Theorem carsgclctunlem2 30381

Description: Lemma for carsgclctun 30383. (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 4584	. . . . 5 ⊢ ∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴) = (𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)
2	1	fveq2i 6194	. . . 4 ⊢ (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) = (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴))
3		iccssxr 12256	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ*
4		carsgval.2	. . . . . 6 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))
5		nnex 11026	. . . . . . . 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 29357	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∩ 𝐴) ∈ 𝒫 𝑂)
10	6, 9	elpwiuncl 29359	. . . . . 6 ⊢ (𝜑 → ∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴) ∈ 𝒫 𝑂)
11	4, 10	ffvelrnd 6360	. . . . 5 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
12	3, 11	sseldi 3601	. . . 4 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ∈ ℝ*)
13	2, 12	syl5eqelr 2706	. . 3 ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
14	4, 7	ffvelrnd 6360	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐸) ∈ (0[,]+∞))
15	3, 14	sseldi 3601	. . . 4 ⊢ (𝜑 → (𝑀‘𝐸) ∈ ℝ*)
16	7	elpwdifcl 29358	. . . . . . 7 ⊢ (𝜑 → (𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴) ∈ 𝒫 𝑂)
17	4, 16	ffvelrnd 6360	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ (0[,]+∞))
18	3, 17	sseldi 3601	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
19	18	xnegcld 12130	. . . 4 ⊢ (𝜑 → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
20	15, 19	xaddcld 12131	. . 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	ffvelrnd 6360	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
23	22	ralrimiva 2966	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
24		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑘ℕ
25	24	esumcl 30092	. . . . . . 7 ⊢ ((ℕ ∈ V ∧ ∀𝑘 ∈ ℕ (𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞)) → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
26	6, 23, 25	syl2anc 693	. . . . . 6 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
27	3, 26	sseldi 3601	. . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)) ∈ ℝ*)
28	9	ralrimiva 2966	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐸 ∩ 𝐴) ∈ 𝒫 𝑂)
29		dfiun3g 5378	. . . . . . . . 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 6195	. . . . . . 7 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) = (𝑀‘∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))))
32		nnct 12780	. . . . . . . . . 10 ⊢ ℕ ≼ ω
33		mptct 9360	. . . . . . . . . 10 ⊢ (ℕ ≼ ω → (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ≼ ω)
34		rnct 9347	. . . . . . . . . 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 2622	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) = (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))
38	37	rnmptss 6392	. . . . . . . . 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 6484	. . . . . . . . . 10 ⊢ (ℕ ∈ V → (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ∈ V)
41		rnexg 7098	. . . . . . . . . 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 4656	. . . . . . . . . . . 12 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → (𝑥 ≼ ω ↔ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ≼ ω))
44		sseq1 3626	. . . . . . . . . . . 12 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → (𝑥 ⊆ 𝒫 𝑂 ↔ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) ⊆ 𝒫 𝑂))
45	43, 44	3anbi23d 1402	. . . . . . . . . . 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 4444	. . . . . . . . . . . . 13 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → ∪ 𝑥 = ∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)))
47	46	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑥 = ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴)) → (𝑀‘∪ 𝑥) = (𝑀‘∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))))
48		esumeq1 30096	. . . . . . . . . . . 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 4666	. . . . . . . . . . 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 334	. . . . . . . . . 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 3266	. . . . . . . . 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 1426	. . . . . . 7 ⊢ (𝜑 → (𝑀‘∪ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))(𝑀‘𝑦))
55	31, 54	eqbrtrd 4675	. . . . . 6 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ≤ Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))(𝑀‘𝑦))
56		fveq2 6191	. . . . . . 7 ⊢ (𝑦 = (𝐸 ∩ 𝐴) → (𝑀‘𝑦) = (𝑀‘(𝐸 ∩ 𝐴)))
57		simpr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐸 ∩ 𝐴) = ∅) → (𝐸 ∩ 𝐴) = ∅)
58	57	fveq2d 6195	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐸 ∩ 𝐴) = ∅) → (𝑀‘(𝐸 ∩ 𝐴)) = (𝑀‘∅))
59		carsgsiga.1	. . . . . . . . 9 ⊢ (𝜑 → (𝑀‘∅) = 0)
60	59	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐸 ∩ 𝐴) = ∅) → (𝑀‘∅) = 0)
61	58, 60	eqtrd 2656	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝐸 ∩ 𝐴) = ∅) → (𝑀‘(𝐸 ∩ 𝐴)) = 0)
62		carsgclctunlem2.1	. . . . . . . . 9 ⊢ (𝜑 → Disj 𝑘 ∈ ℕ 𝐴)
63		disjin 29399	. . . . . . . . 9 ⊢ (Disj 𝑘 ∈ ℕ 𝐴 → Disj 𝑘 ∈ ℕ (𝐴 ∩ 𝐸))
64	62, 63	syl 17	. . . . . . . 8 ⊢ (𝜑 → Disj 𝑘 ∈ ℕ (𝐴 ∩ 𝐸))
65		incom 3805	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝐸) = (𝐸 ∩ 𝐴)
66	65	rgenw 2924	. . . . . . . . 9 ⊢ ∀𝑘 ∈ ℕ (𝐴 ∩ 𝐸) = (𝐸 ∩ 𝐴)
67		disjeq2 4624	. . . . . . . . 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 208	. . . . . . 7 ⊢ (𝜑 → Disj 𝑘 ∈ ℕ (𝐸 ∩ 𝐴))
