carsgclctunlem1 - Mathbox for Thierry Arnoux

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Theorem carsgclctunlem1 34301

Description: Lemma for carsgclctun 34305. (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 4869	. . . . 5 ⊢ (𝑎 = ∅ → ∪ 𝑎 = ∪ ∅)
2	1	ineq2d 4171	. . . 4 ⊢ (𝑎 = ∅ → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ ∅))
3	2	fveq2d 6826	. . 3 ⊢ (𝑎 = ∅ → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ ∅)))
4		esumeq1 34017	. . 3 ⊢ (𝑎 = ∅ → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ ∅(𝑀‘(𝐸 ∩ 𝑦)))
5	3, 4	eqeq12d 2745	. 2 ⊢ (𝑎 = ∅ → ((𝑀‘(𝐸 ∩ ∪ 𝑎)) = Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) ↔ (𝑀‘(𝐸 ∩ ∪ ∅)) = Σ^𝑦 ∈ ∅(𝑀‘(𝐸 ∩ 𝑦))))
6		unieq 4869	. . . . 5 ⊢ (𝑎 = 𝑏 → ∪ 𝑎 = ∪ 𝑏)
7	6	ineq2d 4171	. . . 4 ⊢ (𝑎 = 𝑏 → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ 𝑏))
8	7	fveq2d 6826	. . 3 ⊢ (𝑎 = 𝑏 → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ 𝑏)))
9		esumeq1 34017	. . 3 ⊢ (𝑎 = 𝑏 → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦)))
10	8, 9	eqeq12d 2745	. 2 ⊢ (𝑎 = 𝑏 → ((𝑀‘(𝐸 ∩ ∪ 𝑎)) = Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) ↔ (𝑀‘(𝐸 ∩ ∪ 𝑏)) = Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦))))
11		unieq 4869	. . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → ∪ 𝑎 = ∪ (𝑏 ∪ {𝑥}))
12	11	ineq2d 4171	. . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ (𝑏 ∪ {𝑥})))
13	12	fveq2d 6826	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ (𝑏 ∪ {𝑥}))))
14		esumeq1 34017	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ (𝑏 ∪ {𝑥})(𝑀‘(𝐸 ∩ 𝑦)))
15	13, 14	eqeq12d 2745	. 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 4869	. . . . 5 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴)
17	16	ineq2d 4171	. . . 4 ⊢ (𝑎 = 𝐴 → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ 𝐴))
18	17	fveq2d 6826	. . 3 ⊢ (𝑎 = 𝐴 → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ 𝐴)))
19		esumeq1 34017	. . 3 ⊢ (𝑎 = 𝐴 → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ 𝐴(𝑀‘(𝐸 ∩ 𝑦)))
20	18, 19	eqeq12d 2745	. 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 4886	. . . . . 6 ⊢ ∪ ∅ = ∅
23	22	ineq2i 4168	. . . . 5 ⊢ (𝐸 ∩ ∪ ∅) = (𝐸 ∩ ∅)
24		in0 4346	. . . . 5 ⊢ (𝐸 ∩ ∅) = ∅
25	23, 24	eqtri 2752	. . . 4 ⊢ (𝐸 ∩ ∪ ∅) = ∅
26	25	fveq2i 6825	. . 3 ⊢ (𝑀‘(𝐸 ∩ ∪ ∅)) = (𝑀‘∅)
27		esumnul 34031	. . 3 ⊢ Σ^*𝑦 ∈ ∅(𝑀‘(𝐸 ∩ 𝑦)) = 0
28	21, 26, 27	3eqtr4g 2789	. 2 ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ ∅)) = Σ^*𝑦 ∈ ∅(𝑀‘(𝐸 ∩ 𝑦)))
29		simpr 484	. . . . . 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 2735	. . . . 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 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
32	31	ineq2d 4171	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 = 𝑥) → (𝐸 ∩ 𝑦) = (𝐸 ∩ 𝑥))
33	32	fveq2d 6826	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 = 𝑥) → (𝑀‘(𝐸 ∩ 𝑦)) = (𝑀‘(𝐸 ∩ 𝑥)))
34		simprr 772	. . . . . . 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 32474	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝐸 ∩ 𝑥) ∈ 𝒫 𝑂)
40	36, 39	ffvelcdmd 7019	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑀‘(𝐸 ∩ 𝑥)) ∈ (0[,]+∞))
41	33, 34, 40	esumsn 34048	. . . . . 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 7367	. . . 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 1914	. . . . . 6 ⊢ Ⅎ𝑦(𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏)))
45		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑦𝑏
46		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑦{𝑥}
47		vex 3440	. . . . . . 7 ⊢ 𝑏 ∈ V
48	47	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → 𝑏 ∈ V)
49		vsnex 5373	. . . . . . 7 ⊢ {𝑥} ∈ V
50	49	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → {𝑥} ∈ V)
51	34	eldifbd 3916	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ¬ 𝑥 ∈ 𝑏)
