carsgclctunlem1 - Mathbox for Thierry Arnoux

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Theorem carsgclctunlem1 29706

Description: Lemma for carsgclctun 29710. (Contributed by Thierry Arnoux, 23-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))
fiunelcarsg.1	⊢ (𝜑 → 𝐴 ∈ Fin)
fiunelcarsg.2	⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))
carsgclctunlem1.1	⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦)
carsgclctunlem1.2	⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂)

**Assertion**
Ref	Expression
carsgclctunlem1	⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝐴)) = Σ^*𝑦 ∈ 𝐴(𝑀‘(𝐸 ∩ 𝑦)))

Distinct variable groups: 𝑥,𝐴,𝑦 𝑥,𝐸,𝑦 𝑥,𝑀,𝑦 𝑥,𝑂,𝑦 𝜑,𝑥,𝑦 Allowed substitution hints: 𝑉(𝑥,𝑦)

Proof of Theorem carsgclctunlem1 Dummy variables 𝑎 𝑒 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 4380	. . . . 5 ⊢ (𝑎 = ∅ → ∪ 𝑎 = ∪ ∅)
2	1	ineq2d 3776	. . . 4 ⊢ (𝑎 = ∅ → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ ∅))
3	2	fveq2d 6107	. . 3 ⊢ (𝑎 = ∅ → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ ∅)))
4		esumeq1 29423	. . 3 ⊢ (𝑎 = ∅ → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ ∅(𝑀‘(𝐸 ∩ 𝑦)))
5	3, 4	eqeq12d 2625	. 2 ⊢ (𝑎 = ∅ → ((𝑀‘(𝐸 ∩ ∪ 𝑎)) = Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) ↔ (𝑀‘(𝐸 ∩ ∪ ∅)) = Σ^𝑦 ∈ ∅(𝑀‘(𝐸 ∩ 𝑦))))
6		unieq 4380	. . . . 5 ⊢ (𝑎 = 𝑏 → ∪ 𝑎 = ∪ 𝑏)
7	6	ineq2d 3776	. . . 4 ⊢ (𝑎 = 𝑏 → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ 𝑏))
8	7	fveq2d 6107	. . 3 ⊢ (𝑎 = 𝑏 → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ 𝑏)))
9		esumeq1 29423	. . 3 ⊢ (𝑎 = 𝑏 → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦)))
10	8, 9	eqeq12d 2625	. 2 ⊢ (𝑎 = 𝑏 → ((𝑀‘(𝐸 ∩ ∪ 𝑎)) = Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) ↔ (𝑀‘(𝐸 ∩ ∪ 𝑏)) = Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦))))
11		unieq 4380	. . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → ∪ 𝑎 = ∪ (𝑏 ∪ {𝑥}))
12	11	ineq2d 3776	. . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ (𝑏 ∪ {𝑥})))
13	12	fveq2d 6107	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ (𝑏 ∪ {𝑥}))))
14		esumeq1 29423	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ (𝑏 ∪ {𝑥})(𝑀‘(𝐸 ∩ 𝑦)))
15	13, 14	eqeq12d 2625	. 2 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → ((𝑀‘(𝐸 ∩ ∪ 𝑎)) = Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) ↔ (𝑀‘(𝐸 ∩ ∪ (𝑏 ∪ {𝑥}))) = Σ^𝑦 ∈ (𝑏 ∪ {𝑥})(𝑀‘(𝐸 ∩ 𝑦))))
16		unieq 4380	. . . . 5 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴)
17	16	ineq2d 3776	. . . 4 ⊢ (𝑎 = 𝐴 → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ 𝐴))
18	17	fveq2d 6107	. . 3 ⊢ (𝑎 = 𝐴 → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ 𝐴)))
19		esumeq1 29423	. . 3 ⊢ (𝑎 = 𝐴 → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ 𝐴(𝑀‘(𝐸 ∩ 𝑦)))
20	18, 19	eqeq12d 2625	. 2 ⊢ (𝑎 = 𝐴 → ((𝑀‘(𝐸 ∩ ∪ 𝑎)) = Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) ↔ (𝑀‘(𝐸 ∩ ∪ 𝐴)) = Σ^𝑦 ∈ 𝐴(𝑀‘(𝐸 ∩ 𝑦))))
21		carsgsiga.1	. . 3 ⊢ (𝜑 → (𝑀‘∅) = 0)
22		uni0 4401	. . . . . 6 ⊢ ∪ ∅ = ∅
23	22	ineq2i 3773	. . . . 5 ⊢ (𝐸 ∩ ∪ ∅) = (𝐸 ∩ ∅)
24		in0 3920	. . . . 5 ⊢ (𝐸 ∩ ∅) = ∅
25	23, 24	eqtri 2632	. . . 4 ⊢ (𝐸 ∩ ∪ ∅) = ∅
26	25	fveq2i 6106	. . 3 ⊢ (𝑀‘(𝐸 ∩ ∪ ∅)) = (𝑀‘∅)
27		esumnul 29437	. . 3 ⊢ Σ^*𝑦 ∈ ∅(𝑀‘(𝐸 ∩ 𝑦)) = 0
