carsgclctunlem1 - Mathbox for Thierry Arnoux

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Theorem carsgclctunlem1 34616

Description: Lemma for carsgclctun 34620. (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 4877	. . . . 5 ⊢ (𝑎 = ∅ → ∪ 𝑎 = ∪ ∅)
2	1	ineq2d 4173	. . . 4 ⊢ (𝑎 = ∅ → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ ∅))
3	2	fveq2d 6871	. . 3 ⊢ (𝑎 = ∅ → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ ∅)))
4		esumeq1 34333	. . 3 ⊢ (𝑎 = ∅ → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ ∅(𝑀‘(𝐸 ∩ 𝑦)))
5	3, 4	eqeq12d 2779	. 2 ⊢ (𝑎 = ∅ → ((𝑀‘(𝐸 ∩ ∪ 𝑎)) = Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) ↔ (𝑀‘(𝐸 ∩ ∪ ∅)) = Σ^𝑦 ∈ ∅(𝑀‘(𝐸 ∩ 𝑦))))
6		unieq 4877	. . . . 5 ⊢ (𝑎 = 𝑏 → ∪ 𝑎 = ∪ 𝑏)
7	6	ineq2d 4173	. . . 4 ⊢ (𝑎 = 𝑏 → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ 𝑏))
8	7	fveq2d 6871	. . 3 ⊢ (𝑎 = 𝑏 → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ 𝑏)))
9		esumeq1 34333	. . 3 ⊢ (𝑎 = 𝑏 → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦)))
10	8, 9	eqeq12d 2779	. 2 ⊢ (𝑎 = 𝑏 → ((𝑀‘(𝐸 ∩ ∪ 𝑎)) = Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) ↔ (𝑀‘(𝐸 ∩ ∪ 𝑏)) = Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦))))
11		unieq 4877	. . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → ∪ 𝑎 = ∪ (𝑏 ∪ {𝑥}))
12	11	ineq2d 4173	. . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ (𝑏 ∪ {𝑥})))
13	12	fveq2d 6871	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ (𝑏 ∪ {𝑥}))))
14		esumeq1 34333	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ (𝑏 ∪ {𝑥})(𝑀‘(𝐸 ∩ 𝑦)))
15	13, 14	eqeq12d 2779	. 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 4877	. . . . 5 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴)
17	16	ineq2d 4173	. . . 4 ⊢ (𝑎 = 𝐴 → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ 𝐴))
18	17	fveq2d 6871	. . 3 ⊢ (𝑎 = 𝐴 → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ 𝐴)))
19		esumeq1 34333	. . 3 ⊢ (𝑎 = 𝐴 → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ 𝐴(𝑀‘(𝐸 ∩ 𝑦)))
20	18, 19	eqeq12d 2779	. 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 4895	. . . . . 6 ⊢ ∪ ∅ = ∅
23	22	ineq2i 4170	. . . . 5 ⊢ (𝐸 ∩ ∪ ∅) = (𝐸 ∩ ∅)
24		in0 4350	. . . . 5 ⊢ (𝐸 ∩ ∅) = ∅
25	23, 24	eqtri 2786	. . . 4 ⊢ (𝐸 ∩ ∪ ∅) = ∅
26	25	fveq2i 6870	. . 3 ⊢ (𝑀‘(𝐸 ∩ ∪ ∅)) = (𝑀‘∅)
27		esumnul 34347	. . 3 ⊢ Σ^*𝑦 ∈ ∅(𝑀‘(𝐸 ∩ 𝑦)) = 0
28	21, 26, 27	3eqtr4g 2823	. 2 ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ ∅)) = Σ^*𝑦 ∈ ∅(𝑀‘(𝐸 ∩ 𝑦)))
29		simpr 488	. . . . . 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 2769	. . . . 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 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
32	31	ineq2d 4173	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 = 𝑥) → (𝐸 ∩ 𝑦) = (𝐸 ∩ 𝑥))
33	32	fveq2d 6871	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 = 𝑥) → (𝑀‘(𝐸 ∩ 𝑦)) = (𝑀‘(𝐸 ∩ 𝑥)))
34		simprr 782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → 𝑥 ∈ (𝐴 ∖ 𝑏))
35		carsgval.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))
36	35	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
37		carsgclctunlem1.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂)