52		disjsn 4663	. . . . . . 7 ⊢ ((𝑏 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ 𝑏)
53	51, 52	sylibr 234	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑏 ∩ {𝑥}) = ∅)
54	35	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
55	37	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → 𝐸 ∈ 𝒫 𝑂)
56	55	elpwincl1 32474	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → (𝐸 ∩ 𝑦) ∈ 𝒫 𝑂)
57	54, 56	ffvelcdmd 7019	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → (𝑀‘(𝐸 ∩ 𝑦)) ∈ (0[,]+∞))
58	35	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
59	37	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → 𝐸 ∈ 𝒫 𝑂)
60	59	elpwincl1 32474	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → (𝐸 ∩ 𝑦) ∈ 𝒫 𝑂)
61	58, 60	ffvelcdmd 7019	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → (𝑀‘(𝐸 ∩ 𝑦)) ∈ (0[,]+∞))
62	44, 45, 46, 48, 50, 53, 57, 61	esumsplit 34036	. . . . 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		uniun 4881	. . . . . . . 8 ⊢ ∪ (𝑏 ∪ {𝑥}) = (∪ 𝑏 ∪ ∪ {𝑥})
65		unisnv 4878	. . . . . . . . 9 ⊢ ∪ {𝑥} = 𝑥
66	65	uneq2i 4116	. . . . . . . 8 ⊢ (∪ 𝑏 ∪ ∪ {𝑥}) = (∪ 𝑏 ∪ 𝑥)
67	64, 66	eqtri 2752	. . . . . . 7 ⊢ ∪ (𝑏 ∪ {𝑥}) = (∪ 𝑏 ∪ 𝑥)
68	67	ineq2i 4168	. . . . . 6 ⊢ (𝐸 ∩ ∪ (𝑏 ∪ {𝑥})) = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))
69	68	fveq2i 6825	. . . . 5 ⊢ (𝑀‘(𝐸 ∩ ∪ (𝑏 ∪ {𝑥}))) = (𝑀‘(𝐸 ∩ (∪ 𝑏 ∪ 𝑥)))
70		inass 4179	. . . . . . . . . 10 ⊢ ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏) = (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∩ ∪ 𝑏))
71		indir 4237	. . . . . . . . . . . 12 ⊢ ((∪ 𝑏 ∪ 𝑥) ∩ ∪ 𝑏) = ((∪ 𝑏 ∩ ∪ 𝑏) ∪ (𝑥 ∩ ∪ 𝑏))
72		inidm 4178	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑏 ∩ ∪ 𝑏) = ∪ 𝑏
73	72	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (∪ 𝑏 ∩ ∪ 𝑏) = ∪ 𝑏)
74		incom 4160	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑏 ∩ 𝑥) = (𝑥 ∩ ∪ 𝑏)
75		carsgclctunlem1.1	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦)
76	75	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → Disj 𝑦 ∈ 𝐴 𝑦)
77		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑏 ⊆ 𝐴)
78	77	adantrr 717	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → 𝑏 ⊆ 𝐴)
79	76, 78, 34	disjuniel 32546	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (∪ 𝑏 ∩ 𝑥) = ∅)
80	74, 79	eqtr3id 2778	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑥 ∩ ∪ 𝑏) = ∅)
81	73, 80	uneq12d 4120	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((∪ 𝑏 ∩ ∪ 𝑏) ∪ (𝑥 ∩ ∪ 𝑏)) = (∪ 𝑏 ∪ ∅))
82		un0 4345	. . . . . . . . . . . . 13 ⊢ (∪ 𝑏 ∪ ∅) = ∪ 𝑏
83	81, 82	eqtrdi 2780	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((∪ 𝑏 ∩ ∪ 𝑏) ∪ (𝑥 ∩ ∪ 𝑏)) = ∪ 𝑏)
84	71, 83	eqtrid 2776	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((∪ 𝑏 ∪ 𝑥) ∩ ∪ 𝑏) = ∪ 𝑏)
85	84	ineq2d 4171	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∩ ∪ 𝑏)) = (𝐸 ∩ ∪ 𝑏))
86	70, 85	eqtrid 2776	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏) = (𝐸 ∩ ∪ 𝑏))
87	86	fveq2d 6826	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏)) = (𝑀‘(𝐸 ∩ ∪ 𝑏)))
88		indif2 4232	. . . . . . . . . 10 ⊢ (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∖ ∪ 𝑏)) = ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏)
89		uncom 4109	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑏 ∪ 𝑥) = (𝑥 ∪ ∪ 𝑏)
90	89	difeq1i 4073	. . . . . . . . . . . . 13 ⊢ ((∪ 𝑏 ∪ 𝑥) ∖ ∪ 𝑏) = ((𝑥 ∪ ∪ 𝑏) ∖ ∪ 𝑏)
91		difun2 4432	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∪ ∪ 𝑏) ∖ ∪ 𝑏) = (𝑥 ∖ ∪ 𝑏)
92		disj3 4405	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∩ ∪ 𝑏) = ∅ ↔ 𝑥 = (𝑥 ∖ ∪ 𝑏))
93	92	biimpi 216	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∩ ∪ 𝑏) = ∅ → 𝑥 = (𝑥 ∖ ∪ 𝑏))
94	91, 93	eqtr4id 2783	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∩ ∪ 𝑏) = ∅ → ((𝑥 ∪ ∪ 𝑏) ∖ ∪ 𝑏) = 𝑥)
95	90, 94	eqtrid 2776	. . . . . . . . . . . 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 4171	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∖ ∪ 𝑏)) = (𝐸 ∩ 𝑥))
