carsgclctunlem1 - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > carsgclctunlem1		Structured version Visualization version GIF version

Theorem carsgclctunlem1 32917

Description: Lemma for carsgclctun 32921. (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 4876	. . . . 5 ⊢ (𝑎 = ∅ → ∪ 𝑎 = ∪ ∅)
2	1	ineq2d 4172	. . . 4 ⊢ (𝑎 = ∅ → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ ∅))
3	2	fveq2d 6846	. . 3 ⊢ (𝑎 = ∅ → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ ∅)))
4		esumeq1 32633	. . 3 ⊢ (𝑎 = ∅ → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ ∅(𝑀‘(𝐸 ∩ 𝑦)))
5	3, 4	eqeq12d 2752	. 2 ⊢ (𝑎 = ∅ → ((𝑀‘(𝐸 ∩ ∪ 𝑎)) = Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) ↔ (𝑀‘(𝐸 ∩ ∪ ∅)) = Σ^𝑦 ∈ ∅(𝑀‘(𝐸 ∩ 𝑦))))
6		unieq 4876	. . . . 5 ⊢ (𝑎 = 𝑏 → ∪ 𝑎 = ∪ 𝑏)
7	6	ineq2d 4172	. . . 4 ⊢ (𝑎 = 𝑏 → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ 𝑏))
8	7	fveq2d 6846	. . 3 ⊢ (𝑎 = 𝑏 → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ 𝑏)))
9		esumeq1 32633	. . 3 ⊢ (𝑎 = 𝑏 → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦)))
10	8, 9	eqeq12d 2752	. 2 ⊢ (𝑎 = 𝑏 → ((𝑀‘(𝐸 ∩ ∪ 𝑎)) = Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) ↔ (𝑀‘(𝐸 ∩ ∪ 𝑏)) = Σ^𝑦 ∈ 𝑏(𝑀‘(𝐸 ∩ 𝑦))))
11		unieq 4876	. . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → ∪ 𝑎 = ∪ (𝑏 ∪ {𝑥}))
12	11	ineq2d 4172	. . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ (𝑏 ∪ {𝑥})))
13	12	fveq2d 6846	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ (𝑏 ∪ {𝑥}))))
14		esumeq1 32633	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ (𝑏 ∪ {𝑥})(𝑀‘(𝐸 ∩ 𝑦)))
15	13, 14	eqeq12d 2752	. 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 4876	. . . . 5 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴)
17	16	ineq2d 4172	. . . 4 ⊢ (𝑎 = 𝐴 → (𝐸 ∩ ∪ 𝑎) = (𝐸 ∩ ∪ 𝐴))
18	17	fveq2d 6846	. . 3 ⊢ (𝑎 = 𝐴 → (𝑀‘(𝐸 ∩ ∪ 𝑎)) = (𝑀‘(𝐸 ∩ ∪ 𝐴)))
19		esumeq1 32633	. . 3 ⊢ (𝑎 = 𝐴 → Σ^𝑦 ∈ 𝑎(𝑀‘(𝐸 ∩ 𝑦)) = Σ^𝑦 ∈ 𝐴(𝑀‘(𝐸 ∩ 𝑦)))
20	18, 19	eqeq12d 2752	. 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 4896	. . . . . 6 ⊢ ∪ ∅ = ∅
23	22	ineq2i 4169	. . . . 5 ⊢ (𝐸 ∩ ∪ ∅) = (𝐸 ∩ ∅)
24		in0 4351	. . . . 5 ⊢ (𝐸 ∩ ∅) = ∅
25	23, 24	eqtri 2764	. . . 4 ⊢ (𝐸 ∩ ∪ ∅) = ∅
26	25	fveq2i 6845	. . 3 ⊢ (𝑀‘(𝐸 ∩ ∪ ∅)) = (𝑀‘∅)
27		esumnul 32647	. . 3 ⊢ Σ^*𝑦 ∈ ∅(𝑀‘(𝐸 ∩ 𝑦)) = 0
28	21, 26, 27	3eqtr4g 2801	. 2 ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ ∅)) = Σ^*𝑦 ∈ ∅(𝑀‘(𝐸 ∩ 𝑦)))
29		simpr 485	. . . . . 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 2742	. . . . 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 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
32	31	ineq2d 4172	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 = 𝑥) → (𝐸 ∩ 𝑦) = (𝐸 ∩ 𝑥))
33	32	fveq2d 6846	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 = 𝑥) → (𝑀‘(𝐸 ∩ 𝑦)) = (𝑀‘(𝐸 ∩ 𝑥)))
34		simprr 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → 𝑥 ∈ (𝐴 ∖ 𝑏))
35		carsgval.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))
36	35	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
37		carsgclctunlem1.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑂)
38	37	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → 𝐸 ∈ 𝒫 𝑂)
39	38	elpwincl1 31453	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝐸 ∩ 𝑥) ∈ 𝒫 𝑂)
40	36, 39	ffvelcdmd 7036	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑀‘(𝐸 ∩ 𝑥)) ∈ (0[,]+∞))
41	33, 34, 40	esumsn 32664	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → Σ^*𝑦 ∈ {𝑥} (𝑀‘(𝐸 ∩ 𝑦)) = (𝑀‘(𝐸 ∩ 𝑥)))
42	41	adantr 481	. . . . 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 7375	. . . 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 1917	. . . . . 6 ⊢ Ⅎ𝑦(𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏)))
45		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑦𝑏
46		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑦{𝑥}
47		vex 3449	. . . . . . 7 ⊢ 𝑏 ∈ V
48	47	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → 𝑏 ∈ V)
49		vsnex 5386	. . . . . . 7 ⊢ {𝑥} ∈ V