38	37	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → 𝐸 ∈ 𝒫 𝑂)
39	38	elpwincl1 32730	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝐸 ∩ 𝑥) ∈ 𝒫 𝑂)
40	36, 39	ffvelcdmd 7066	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑀‘(𝐸 ∩ 𝑥)) ∈ (0[,]+∞))
41	33, 34, 40	esumsn 34364	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → Σ^*𝑦 ∈ {𝑥} (𝑀‘(𝐸 ∩ 𝑦)) = (𝑀‘(𝐸 ∩ 𝑥)))
42	41	adantr 484	. . . . 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 7414	. . . 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 1935	. . . . . 6 ⊢ Ⅎ𝑦(𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏)))
45		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑦𝑏
46		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑦{𝑥}
47		vex 3459	. . . . . . 7 ⊢ 𝑏 ∈ V
48	47	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → 𝑏 ∈ V)
49		vsnex 5393	. . . . . . 7 ⊢ {𝑥} ∈ V
50	49	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → {𝑥} ∈ V)
51	34	eldifbd 3918	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ¬ 𝑥 ∈ 𝑏)
52		disjsn 4671	. . . . . . 7 ⊢ ((𝑏 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ 𝑏)
53	51, 52	sylibr 236	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑏 ∩ {𝑥}) = ∅)
54	35	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
55	37	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → 𝐸 ∈ 𝒫 𝑂)
56	55	elpwincl1 32730	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → (𝐸 ∩ 𝑦) ∈ 𝒫 𝑂)
57	54, 56	ffvelcdmd 7066	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → (𝑀‘(𝐸 ∩ 𝑦)) ∈ (0[,]+∞))
58	35	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
59	37	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → 𝐸 ∈ 𝒫 𝑂)
60	59	elpwincl1 32730	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → (𝐸 ∩ 𝑦) ∈ 𝒫 𝑂)
61	58, 60	ffvelcdmd 7066	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → (𝑀‘(𝐸 ∩ 𝑦)) ∈ (0[,]+∞))
62	44, 45, 46, 48, 50, 53, 57, 61	esumsplit 34352	. . . . 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 484	. . . 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		uniun 4889	. . . . . . . 8 ⊢ ∪ (𝑏 ∪ {𝑥}) = (∪ 𝑏 ∪ ∪ {𝑥})
65		unisnv 4886	. . . . . . . . 9 ⊢ ∪ {𝑥} = 𝑥
66	65	uneq2i 4119	. . . . . . . 8 ⊢ (∪ 𝑏 ∪ ∪ {𝑥}) = (∪ 𝑏 ∪ 𝑥)
67	64, 66	eqtri 2786	. . . . . . 7 ⊢ ∪ (𝑏 ∪ {𝑥}) = (∪ 𝑏 ∪ 𝑥)
68	67	ineq2i 4170	. . . . . 6 ⊢ (𝐸 ∩ ∪ (𝑏 ∪ {𝑥})) = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))
69	68	fveq2i 6870	. . . . 5 ⊢ (𝑀‘(𝐸 ∩ ∪ (𝑏 ∪ {𝑥}))) = (𝑀‘(𝐸 ∩ (∪ 𝑏 ∪ 𝑥)))
70		inass 4180	. . . . . . . . . 10 ⊢ ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏) = (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∩ ∪ 𝑏))
71		indir 4239	. . . . . . . . . . . 12 ⊢ ((∪ 𝑏 ∪ 𝑥) ∩ ∪ 𝑏) = ((∪ 𝑏 ∩ ∪ 𝑏) ∪ (𝑥 ∩ ∪ 𝑏))
72		inidm 4179	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑏 ∩ ∪ 𝑏) = ∪ 𝑏
73	72	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (∪ 𝑏 ∩ ∪ 𝑏) = ∪ 𝑏)
74		incom 4162	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑏 ∩ 𝑥) = (𝑥 ∩ ∪ 𝑏)
75		carsgclctunlem1.1	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦)
76	75	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → Disj 𝑦 ∈ 𝐴 𝑦)