28	21, 26, 27	3eqtr4g 2669	. 2 ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ ∅)) = Σ^*𝑦 ∈ ∅(𝑀‘(𝐸 ∩ 𝑦)))
29		simpr 476	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ (𝑀‘(𝐸 ∩ ∪ 𝑏)) = Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦))) → (𝑀‘(𝐸 ∩ ∪ 𝑏)) = Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦)))
30	29	eqcomd 2616	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ (𝑀‘(𝐸 ∩ ∪ 𝑏)) = Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦))) → Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦)) = (𝑀‘(𝐸 ∩ ∪ 𝑏)))
31		simpr 476	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
32	31	ineq2d 3776	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 = 𝑥) → (𝐸 ∩ 𝑦) = (𝐸 ∩ 𝑥))
33	32	fveq2d 6107	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 = 𝑥) → (𝑀‘(𝐸 ∩ 𝑦)) = (𝑀‘(𝐸 ∩ 𝑥)))
34		simprr 792	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → 𝑥 ∈ (𝐴 ∖ 𝑏))
35		carsgval.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))
36	35	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
37		carsgclctunlem1.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂)
38	37	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → 𝐸 ∈ 𝒫 𝑂)
39	38	elpwincl1 28741	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝐸 ∩ 𝑥) ∈ 𝒫 𝑂)
40	36, 39	ffvelrnd 6268	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑀‘(𝐸 ∩ 𝑥)) ∈ (0[,]+∞))
41	33, 34, 40	esumsn 29454	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → Σ^*𝑦 ∈ {𝑥} (𝑀‘(𝐸 ∩ 𝑦)) = (𝑀‘(𝐸 ∩ 𝑥)))
42	41	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ (𝑀‘(𝐸 ∩ ∪ 𝑏)) = Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦))) → Σ^𝑦 ∈ {𝑥} (𝑀‘(𝐸 ∩ 𝑦)) = (𝑀‘(𝐸 ∩ 𝑥)))
43	30, 42	oveq12d 6567	. . . 4 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ (𝑀‘(𝐸 ∩ ∪ 𝑏)) = Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦))) → (Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦)) +_𝑒 Σ^*𝑦 ∈ {𝑥} (𝑀‘(𝐸 ∩ 𝑦))) = ((𝑀‘(𝐸 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝐸 ∩ 𝑥))))
44		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑦(𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏)))
45		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑦𝑏
46		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑦{𝑥}
47		vex 3176	. . . . . . 7 ⊢ 𝑏 ∈ V
48	47	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → 𝑏 ∈ V)
49		snex 4835	. . . . . . 7 ⊢ {𝑥} ∈ V
50	49	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → {𝑥} ∈ V)
51	34	eldifbd 3553	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ¬ 𝑥 ∈ 𝑏)
52		disjsn 4192	. . . . . . 7 ⊢ ((𝑏 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ 𝑏)
53	51, 52	sylibr 223	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑏 ∩ {𝑥}) = ∅)
54	35	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
55	37	ad2antrr 758	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → 𝐸 ∈ 𝒫 𝑂)
56	55	elpwincl1 28741	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → (𝐸 ∩ 𝑦) ∈ 𝒫 𝑂)
57	54, 56	ffvelrnd 6268	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → (𝑀‘(𝐸 ∩ 𝑦)) ∈ (0[,]+∞))
58	35	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
59	37	ad2antrr 758	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → 𝐸 ∈ 𝒫 𝑂)
60	59	elpwincl1 28741	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → (𝐸 ∩ 𝑦) ∈ 𝒫 𝑂)