70	56, 6, 22, 9, 61, 69	esumrnmpt2 30130	. . . . . 6 ⊢ (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ ℕ ↦ (𝐸 ∩ 𝐴))(𝑀‘𝑦) = Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)))
71	55, 70	breqtrd 4679	. . . . 5 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ≤ Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)))
72		carsgval.1	. . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ 𝑉)
73		difssd 3738	. . . . . . . 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 30376	. . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀‘𝐸))
76	14, 17, 75	xrge0subcld 29528	. . . . . 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 29357	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴) ∈ 𝒫 𝑂)
80	77, 79	ffvelrnd 6360	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞))
81	3, 80	sseldi 3601	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
82		xrge0neqmnf 12276	. . . . . . . . . . 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 29358	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴) ∈ 𝒫 𝑂)
85	77, 84	ffvelrnd 6360	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ (0[,]+∞))
86	3, 85	sseldi 3601	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
87		xrge0neqmnf 12276	. . . . . . . . . . 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 12130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ∈ ℝ*)
90		xnegneg 12045	. . . . . . . . . . . . . . . . 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 12038	. . . . . . . . . . . . . . . . 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 12041	. . . . . . . . . . . . . . . 16 ⊢ -𝑒-∞ = +∞
96	94, 95	syl6eq 2672	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → -𝑒-𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = +∞)
97	92, 96	eqtr3d 2658	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = +∞)
98	97	oveq2d 6666	. . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝜑)
100		fz1ssnn 12372	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1...𝑛) ⊆ ℕ
101	100	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ⊆ ℕ)
102	101	sselda 3603	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝑘 ∈ ℕ)
103		carsgclctunlem2.2	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (toCaraSiga‘𝑀))
104	99, 102, 103	syl2anc 693	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → 𝐴 ∈ (toCaraSiga‘𝑀))
105	104	ralrimiva 2966	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀))
106		dfiun3g 5378	. . . . . . . . . . . . . . . . . . 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 1319	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))
111		fzfi 12771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1...𝑛) ∈ Fin
112		mptfi 8265	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1...𝑛) ∈ Fin → (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ Fin)
113		rnfi 8249	. . . . . . . . . . . . . . . . . . . . 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 2622	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (1...𝑛) ↦ 𝐴) = (𝑘 ∈ (1...𝑛) ↦ 𝐴)
117	116	rnmptss 6392	. . . . . . . . . . . . . . . . . . . 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 30378	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴) ∈ (toCaraSiga‘𝑀))
120	107, 119	eqeltrd 2701	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∪ 𝑘 ∈ (1...𝑛)𝐴 ∈ (toCaraSiga‘𝑀))
121	108, 77	elcarsg 30367	. . . . . . . . . . . . . . . . 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 222	. . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) +𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))) = (𝑀‘𝑒))
124		ineq1 3807	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝐸 → (𝑒 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴))
125	124	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = 𝐸 → (𝑀‘(𝑒 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)))
126		difeq1 3721	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = 𝐸 → (𝑒 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))
127	126	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 = 𝐸 → (𝑀‘(𝑒 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)))
128	125, 127	oveq12d 6668	. . . . . . . . . . . . . . . . 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 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = 𝐸 → (𝑀‘𝑒) = (𝑀‘𝐸))
130	128, 129	eqeq12d 2637	. . . . . . . . . . . . . . . 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 3305	. . . . . . . . . . . . . . 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 12057	. . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . 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 2664	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀‘𝐸) = +∞)
138		carsgclctunlem2.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀‘𝐸) ≠ +∞)
139	138	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → (𝑀‘𝐸) ≠ +∞)
140	139	neneqd 2799	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞) → ¬ (𝑀‘𝐸) = +∞)
141	137, 140	pm2.65da 600	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = -∞)
142	141	neqned 2801	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≠ -∞)