77		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑏 ⊆ 𝐴)
78	77	adantrr 727	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → 𝑏 ⊆ 𝐴)
79	76, 78, 34	disjuniel 32803	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (∪ 𝑏 ∩ 𝑥) = ∅)
80	74, 79	eqtr3id 2812	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑥 ∩ ∪ 𝑏) = ∅)
81	73, 80	uneq12d 4123	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((∪ 𝑏 ∩ ∪ 𝑏) ∪ (𝑥 ∩ ∪ 𝑏)) = (∪ 𝑏 ∪ ∅))
82		un0 4349	. . . . . . . . . . . . 13 ⊢ (∪ 𝑏 ∪ ∅) = ∪ 𝑏
83	81, 82	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((∪ 𝑏 ∩ ∪ 𝑏) ∪ (𝑥 ∩ ∪ 𝑏)) = ∪ 𝑏)
84	71, 83	eqtrid 2810	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((∪ 𝑏 ∪ 𝑥) ∩ ∪ 𝑏) = ∪ 𝑏)
85	84	ineq2d 4173	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∩ ∪ 𝑏)) = (𝐸 ∩ ∪ 𝑏))
86	70, 85	eqtrid 2810	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏) = (𝐸 ∩ ∪ 𝑏))
87	86	fveq2d 6871	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏)) = (𝑀‘(𝐸 ∩ ∪ 𝑏)))
88		indif2 4234	. . . . . . . . . 10 ⊢ (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∖ ∪ 𝑏)) = ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏)
89		uncom 4112	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑏 ∪ 𝑥) = (𝑥 ∪ ∪ 𝑏)
90	89	difeq1i 4077	. . . . . . . . . . . . 13 ⊢ ((∪ 𝑏 ∪ 𝑥) ∖ ∪ 𝑏) = ((𝑥 ∪ ∪ 𝑏) ∖ ∪ 𝑏)
91		difun2 4436	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∪ ∪ 𝑏) ∖ ∪ 𝑏) = (𝑥 ∖ ∪ 𝑏)
92		disj3 4409	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∩ ∪ 𝑏) = ∅ ↔ 𝑥 = (𝑥 ∖ ∪ 𝑏))
93	92	biimpi 218	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∩ ∪ 𝑏) = ∅ → 𝑥 = (𝑥 ∖ ∪ 𝑏))
94	91, 93	eqtr4id 2817	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∩ ∪ 𝑏) = ∅ → ((𝑥 ∪ ∪ 𝑏) ∖ ∪ 𝑏) = 𝑥)
95	90, 94	eqtrid 2810	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ ∪ 𝑏) = ∅ → ((∪ 𝑏 ∪ 𝑥) ∖ ∪ 𝑏) = 𝑥)
96	80, 95	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((∪ 𝑏 ∪ 𝑥) ∖ ∪ 𝑏) = 𝑥)
97	96	ineq2d 4173	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∖ ∪ 𝑏)) = (𝐸 ∩ 𝑥))
98	88, 97	eqtr3id 2812	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏) = (𝐸 ∩ 𝑥))
99	98	fveq2d 6871	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏)) = (𝑀‘(𝐸 ∩ 𝑥)))
100	87, 99	oveq12d 7414	. . . . . . 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))))
101		carsgval.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑂 ∈ 𝑉)
102	101	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑂 ∈ 𝑉)
103	35	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
104	21	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → (𝑀‘∅) = 0)
105		carsgsiga.2	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ^*𝑦 ∈ 𝑥(𝑀‘𝑦))
106	105	3adant1r 1192	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ⊆ 𝐴) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ^*𝑦 ∈ 𝑥(𝑀‘𝑦))
107		fiunelcarsg.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ Fin)
108		ssfi 9141	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑏 ⊆ 𝐴) → 𝑏 ∈ Fin)
109	107, 108	sylan 589	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑏 ∈ Fin)
110		fiunelcarsg.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))
