneg 13193	. . . . . . . . . 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 477	. . . . . . . . 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 836	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≤ -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)))
196		xleadd2a 13229	. . . . . . . 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 1373	. . . . . . 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 5169	. . . . . 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 33076	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)) ≤ ((𝑀‘𝐸) +_𝑒 -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
200	12, 27, 20, 71, 199	xrletrd 13137	. . . 4 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ≤ ((𝑀‘𝐸) +_𝑒 -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
201	2, 200	eqbrtrrid 5183	. . 3 ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) ≤ ((𝑀‘𝐸) +_𝑒 -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
202		xleadd1a 13228	. . 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 1373	. 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 32182	. . 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 1371	. 2 ⊢ (𝜑 → (((𝑀‘𝐸) +_𝑒 -_𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) +_𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) = (𝑀‘𝐸))
206	203, 205	breqtrd 5173	1 ⊢ (𝜑 → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) +_𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) ≤ (𝑀‘𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 Vcvv 3474 ∖ cdif 3944 ∩ cin 3946 ⊆ wss 3947 ∅c0 4321 𝒫 cpw 4601 ∪ cuni 4907 ∪ ciun 4996 Disj wdisj 5112 class class class wbr 5147 ↦ cmpt 5230 ran crn 5676 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ωcom 7851 ≼ cdom 8933 Fincfn 8935 0cc0 11106 1c1 11107 +∞cpnf 11241 -∞cmnf 11242 ℝ^*cxr 11243 ≤ cle 11245 ℕcn 12208 -_𝑒cxne 13085 +_𝑒 cxad 13086 [,]cicc 13323 ...cfz 13480 Σ^*cesum 33013 toCaraSigaccarsg 33288

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-ac2 10454 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-acn 9933 df-ac 10107 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-ordt 17443 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-ps 18515 df-tsr 18516 df-plusf 18556 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-cntz 19175 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-cring 20052 df-subrg 20353 df-abv 20417 df-lmod 20465 df-scaf 20466 df-sra 20777 df-rgmod 20778 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-tmd 23567 df-tgp 23568 df-tsms 23622 df-trg 23655 df-xms 23817 df-ms 23818 df-tms 23819 df-nm 24082 df-ngp 24083 df-nrg 24085 df-nlm 24086 df-ii 24384 df-cncf 24385 df-limc 25374 df-dv 25375 df-log 26056 df-esum 33014 df-carsg 33289

This theorem is referenced by: carsgclctunlem3 33307

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