carsgclctunlem1 - Mathbox for Thierry Arnoux

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Theorem carsgclctunlem1 34282

Description: Lemma for carsgclctun 34286. (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 4942	. . . . 5 ⊢ (𝑎 = ∅ → ∪ 𝑎 = ∪ ∅)
2	1	ineq2d 4241	. . . 4 ⊢ (𝑎 = ∅ → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ ∅))
3	2	fveq2d 6924	. . 3 ⊢ (𝑎 = ∅ → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ ∅)))
4		esumeq1 33998	. . 3 ⊢ (𝑎 = ∅ → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ ∅(𝑀‘(𝐸 ∩ 𝑦)))
5	3, 4	eqeq12d 2756	. 2 ⊢ (𝑎 = ∅ → ((𝑀‘(𝐸 ∩ ∪ 𝑎)) = Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) ↔ (𝑀‘(𝐸 ∩ ∪ ∅)) = Σ^𝑦 ∈ ∅(𝑀‘(𝐸 ∩ 𝑦))))
6		unieq 4942	. . . . 5 ⊢ (𝑎 = 𝑏 → ∪ 𝑎 = ∪ 𝑏)
7	6	ineq2d 4241	. . . 4 ⊢ (𝑎 = 𝑏 → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ 𝑏))
8	7	fveq2d 6924	. . 3 ⊢ (𝑎 = 𝑏 → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ 𝑏)))
9		esumeq1 33998	. . 3 ⊢ (𝑎 = 𝑏 → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦)))
10	8, 9	eqeq12d 2756	. 2 ⊢ (𝑎 = 𝑏 → ((𝑀‘(𝐸 ∩ ∪ 𝑎)) = Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) ↔ (𝑀‘(𝐸 ∩ ∪ 𝑏)) = Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦))))
11		unieq 4942	. . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → ∪ 𝑎 = ∪ (𝑏 ∪ {𝑥}))
12	11	ineq2d 4241	. . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ (𝑏 ∪ {𝑥})))
13	12	fveq2d 6924	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ (𝑏 ∪ {𝑥}))))
14		esumeq1 33998	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ (𝑏 ∪ {𝑥})(𝑀‘(𝐸 ∩ 𝑦)))
15	13, 14	eqeq12d 2756	. 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 4942	. . . . 5 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴)
17	16	ineq2d 4241	. . . 4 ⊢ (𝑎 = 𝐴 → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ 𝐴))
18	17	fveq2d 6924	. . 3 ⊢ (𝑎 = 𝐴 → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ 𝐴)))
19		esumeq1 33998	. . 3 ⊢ (𝑎 = 𝐴 → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ 𝐴(𝑀‘(𝐸 ∩ 𝑦)))
20	18, 19	eqeq12d 2756	. 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 4959	. . . . . 6 ⊢ ∪ ∅ = ∅
23	22	ineq2i 4238	. . . . 5 ⊢ (𝐸 ∩ ∪ ∅) = (𝐸 ∩ ∅)
24		in0 4418	. . . . 5 ⊢ (𝐸 ∩ ∅) = ∅
25	23, 24	eqtri 2768	. . . 4 ⊢ (𝐸 ∩ ∪ ∅) = ∅
26	25	fveq2i 6923	. . 3 ⊢ (𝑀‘(𝐸 ∩ ∪ ∅)) = (𝑀‘∅)
27		esumnul 34012	. . 3 ⊢ Σ^*𝑦 ∈ ∅(𝑀‘(𝐸 ∩ 𝑦)) = 0
28	21, 26, 27	3eqtr4g 2805	. 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 2746	. . . . 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 4241	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 = 𝑥) → (𝐸 ∩ 𝑦) = (𝐸 ∩ 𝑥))
33	32	fveq2d 6924	. . . . . . 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 32555	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝐸 ∩ 𝑥) ∈ 𝒫 𝑂)
40	36, 39	ffvelcdmd 7119	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑀‘(𝐸 ∩ 𝑥)) ∈ (0[,]+∞))
41	33, 34, 40	esumsn 34029	. . . . . 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 7466	. . . 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 1913	. . . . . 6 ⊢ Ⅎ𝑦(𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏)))
45		nfcv 2908	. . . . . 6 ⊢ Ⅎ𝑦𝑏
46		nfcv 2908	. . . . . 6 ⊢ Ⅎ𝑦{𝑥}
47		vex 3492	. . . . . . 7 ⊢ 𝑏 ∈ V
48	47	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → 𝑏 ∈ V)
49		vsnex 5449	. . . . . . 7 ⊢ {𝑥} ∈ V
50	49	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → {𝑥} ∈ V)
51	34	eldifbd 3989	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ¬ 𝑥 ∈ 𝑏)
52		disjsn 4736	. . . . . . 7 ⊢ ((𝑏 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ 𝑏)