111	110	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝐴 ⊆ (toCaraSiga‘𝑀))
112	77, 111	sstrd 3947	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑏 ⊆ (toCaraSiga‘𝑀))
113	102, 103, 104, 106, 109, 112	fiunelcarsg 34615	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → ∪ 𝑏 ∈ (toCaraSiga‘𝑀))
114	101, 35	elcarsg 34604	. . . . . . . . . . . 12 ⊢ (𝜑 → (∪ 𝑏 ∈ (toCaraSiga‘𝑀) ↔ (∪ 𝑏 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒))))
115	114	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → (∪ 𝑏 ∈ (toCaraSiga‘𝑀) ↔ (∪ 𝑏 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒))))
116	113, 115	mpbid 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → (∪ 𝑏 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒)))
117	116	simprd 499	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒))
118	117	adantrr 727	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒))
119	38	elpwincl1 32730	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∈ 𝒫 𝑂)
120		simpr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥)))
121	120	ineq1d 4172	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑒 ∩ ∪ 𝑏) = ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏))
122	121	fveq2d 6871	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑀‘(𝑒 ∩ ∪ 𝑏)) = (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏)))
123	120	difeq1d 4080	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑒 ∖ ∪ 𝑏) = ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏))
124	123	fveq2d 6871	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑀‘(𝑒 ∖ ∪ 𝑏)) = (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏)))
125	122, 124	oveq12d 7414	. . . . . . . . . 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))))
126	120	fveq2d 6871	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑀‘𝑒) = (𝑀‘(𝐸 ∩ (∪ 𝑏 ∪ 𝑥))))
127	125, 126	eqeq12d 2779	. . . . . . . . 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)))))
128	119, 127	rspcdv 3574	. . . . . . . 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)))))
129	118, 128	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏)) +_𝑒 (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏))) = (𝑀‘(𝐸 ∩ (∪ 𝑏 ∪ 𝑥))))
130	100, 129	eqtr3d 2800	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((𝑀‘(𝐸 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝐸 ∩ 𝑥))) = (𝑀‘(𝐸 ∩ (∪ 𝑏 ∪ 𝑥))))
131	130	adantr 484	. . . . 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))))
132	69, 131	eqtr4id 2817	. . . 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))))
133	43, 63, 132	3eqtr4rd 2809	. . 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)))
134	133	ex 416	. 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))))
135	5, 10, 15, 20, 28, 134, 107	findcard2d 9135	1 ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝐴)) = Σ^*𝑦 ∈ 𝐴(𝑀‘(𝐸 ∩ 𝑦)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1099 = wceq 1561 ∈ wcel 2143 ∀wral 3077 Vcvv 3455 ∖ cdif 3902 ∪ cun 3903 ∩ cin 3904 ⊆ wss 3905 ∅c0 4286 𝒫 cpw 4556 {csn 4583 ∪ cuni 4866 Disj wdisj 5068 class class class wbr 5101 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 ωcom 7846 ≼ cdom 8925 Fincfn 8927 0cc0 11084 +∞cpnf 11224 ≤ cle 11228 +_𝑒 cxad 13122 [,]cicc 13362 Σ^*cesum 34326 toCaraSigaccarsg 34600