98	88, 97	eqtr3id 2778	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏) = (𝐸 ∩ 𝑥))
99	98	fveq2d 6826	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏)) = (𝑀‘(𝐸 ∩ 𝑥)))
100	87, 99	oveq12d 7367	. . . . . . 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 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑂 ∈ 𝑉)
103	35	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
104	21	adantr 480	. . . . . . . . . . . 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 1178	. . . . . . . . . . . 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 9087	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑏 ⊆ 𝐴) → 𝑏 ∈ Fin)
109	107, 108	sylan 580	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑏 ∈ Fin)
110		fiunelcarsg.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))
111	110	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝐴 ⊆ (toCaraSiga‘𝑀))
112	77, 111	sstrd 3946	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑏 ⊆ (toCaraSiga‘𝑀))
113	102, 103, 104, 106, 109, 112	fiunelcarsg 34300	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → ∪ 𝑏 ∈ (toCaraSiga‘𝑀))
114	101, 35	elcarsg 34289	. . . . . . . . . . . 12 ⊢ (𝜑 → (∪ 𝑏 ∈ (toCaraSiga‘𝑀) ↔ (∪ 𝑏 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒))))
115	114	adantr 480	. . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → (∪ 𝑏 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒)))
117	116	simprd 495	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒))
118	117	adantrr 717	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒))
119	38	elpwincl1 32474	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∈ 𝒫 𝑂)
120		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥)))
121	120	ineq1d 4170	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑒 ∩ ∪ 𝑏) = ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏))
122	121	fveq2d 6826	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑀‘(𝑒 ∩ ∪ 𝑏)) = (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏)))
123	120	difeq1d 4076	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑒 ∖ ∪ 𝑏) = ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏))
124	123	fveq2d 6826	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑀‘(𝑒 ∖ ∪ 𝑏)) = (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏)))
125	122, 124	oveq12d 7367	. . . . . . . . . 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 6826	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑀‘𝑒) = (𝑀‘(𝐸 ∩ (∪ 𝑏 ∪ 𝑥))))
127	125, 126	eqeq12d 2745	. . . . . . . . 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 3569	. . . . . . . 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 2766	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((𝑀‘(𝐸 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝐸 ∩ 𝑥))) = (𝑀‘(𝐸 ∩ (∪ 𝑏 ∪ 𝑥))))
131	130	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))))
132	69, 131	eqtr4id 2783	. . . 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 2775	. . 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 412	. 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 9080	1 ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝐴)) = Σ^*𝑦 ∈ 𝐴(𝑀‘(𝐸 ∩ 𝑦)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 Vcvv 3436 ∖ cdif 3900 ∪ cun 3901 ∩ cin 3902 ⊆ wss 3903 ∅c0 4284 𝒫 cpw 4551 {csn 4577 ∪ cuni 4858 Disj wdisj 5059 class class class wbr 5092 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ωcom 7799 ≼ cdom 8870 Fincfn 8872 0cc0 11009 +∞cpnf 11146 ≤ cle 11150 +_𝑒 cxad 13012 [,]cicc 13251 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-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-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: carsggect 34302 carsgclctunlem2 34303

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