61	58, 60	ffvelrnd 6268	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → (𝑀‘(𝐸 ∩ 𝑦)) ∈ (0[,]+∞))
62	44, 45, 46, 48, 50, 53, 57, 61	esumsplit 29442	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → Σ^𝑦 ∈ (𝑏 ∪ {𝑥})(𝑀‘(𝐸 ∩ 𝑦)) = (Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦)) +_𝑒 Σ^*𝑦 ∈ {𝑥} (𝑀‘(𝐸 ∩ 𝑦))))
63	62	adantr 480	. . . 4 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ (𝑀‘(𝐸 ∩ ∪ 𝑏)) = Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦))) → Σ^𝑦 ∈ (𝑏 ∪ {𝑥})(𝑀‘(𝐸 ∩ 𝑦)) = (Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦)) +_𝑒 Σ^𝑦 ∈ {𝑥} (𝑀‘(𝐸 ∩ 𝑦))))
64		inass 3785	. . . . . . . . . 10 ⊢ ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏) = (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∩ ∪ 𝑏))
65		indir 3834	. . . . . . . . . . . 12 ⊢ ((∪ 𝑏 ∪ 𝑥) ∩ ∪ 𝑏) = ((∪ 𝑏 ∩ ∪ 𝑏) ∪ (𝑥 ∩ ∪ 𝑏))
66		inidm 3784	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑏 ∩ ∪ 𝑏) = ∪ 𝑏
67	66	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (∪ 𝑏 ∩ ∪ 𝑏) = ∪ 𝑏)
68		incom 3767	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑏 ∩ 𝑥) = (𝑥 ∩ ∪ 𝑏)
69		carsgclctunlem1.1	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦)
70	69	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → Disj 𝑦 ∈ 𝐴 𝑦)
71		simpr 476	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑏 ⊆ 𝐴)
72	71	adantrr 749	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → 𝑏 ⊆ 𝐴)
73	70, 72, 34	disjuniel 28792	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (∪ 𝑏 ∩ 𝑥) = ∅)
74	68, 73	syl5eqr 2658	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑥 ∩ ∪ 𝑏) = ∅)
75	67, 74	uneq12d 3730	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((∪ 𝑏 ∩ ∪ 𝑏) ∪ (𝑥 ∩ ∪ 𝑏)) = (∪ 𝑏 ∪ ∅))
76		un0 3919	. . . . . . . . . . . . 13 ⊢ (∪ 𝑏 ∪ ∅) = ∪ 𝑏
77	75, 76	syl6eq 2660	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((∪ 𝑏 ∩ ∪ 𝑏) ∪ (𝑥 ∩ ∪ 𝑏)) = ∪ 𝑏)
78	65, 77	syl5eq 2656	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((∪ 𝑏 ∪ 𝑥) ∩ ∪ 𝑏) = ∪ 𝑏)
79	78	ineq2d 3776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∩ ∪ 𝑏)) = (𝐸 ∩ ∪ 𝑏))
80	64, 79	syl5eq 2656	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏) = (𝐸 ∩ ∪ 𝑏))
81	80	fveq2d 6107	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏)) = (𝑀‘(𝐸 ∩ ∪ 𝑏)))
82		indif2 3829	. . . . . . . . . 10 ⊢ (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∖ ∪ 𝑏)) = ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏)
83		uncom 3719	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑏 ∪ 𝑥) = (𝑥 ∪ ∪ 𝑏)
84	83	difeq1i 3686	. . . . . . . . . . . . 13 ⊢ ((∪ 𝑏 ∪ 𝑥) ∖ ∪ 𝑏) = ((𝑥 ∪ ∪ 𝑏) ∖ ∪ 𝑏)
85		disj3 3973	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∩ ∪ 𝑏) = ∅ ↔ 𝑥 = (𝑥 ∖ ∪ 𝑏))
86	85	biimpi 205	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∩ ∪ 𝑏) = ∅ → 𝑥 = (𝑥 ∖ ∪ 𝑏))
87		difun2 4000	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∪ ∪ 𝑏) ∖ ∪ 𝑏) = (𝑥 ∖ ∪ 𝑏)
88	86, 87	syl6reqr 2663	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∩ ∪ 𝑏) = ∅ → ((𝑥 ∪ ∪ 𝑏) ∖ ∪ 𝑏) = 𝑥)
89	84, 88	syl5eq 2656	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ ∪ 𝑏) = ∅ → ((∪ 𝑏 ∪ 𝑥) ∖ ∪ 𝑏) = 𝑥)
90	74, 89	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((∪ 𝑏 ∪ 𝑥) ∖ ∪ 𝑏) = 𝑥)
91	90	ineq2d 3776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∖ ∪ 𝑏)) = (𝐸 ∩ 𝑥))