143		xaddass 12079	. . . . . . . . . 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 1342	. . . . . . . . 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 12069	. . . . . . . . . . 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 6666	. . . . . . . . 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		xaddid1 12072	. . . . . . . . . 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 2660	. . . . . . . 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 6665	. . . . . . . 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 3814	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴) = (𝐸 ∩ ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)))
153	152	fveq2d 6195	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = (𝑀‘(𝐸 ∩ ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))))
154		mptss 5454	. . . . . . . . . . . . 13 ⊢ ((1...𝑛) ⊆ ℕ → (𝑘 ∈ (1...𝑛) ↦ 𝐴) ⊆ (𝑘 ∈ ℕ ↦ 𝐴))
155		rnss 5354	. . . . . . . . . . . . 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 29398	. . . . . . . . . . . . 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 4626	. . . . . . . . . . 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 30379	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴))) = Σ*𝑦 ∈ ran (𝑘 ∈ (1...𝑛) ↦ 𝐴)(𝑀‘(𝐸 ∩ 𝑦)))
164		ineq2 3808	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝐸 ∩ 𝑦) = (𝐸 ∩ 𝐴))
165	164	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑀‘(𝐸 ∩ 𝑦)) = (𝑀‘(𝐸 ∩ 𝐴)))
166	111	elexi 3213	. . . . . . . . . . 11 ⊢ (1...𝑛) ∈ V
167	166	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1...𝑛) ∈ V)
168	99, 102, 22	syl2anc 693	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) → (𝑀‘(𝐸 ∩ 𝐴)) ∈ (0[,]+∞))
169		inss2 3834	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∩ 𝐴) ⊆ 𝐴
170		sseq2 3627	. . . . . . . . . . . . . . 15 ⊢ (𝐴 = ∅ → ((𝐸 ∩ 𝐴) ⊆ 𝐴 ↔ (𝐸 ∩ 𝐴) ⊆ ∅))
171	169, 170	mpbii 223	. . . . . . . . . . . . . 14 ⊢ (𝐴 = ∅ → (𝐸 ∩ 𝐴) ⊆ ∅)
172		ss0 3974	. . . . . . . . . . . . . 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 6195	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘(𝐸 ∩ 𝐴)) = (𝑀‘∅))
176	109	ad2antrr 762	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (1...𝑛)) ∧ 𝐴 = ∅) → (𝑀‘∅) = 0)
177	175, 176	eqtrd 2656	. . . . . . . . . 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 4626	. . . . . . . . . . 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 30130	. . . . . . . . 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 2660	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ (1...𝑛)𝐴)) = Σ*𝑘 ∈ (1...𝑛)(𝑀‘(𝐸 ∩ 𝐴)))
183	150, 151, 182	3eqtr3rd 2665	. . . . . . 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	sseldi 3601	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
186	185	xnegcld 12130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ∈ ℝ*)
187	15	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘𝐸) ∈ ℝ*)
188		iunss1 4532	. . . . . . . . . . . 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 3747	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴) ⊆ (𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴))
191	74	3adant1r 1319	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))
192	108, 77, 190, 84, 191	carsgmon 30376	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)) ≤ (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)))
193		xleneg 12049	. . . . . . . . . 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 501	. . . . . . . . 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 1325	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ (1...𝑛)𝐴)) ≤ -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴)))
196		xleadd2a 12084	. . . . . . . 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 1329	. . . . . . 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 4675	. . . . . 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 30152	. . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ ℕ(𝑀‘(𝐸 ∩ 𝐴)) ≤ ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
200	12, 27, 20, 71, 199	xrletrd 11993	. . . 4 ⊢ (𝜑 → (𝑀‘∪ 𝑘 ∈ ℕ (𝐸 ∩ 𝐴)) ≤ ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
201	2, 200	syl5eqbrr 4689	. . 3 ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) ≤ ((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))))
202		xleadd1a 12083	. . 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 1329	. 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 29694	. . 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 1326	. 2 ⊢ (𝜑 → (((𝑀‘𝐸) +𝑒 -𝑒(𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) = (𝑀‘𝐸))
206	203, 205	breqtrd 4679	1 ⊢ (𝜑 → ((𝑀‘(𝐸 ∩ ∪ 𝑘 ∈ ℕ 𝐴)) +𝑒 (𝑀‘(𝐸 ∖ ∪ 𝑘 ∈ ℕ 𝐴))) ≤ (𝑀‘𝐸))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  ∪ cuni 4436  ∪ ciun 4520  Disj wdisj 4620   class class class wbr 4653   ↦ cmpt 4729  ran crn 5115  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ωcom 7065   ≼ cdom 7953  Fincfn 7955  0cc0 9936  1c1 9937  +∞cpnf 10071  -∞cmnf 10072  ℝ^*cxr 10073   ≤ cle 10075  ℕcn 11020  -_𝑒cxne 11943   +_𝑒 cxad 11944  [,]cicc 12178  ...cfz 12326  Σ^*cesum 30089  toCaraSigaccarsg 30363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-ordt 16161  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-ps 17200  df-tsr 17201  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-abv 18817  df-lmod 18865  df-scaf 18866  df-sra 19172  df-rgmod 19173  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-tmd 21876  df-tgp 21877  df-tsms 21930  df-trg 21963  df-xms 22125  df-ms 22126  df-tms 22127  df-nm 22387  df-ngp 22388  df-nrg 22390  df-nlm 22391  df-ii 22680  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-esum 30090  df-carsg 30364

This theorem is referenced by:  carsgclctunlem3  30382