53	51, 52	sylibr 234	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑏 ∩ {𝑥}) = ∅)
54	35	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
55	37	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → 𝐸 ∈ 𝒫 𝑂)
56	55	elpwincl1 32555	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → (𝐸 ∩ 𝑦) ∈ 𝒫 𝑂)
57	54, 56	ffvelcdmd 7119	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → (𝑀‘(𝐸 ∩ 𝑦)) ∈ (0[,]+∞))
58	35	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
59	37	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → 𝐸 ∈ 𝒫 𝑂)
60	59	elpwincl1 32555	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → (𝐸 ∩ 𝑦) ∈ 𝒫 𝑂)
61	58, 60	ffvelcdmd 7119	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → (𝑀‘(𝐸 ∩ 𝑦)) ∈ (0[,]+∞))
62	44, 45, 46, 48, 50, 53, 57, 61	esumsplit 34017	. . . . 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 4954	. . . . . . . 8 ⊢ ∪ (𝑏 ∪ {𝑥}) = (∪ 𝑏 ∪ ∪ {𝑥})
65		unisnv 4951	. . . . . . . . 9 ⊢ ∪ {𝑥} = 𝑥
66	65	uneq2i 4188	. . . . . . . 8 ⊢ (∪ 𝑏 ∪ ∪ {𝑥}) = (∪ 𝑏 ∪ 𝑥)
67	64, 66	eqtri 2768	. . . . . . 7 ⊢ ∪ (𝑏 ∪ {𝑥}) = (∪ 𝑏 ∪ 𝑥)
68	67	ineq2i 4238	. . . . . 6 ⊢ (𝐸 ∩ ∪ (𝑏 ∪ {𝑥})) = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))
69	68	fveq2i 6923	. . . . 5 ⊢ (𝑀‘(𝐸 ∩ ∪ (𝑏 ∪ {𝑥}))) = (𝑀‘(𝐸 ∩ (∪ 𝑏 ∪ 𝑥)))
70		inass 4249	. . . . . . . . . 10 ⊢ ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏) = (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∩ ∪ 𝑏))
71		indir 4305	. . . . . . . . . . . 12 ⊢ ((∪ 𝑏 ∪ 𝑥) ∩ ∪ 𝑏) = ((∪ 𝑏 ∩ ∪ 𝑏) ∪ (𝑥 ∩ ∪ 𝑏))
72		inidm 4248	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑏 ∩ ∪ 𝑏) = ∪ 𝑏
73	72	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (∪ 𝑏 ∩ ∪ 𝑏) = ∪ 𝑏)
74		incom 4230	. . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → 𝑏 ⊆ 𝐴)
79	76, 78, 34	disjuniel 32619	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (∪ 𝑏 ∩ 𝑥) = ∅)
80	74, 79	eqtr3id 2794	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑥 ∩ ∪ 𝑏) = ∅)
81	73, 80	uneq12d 4192	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((∪ 𝑏 ∩ ∪ 𝑏) ∪ (𝑥 ∩ ∪ 𝑏)) = (∪ 𝑏 ∪ ∅))
82		un0 4417	. . . . . . . . . . . . 13 ⊢ (∪ 𝑏 ∪ ∅) = ∪ 𝑏
83	81, 82	eqtrdi 2796	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((∪ 𝑏 ∩ ∪ 𝑏) ∪ (𝑥 ∩ ∪ 𝑏)) = ∪ 𝑏)
84	71, 83	eqtrid 2792	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((∪ 𝑏 ∪ 𝑥) ∩ ∪ 𝑏) = ∪ 𝑏)
85	84	ineq2d 4241	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∩ ∪ 𝑏)) = (𝐸 ∩ ∪ 𝑏))
86	70, 85	eqtrid 2792	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏) = (𝐸 ∩ ∪ 𝑏))
87	86	fveq2d 6924	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏)) = (𝑀‘(𝐸 ∩ ∪ 𝑏)))
88		indif2 4300	. . . . . . . . . 10 ⊢ (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∖ ∪ 𝑏)) = ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏)
89		uncom 4181	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑏 ∪ 𝑥) = (𝑥 ∪ ∪ 𝑏)
90	89	difeq1i 4145	. . . . . . . . . . . . 13 ⊢ ((∪ 𝑏 ∪ 𝑥) ∖ ∪ 𝑏) = ((𝑥 ∪ ∪ 𝑏) ∖ ∪ 𝑏)
91		difun2 4504	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∪ ∪ 𝑏) ∖ ∪ 𝑏) = (𝑥 ∖ ∪ 𝑏)
92		disj3 4477	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∩ ∪ 𝑏) = ∅ ↔ 𝑥 = (𝑥 ∖ ∪ 𝑏))
93	92	biimpi 216	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∩ ∪ 𝑏) = ∅ → 𝑥 = (𝑥 ∖ ∪ 𝑏))
94	91, 93	eqtr4id 2799	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∩ ∪ 𝑏) = ∅ → ((𝑥 ∪ ∪ 𝑏) ∖ ∪ 𝑏) = 𝑥)
95	90, 94	eqtrid 2792	. . . . . . . . . . . 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 4241	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∖ ∪ 𝑏)) = (𝐸 ∩ 𝑥))