92	82, 91	syl5eqr 2658	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏) = (𝐸 ∩ 𝑥))
93	92	fveq2d 6107	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏)) = (𝑀‘(𝐸 ∩ 𝑥)))
94	81, 93	oveq12d 6567	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏)) +_𝑒 (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏))) = ((𝑀‘(𝐸 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝐸 ∩ 𝑥))))
95		carsgval.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑂 ∈ 𝑉)
96	95	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑂 ∈ 𝑉)
97	35	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
98	21	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → (𝑀‘∅) = 0)
99		carsgsiga.2	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ^*𝑦 ∈ 𝑥(𝑀‘𝑦))
100	99	3adant1r 1311	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ^*𝑦 ∈ 𝑥(𝑀‘𝑦))
101		fiunelcarsg.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ Fin)
102		ssfi 8065	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑏 ⊆ 𝐴) → 𝑏 ∈ Fin)
103	101, 102	sylan 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑏 ∈ Fin)
104		fiunelcarsg.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))
105	104	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝐴 ⊆ (toCaraSiga‘𝑀))
106	71, 105	sstrd 3578	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑏 ⊆ (toCaraSiga‘𝑀))
107	96, 97, 98, 100, 103, 106	fiunelcarsg 29705	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → ∪ 𝑏 ∈ (toCaraSiga‘𝑀))
108	95, 35	elcarsg 29694	. . . . . . . . . . . 12 ⊢ (𝜑 → (∪ 𝑏 ∈ (toCaraSiga‘𝑀) ↔ (∪ 𝑏 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒))))
109	108	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → (∪ 𝑏 ∈ (toCaraSiga‘𝑀) ↔ (∪ 𝑏 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒))))
110	107, 109	mpbid 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → (∪ 𝑏 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒)))
111	110	simprd 478	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒))
112	111	adantrr 749	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒))
113	38	elpwincl1 28741	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∈ 𝒫 𝑂)
114		simpr 476	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥)))
115	114	ineq1d 3775	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑒 ∩ ∪ 𝑏) = ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏))
116	115	fveq2d 6107	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑀‘(𝑒 ∩ ∪ 𝑏)) = (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏)))
117	114	difeq1d 3689	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑒 ∖ ∪ 𝑏) = ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏))
118	117	fveq2d 6107	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑀‘(𝑒 ∖ ∪ 𝑏)) = (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏)))
119	116, 118	oveq12d 6567	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → ((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = ((𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏)) +_𝑒 (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏))))
120	114	fveq2d 6107	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑀‘𝑒) = (𝑀‘(𝐸 ∩ (∪ 𝑏 ∪ 𝑥))))