50	49	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → {𝑥} ∈ V)
51	34	eldifbd 3923	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ¬ 𝑥 ∈ 𝑏)
52		disjsn 4672	. . . . . . 7 ⊢ ((𝑏 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ 𝑏)
53	51, 52	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑏 ∩ {𝑥}) = ∅)
54	35	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
55	37	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → 𝐸 ∈ 𝒫 𝑂)
56	55	elpwincl1 31453	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → (𝐸 ∩ 𝑦) ∈ 𝒫 𝑂)
57	54, 56	ffvelcdmd 7036	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ 𝑏) → (𝑀‘(𝐸 ∩ 𝑦)) ∈ (0[,]+∞))
58	35	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
59	37	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → 𝐸 ∈ 𝒫 𝑂)
60	59	elpwincl1 31453	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → (𝐸 ∩ 𝑦) ∈ 𝒫 𝑂)
61	58, 60	ffvelcdmd 7036	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑦 ∈ {𝑥}) → (𝑀‘(𝐸 ∩ 𝑦)) ∈ (0[,]+∞))
62	44, 45, 46, 48, 50, 53, 57, 61	esumsplit 32652	. . . . 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 481	. . . 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 4891	. . . . . . . 8 ⊢ ∪ (𝑏 ∪ {𝑥}) = (∪ 𝑏 ∪ ∪ {𝑥})
65		unisnv 4888	. . . . . . . . 9 ⊢ ∪ {𝑥} = 𝑥
66	65	uneq2i 4120	. . . . . . . 8 ⊢ (∪ 𝑏 ∪ ∪ {𝑥}) = (∪ 𝑏 ∪ 𝑥)
67	64, 66	eqtri 2764	. . . . . . 7 ⊢ ∪ (𝑏 ∪ {𝑥}) = (∪ 𝑏 ∪ 𝑥)
68	67	ineq2i 4169	. . . . . 6 ⊢ (𝐸 ∩ ∪ (𝑏 ∪ {𝑥})) = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))
69	68	fveq2i 6845	. . . . 5 ⊢ (𝑀‘(𝐸 ∩ ∪ (𝑏 ∪ {𝑥}))) = (𝑀‘(𝐸 ∩ (∪ 𝑏 ∪ 𝑥)))
70		inass 4179	. . . . . . . . . 10 ⊢ ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏) = (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∩ ∪ 𝑏))
71		indir 4235	. . . . . . . . . . . 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 4161	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑏 ∩ 𝑥) = (𝑥 ∩ ∪ 𝑏)
75		carsgclctunlem1.1	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → Disj 𝑦 ∈ 𝐴 𝑦)
76	75	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → Disj 𝑦 ∈ 𝐴 𝑦)
77		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑏 ⊆ 𝐴)
78	77	adantrr 715	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → 𝑏 ⊆ 𝐴)
79	76, 78, 34	disjuniel 31515	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (∪ 𝑏 ∩ 𝑥) = ∅)
80	74, 79	eqtr3id 2790	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑥 ∩ ∪ 𝑏) = ∅)
81	73, 80	uneq12d 4124	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((∪ 𝑏 ∩ ∪ 𝑏) ∪ (𝑥 ∩ ∪ 𝑏)) = (∪ 𝑏 ∪ ∅))
82		un0 4350	. . . . . . . . . . . . 13 ⊢ (∪ 𝑏 ∪ ∅) = ∪ 𝑏
83	81, 82	eqtrdi 2792	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((∪ 𝑏 ∩ ∪ 𝑏) ∪ (𝑥 ∩ ∪ 𝑏)) = ∪ 𝑏)
84	71, 83	eqtrid 2788	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((∪ 𝑏 ∪ 𝑥) ∩ ∪ 𝑏) = ∪ 𝑏)
85	84	ineq2d 4172	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∩ ∪ 𝑏)) = (𝐸 ∩ ∪ 𝑏))
86	70, 85	eqtrid 2788	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏) = (𝐸 ∩ ∪ 𝑏))
87	86	fveq2d 6846	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏)) = (𝑀‘(𝐸 ∩ ∪ 𝑏)))
88		indif2 4230	. . . . . . . . . 10 ⊢ (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∖ ∪ 𝑏)) = ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏)
89		uncom 4113	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑏 ∪ 𝑥) = (𝑥 ∪ ∪ 𝑏)
90	89	difeq1i 4078	. . . . . . . . . . . . 13 ⊢ ((∪ 𝑏 ∪ 𝑥) ∖ ∪ 𝑏) = ((𝑥 ∪ ∪ 𝑏) ∖ ∪ 𝑏)
91		difun2 4440	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∪ ∪ 𝑏) ∖ ∪ 𝑏) = (𝑥 ∖ ∪ 𝑏)
92		disj3 4413	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∩ ∪ 𝑏) = ∅ ↔ 𝑥 = (𝑥 ∖ ∪ 𝑏))
93	92	biimpi 215	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∩ ∪ 𝑏) = ∅ → 𝑥 = (𝑥 ∖ ∪ 𝑏))
94	91, 93	eqtr4id 2795	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∩ ∪ 𝑏) = ∅ → ((𝑥 ∪ ∪ 𝑏) ∖ ∪ 𝑏) = 𝑥)
95	90, 94	eqtrid 2788	. . . . . . . . . . . 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 4172	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝐸 ∩ ((∪ 𝑏 ∪ 𝑥) ∖ ∪ 𝑏)) = (𝐸 ∩ 𝑥))