98	88, 97	eqtr3id 2794	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏) = (𝐸 ∩ 𝑥))
99	98	fveq2d 6924	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏)) = (𝑀‘(𝐸 ∩ 𝑥)))
100	87, 99	oveq12d 7466	. . . . . . 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 1177	. . . . . . . . . . . 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 9240	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑏 ⊆ 𝐴) → 𝑏 ∈ Fin)
109	107, 108	sylan 579	. . . . . . . . . . . 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 4019	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑏 ⊆ (toCaraSiga‘𝑀))
113	102, 103, 104, 106, 109, 112	fiunelcarsg 34281	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → ∪ 𝑏 ∈ (toCaraSiga‘𝑀))
114	101, 35	elcarsg 34270	. . . . . . . . . . . 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 716	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒))
119	38	elpwincl1 32555	. . . . . . . . 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 4240	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑒 ∩ ∪ 𝑏) = ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏))
122	121	fveq2d 6924	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑀‘(𝑒 ∩ ∪ 𝑏)) = (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏)))
123	120	difeq1d 4148	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑒 ∖ ∪ 𝑏) = ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏))
124	123	fveq2d 6924	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑀‘(𝑒 ∖ ∪ 𝑏)) = (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏)))
125	122, 124	oveq12d 7466	. . . . . . . . . 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 6924	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑀‘𝑒) = (𝑀‘(𝐸 ∩ (∪ 𝑏 ∪ 𝑥))))
127	125, 126	eqeq12d 2756	. . . . . . . . 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 3627	. . . . . . . 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 2782	. . . . . 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 2799	. . . 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 2791	. . 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 9232	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 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 Vcvv 3488 ∖ cdif 3973 ∪ cun 3974 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 𝒫 cpw 4622 {csn 4648 ∪ cuni 4931 Disj wdisj 5133 class class class wbr 5166 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ωcom 7903 ≼ cdom 9001 Fincfn 9003 0cc0 11184 +∞cpnf 11321 ≤ cle 11325 +_𝑒 cxad 13173 [,]cicc 13410 Σ^*cesum 33991 toCaraSigaccarsg 34266

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-ef 16115 df-sin 16117 df-cos 16118 df-pi 16120 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-ordt 17561 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-ps 18636 df-tsr 18637 df-plusf 18677 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-subrng 20572 df-subrg 20597 df-abv 20832 df-lmod 20882 df-scaf 20883 df-sra 21195 df-rgmod 21196 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-tmd 24101 df-tgp 24102 df-tsms 24156 df-trg 24189 df-xms 24351 df-ms 24352 df-tms 24353 df-nm 24616 df-ngp 24617 df-nrg 24619 df-nlm 24620 df-ii 24922 df-cncf 24923 df-limc 25921 df-dv 25922 df-log 26616 df-esum 33992 df-carsg 34267

This theorem is referenced by: carsggect 34283 carsgclctunlem2 34284

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