121	119, 120	eqeq12d 2625	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒) ↔ ((𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏)) +_𝑒 (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏))) = (𝑀‘(𝐸 ∩ (∪ 𝑏 ∪ 𝑥)))))
122	113, 121	rspcdv 3285	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒) → ((𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏)) +_𝑒 (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏))) = (𝑀‘(𝐸 ∩ (∪ 𝑏 ∪ 𝑥)))))
123	112, 122	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏)) +_𝑒 (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏))) = (𝑀‘(𝐸 ∩ (∪ 𝑏 ∪ 𝑥))))
124	94, 123	eqtr3d 2646	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((𝑀‘(𝐸 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝐸 ∩ 𝑥))) = (𝑀‘(𝐸 ∩ (∪ 𝑏 ∪ 𝑥))))
125	124	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ (𝑀‘(𝐸 ∩ ∪ 𝑏)) = Σ^*𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦))) → ((𝑀‘(𝐸 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝐸 ∩ 𝑥))) = (𝑀‘(𝐸 ∩ (∪ 𝑏 ∪ 𝑥))))
126		uniun 4392	. . . . . . . 8 ⊢ ∪ (𝑏 ∪ {𝑥}) = (∪ 𝑏 ∪ ∪ {𝑥})
127		vex 3176	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
128	127	unisn 4387	. . . . . . . . 9 ⊢ ∪ {𝑥} = 𝑥
129	128	uneq2i 3726	. . . . . . . 8 ⊢ (∪ 𝑏 ∪ ∪ {𝑥}) = (∪ 𝑏 ∪ 𝑥)
130	126, 129	eqtri 2632	. . . . . . 7 ⊢ ∪ (𝑏 ∪ {𝑥}) = (∪ 𝑏 ∪ 𝑥)
131	130	ineq2i 3773	. . . . . 6 ⊢ (𝐸 ∩ ∪ (𝑏 ∪ {𝑥})) = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))
132	131	fveq2i 6106	. . . . 5 ⊢ (𝑀‘(𝐸 ∩ ∪ (𝑏 ∪ {𝑥}))) = (𝑀‘(𝐸 ∩ (∪ 𝑏 ∪ 𝑥)))
133	125, 132	syl6reqr 2663	. . . 4 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ (𝑀‘(𝐸 ∩ ∪ 𝑏)) = Σ^*𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦))) → (𝑀‘(𝐸 ∩ ∪ (𝑏 ∪ {𝑥}))) = ((𝑀‘(𝐸 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝐸 ∩ 𝑥))))
134	43, 63, 133	3eqtr4rd 2655	. . 3 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ (𝑀‘(𝐸 ∩ ∪ 𝑏)) = Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦))) → (𝑀‘(𝐸 ∩ ∪ (𝑏 ∪ {𝑥}))) = Σ^𝑦 ∈ (𝑏 ∪ {𝑥})(𝑀‘(𝐸 ∩ 𝑦)))
135	134	ex 449	. 2 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((𝑀‘(𝐸 ∩ ∪ 𝑏)) = Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦)) → (𝑀‘(𝐸 ∩ ∪ (𝑏 ∪ {𝑥}))) = Σ^𝑦 ∈ (𝑏 ∪ {𝑥})(𝑀‘(𝐸 ∩ 𝑦))))
136	5, 10, 15, 20, 28, 135, 101	findcard2d 8087	1 ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝐴)) = Σ^*𝑦 ∈ 𝐴(𝑀‘(𝐸 ∩ 𝑦)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 ∪ cuni 4372 Disj wdisj 4553 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ωcom 6957 ≼ cdom 7839 Fincfn 7841 0cc0 9815 +∞cpnf 9950 ≤ cle 9954 +_𝑒 cxad 11820 [,]cicc 12049 Σ^*cesum 29416 toCaraSigaccarsg 29690

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-ordt 15984 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-ps 17023 df-tsr 17024 df-plusf 17064 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-subrg 18601 df-abv 18640 df-lmod 18688 df-scaf 18689 df-sra 18993 df-rgmod 18994 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-tmd 21686 df-tgp 21687 df-tsms 21740 df-trg 21773 df-xms 21935 df-ms 21936 df-tms 21937 df-nm 22197 df-ngp 22198 df-nrg 22200 df-nlm 22201 df-ii 22488 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 df-esum 29417 df-carsg 29691

This theorem is referenced by: carsggect 29707 carsgclctunlem2 29708

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