98	88, 97	eqtr3id 2790	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏) = (𝐸 ∩ 𝑥))
99	98	fveq2d 6846	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏)) = (𝑀‘(𝐸 ∩ 𝑥)))
100	87, 99	oveq12d 7375	. . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑂 ∈ 𝑉)
103	35	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
104	21	adantr 481	. . . . . . . . . . . 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 9117	. . . . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝐴 ⊆ (toCaraSiga‘𝑀))
112	77, 111	sstrd 3954	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → 𝑏 ⊆ (toCaraSiga‘𝑀))
113	102, 103, 104, 106, 109, 112	fiunelcarsg 32916	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → ∪ 𝑏 ∈ (toCaraSiga‘𝑀))
114	101, 35	elcarsg 32905	. . . . . . . . . . . 12 ⊢ (𝜑 → (∪ 𝑏 ∈ (toCaraSiga‘𝑀) ↔ (∪ 𝑏 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒))))
115	114	adantr 481	. . . . . . . . . . 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 231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → (∪ 𝑏 ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒)))
117	116	simprd 496	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ⊆ 𝐴) → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒))
118	117	adantrr 715	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝑒 ∖ ∪ 𝑏))) = (𝑀‘𝑒))
119	38	elpwincl1 31453	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∈ 𝒫 𝑂)
120		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥)))
121	120	ineq1d 4171	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑒 ∩ ∪ 𝑏) = ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏))
122	121	fveq2d 6846	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑀‘(𝑒 ∩ ∪ 𝑏)) = (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∩ ∪ 𝑏)))
123	120	difeq1d 4081	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑒 ∖ ∪ 𝑏) = ((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏))
124	123	fveq2d 6846	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑀‘(𝑒 ∖ ∪ 𝑏)) = (𝑀‘((𝐸 ∩ (∪ 𝑏 ∪ 𝑥)) ∖ ∪ 𝑏)))
125	122, 124	oveq12d 7375	. . . . . . . . . 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 6846	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑒 = (𝐸 ∩ (∪ 𝑏 ∪ 𝑥))) → (𝑀‘𝑒) = (𝑀‘(𝐸 ∩ (∪ 𝑏 ∪ 𝑥))))
127	125, 126	eqeq12d 2752	. . . . . . . . 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 3573	. . . . . . . 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 2778	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → ((𝑀‘(𝐸 ∩ ∪ 𝑏)) +_𝑒 (𝑀‘(𝐸 ∩ 𝑥))) = (𝑀‘(𝐸 ∩ (∪ 𝑏 ∪ 𝑥))))
131	130	adantr 481	. . . . 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 2795	. . . 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 2787	. . 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 413	. 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 9110	1 ⊢ (𝜑 → (𝑀‘(𝐸 ∩ ∪ 𝐴)) = Σ^*𝑦 ∈ 𝐴(𝑀‘(𝐸 ∩ 𝑦)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 Vcvv 3445 ∖ cdif 3907 ∪ cun 3908 ∩ cin 3909 ⊆ wss 3910 ∅c0 4282 𝒫 cpw 4560 {csn 4586 ∪ cuni 4865 Disj wdisj 5070 class class class wbr 5105 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ωcom 7802 ≼ cdom 8881 Fincfn 8883 0cc0 11051 +∞cpnf 11186 ≤ cle 11190 +_𝑒 cxad 13031 [,]cicc 13267 Σ^*cesum 32626 toCaraSigaccarsg 32901

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 df-cos 15953 df-pi 15955 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-ordt 17383 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-ps 18455 df-tsr 18456 df-plusf 18496 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-subrg 20220 df-abv 20276 df-lmod 20324 df-scaf 20325 df-sra 20633 df-rgmod 20634 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-tmd 23423 df-tgp 23424 df-tsms 23478 df-trg 23511 df-xms 23673 df-ms 23674 df-tms 23675 df-nm 23938 df-ngp 23939 df-nrg 23941 df-nlm 23942 df-ii 24240 df-cncf 24241 df-limc 25230 df-dv 25231 df-log 25912 df-esum 32627 df-carsg 32902

This theorem is referenced by: carsggect 32918 carsgclctunlem2 32919

W3C validator