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7718 ax-inf2 9594 ax-cnex 11140 ax-resscn 11141 ax-1cn 11142 ax-icn 11143 ax-addcl 11144 ax-addrcl 11145 ax-mulcl 11146 ax-mulrcl 11147 ax-mulcom 11148 ax-addass 11149 ax-mulass 11150 ax-distr 11151 ax-i2m1 11152 ax-1ne0 11153 ax-1rid 11154 ax-rnegex 11155 ax-rrecex 11156 ax-cnre 11157 ax-pre-lttri 11158 ax-pre-lttrn 11159 ax-pre-ltadd 11160 ax-pre-mulgt0 11161 ax-pre-sup 11162 ax-addf 11163 ax-mulf 11164

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-disj 5069 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-se 5602 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9306 df-fi 9355 df-sup 9386 df-inf 9387 df-oi 9456 df-dju 9871 df-card 9909 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11427 df-neg 11428 df-div 11856 df-nn 12221 df-2 12290 df-3 12291 df-4 12292 df-5 12293 df-6 12294 df-7 12295 df-8 12296 df-9 12297 df-n0 12492 df-z 12579 df-dec 12699 df-uz 12850 df-q 12960 df-rp 13004 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13363 df-ioc 13364 df-ico 13365 df-icc 13366 df-fz 13523 df-fzo 13670 df-fl 13812 df-mod 13890 df-seq 14025 df-exp 14085 df-fac 14297 df-bc 14326 df-hash 14354 df-shft 15090 df-cj 15136 df-re 15137 df-im 15138 df-sqrt 15272 df-abs 15273 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-ef 16107 df-sin 16109 df-cos 16110 df-pi 16112 df-struct 17193 df-sets 17210 df-slot 17228 df-ndx 17240 df-base 17256 df-ress 17277 df-plusg 17309 df-mulr 17310 df-starv 17311 df-sca 17312 df-vsca 17313 df-ip 17314 df-tset 17315 df-ple 17316 df-ds 17318 df-unif 17319 df-hom 17320 df-cco 17321 df-rest 17461 df-topn 17462 df-0g 17480 df-gsum 17481 df-topgen 17482 df-pt 17483 df-prds 17486 df-ordt 17541 df-xrs 17542 df-qtop 17547 df-imas 17548 df-xps 17550 df-mre 17624 df-mrc 17625 df-acs 17627 df-ps 18608 df-tsr 18609 df-plusf 18683 df-mgm 18684 df-sgrp 18763 df-mnd 18779 df-mhm 18827 df-submnd 18828 df-grp 18988 df-minusg 18989 df-sbg 18990 df-mulg 19120 df-subg 19175 df-cntz 19367 df-cmn 19832 df-abl 19833 df-mgp 20197 df-rng 20209 df-ur 20242 df-ring 20295 df-cring 20296 df-subrng 20606 df-subrg 20630 df-abv 20865 df-lmod 20936 df-scaf 20937 df-sra 21247 df-rgmod 21248 df-psmet 21423 df-xmet 21424 df-met 21425 df-bl 21426 df-mopn 21427 df-fbas 21428 df-fg 21429 df-cnfld 21432 df-top 22961 df-topon 22978 df-topsp 23000 df-bases 23013 df-cld 23086 df-ntr 23087 df-cls 23088 df-nei 23165 df-lp 23203 df-perf 23204 df-cn 23294 df-cnp 23295 df-haus 23382 df-tx 23629 df-hmeo 23822 df-fil 23913 df-fm 24005 df-flim 24006 df-flf 24007 df-tmd 24139 df-tgp 24140 df-tsms 24194 df-trg 24227 df-xms 24387 df-ms 24388 df-tms 24389 df-nm 24649 df-ngp 24650 df-nrg 24652 df-nlm 24653 df-ii 24946 df-cncf 24947 df-limc 25935 df-dv 25936 df-log 26628 df-esum 34327 df-carsg 34601

This theorem is referenced by: carsggect 34617 carsgclctunlem2 34618

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