Theorem esum2dlem 31353

Description: Lemma for esum2d 31354 (finite case). (Contributed by Thierry Arnoux, 17-May-2020.) (Proof shortened by AV, 17-Sep-2021.)

Hypotheses

Ref	Expression
esum2d.0	⊢ Ⅎ𝑘𝐹
esum2d.1	⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐹 = 𝐶)
esum2d.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
esum2d.3	⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)
esum2d.4	⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞))
esum2dlem.e	⊢ (𝜑 → 𝐴 ∈ Fin)

Assertion

Ref	Expression
esum2dlem	⊢ (𝜑 → Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)

Distinct variable groups:   𝑗,𝑘,𝐴,𝑧   𝑧,𝐶   𝐵,𝑘,𝑧   𝑗,𝐹   𝑗,𝑊,𝑘   𝜑,𝑗,𝑘,𝑧

Allowed substitution hints:   𝐵(𝑗)   𝐶(𝑗,𝑘)   𝐹(𝑧,𝑘)   𝑉(𝑧,𝑗,𝑘)   𝑊(𝑧)

Proof of Theorem esum2dlem

Dummy variables 𝑖 𝑙 𝑡 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		esumeq1 31295	. . 3 ⊢ (𝑎 = ∅ → Σ*𝑗 ∈ 𝑎Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑗 ∈ ∅Σ*𝑘 ∈ 𝐵𝐶)
2		nfv 1915	. . . 4 ⊢ Ⅎ𝑧 𝑎 = ∅
3		iuneq1 4937	. . . 4 ⊢ (𝑎 = ∅ → ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵))
4	2, 3	esumeq1d 31296	. . 3 ⊢ (𝑎 = ∅ → Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 = Σ*𝑧 ∈ ∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵)𝐹)
5	1, 4	eqeq12d 2839	. 2 ⊢ (𝑎 = ∅ → (Σ*𝑗 ∈ 𝑎Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 ↔ Σ*𝑗 ∈ ∅Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵)𝐹))
6		esumeq1 31295	. . 3 ⊢ (𝑎 = 𝑏 → Σ*𝑗 ∈ 𝑎Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶)
7		nfv 1915	. . . 4 ⊢ Ⅎ𝑧 𝑎 = 𝑏
8		iuneq1 4937	. . . 4 ⊢ (𝑎 = 𝑏 → ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵))
9	7, 8	esumeq1d 31296	. . 3 ⊢ (𝑎 = 𝑏 → Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹)
10	6, 9	eqeq12d 2839	. 2 ⊢ (𝑎 = 𝑏 → (Σ*𝑗 ∈ 𝑎Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 ↔ Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹))
11		esumeq1 31295	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑙}) → Σ*𝑗 ∈ 𝑎Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑗 ∈ (𝑏 ∪ {𝑙})Σ*𝑘 ∈ 𝐵𝐶)
12		nfv 1915	. . . 4 ⊢ Ⅎ𝑧 𝑎 = (𝑏 ∪ {𝑙})
13		iuneq1 4937	. . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑙}) → ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵))
14	12, 13	esumeq1d 31296	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑙}) → Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 = Σ*𝑧 ∈ ∪ 𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵)𝐹)
15	11, 14	eqeq12d 2839	. 2 ⊢ (𝑎 = (𝑏 ∪ {𝑙}) → (Σ*𝑗 ∈ 𝑎Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 ↔ Σ*𝑗 ∈ (𝑏 ∪ {𝑙})Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵)𝐹))
16		esumeq1 31295	. . 3 ⊢ (𝑎 = 𝐴 → Σ*𝑗 ∈ 𝑎Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶)
17		nfv 1915	. . . 4 ⊢ Ⅎ𝑧 𝑎 = 𝐴
18		iuneq1 4937	. . . 4 ⊢ (𝑎 = 𝐴 → ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵) = ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
19	17, 18	esumeq1d 31296	. . 3 ⊢ (𝑎 = 𝐴 → Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)
20	16, 19	eqeq12d 2839	. 2 ⊢ (𝑎 = 𝐴 → (Σ*𝑗 ∈ 𝑎Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑎 ({𝑗} × 𝐵)𝐹 ↔ Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹))
21		esumnul 31309	. . . 4 ⊢ Σ*𝑧 ∈ ∅𝐹 = 0
22		0iun 4988	. . . . 5 ⊢ ∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵) = ∅
23		esumeq1 31295	. . . . 5 ⊢ (∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵) = ∅ → Σ*𝑧 ∈ ∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵)𝐹 = Σ*𝑧 ∈ ∅𝐹)
24	22, 23	ax-mp 5	. . . 4 ⊢ Σ*𝑧 ∈ ∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵)𝐹 = Σ*𝑧 ∈ ∅𝐹
25		esumnul 31309	. . . 4 ⊢ Σ*𝑗 ∈ ∅Σ*𝑘 ∈ 𝐵𝐶 = 0
26	21, 24, 25	3eqtr4ri 2857	. . 3 ⊢ Σ*𝑗 ∈ ∅Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵)𝐹
27	26	a1i 11	. 2 ⊢ (𝜑 → Σ*𝑗 ∈ ∅Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ ∅ ({𝑗} × 𝐵)𝐹)
28		simpr 487	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹) → Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹)
29		nfcsb1v 3909	. . . . . . . . 9 ⊢ Ⅎ𝑗⦋𝑙 / 𝑗⦌𝐵
30		nfcsb1v 3909	. . . . . . . . 9 ⊢ Ⅎ𝑗⦋𝑙 / 𝑗⦌𝐶
31	29, 30	nfesum2 31302	. . . . . . . 8 ⊢ Ⅎ𝑗Σ*𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶
32		csbeq1a 3899	. . . . . . . . . 10 ⊢ (𝑗 = 𝑙 → 𝐵 = ⦋𝑙 / 𝑗⦌𝐵)
33		csbeq1a 3899	. . . . . . . . . 10 ⊢ (𝑗 = 𝑙 → 𝐶 = ⦋𝑙 / 𝑗⦌𝐶)
34	32, 33	esumeq12d 31294	. . . . . . . . 9 ⊢ (𝑗 = 𝑙 → Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶)
35	34	adantl 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 = 𝑙) → Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶)
36		simprr 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → 𝑙 ∈ (𝐴 ∖ 𝑏))
37	36	eldifad 3950	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → 𝑙 ∈ 𝐴)
38		esum2d.3	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)
39	38	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)
40	39	ralrimiva 3184	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ∀𝑗 ∈ 𝐴 𝐵 ∈ 𝑊)
41		rspcsbela 4389	. . . . . . . . . 10 ⊢ ((𝑙 ∈ 𝐴 ∧ ∀𝑗 ∈ 𝐴 𝐵 ∈ 𝑊) → ⦋𝑙 / 𝑗⦌𝐵 ∈ 𝑊)
42	37, 40, 41	syl2anc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ⦋𝑙 / 𝑗⦌𝐵 ∈ 𝑊)
43		simpll 765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵) → 𝜑)
44	37	adantr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵) → 𝑙 ∈ 𝐴)
45		simpr 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵) → 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵)
46		esum2d.4	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ (0[,]+∞))
47	46	ex 415	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞)))
48	47	sbcimdv 3845	. . . . . . . . . . . . 13 ⊢ (𝜑 → ([𝑙 / 𝑗](𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → [𝑙 / 𝑗]𝐶 ∈ (0[,]+∞)))
49		sbcan 3823	. . . . . . . . . . . . . 14 ⊢ ([𝑙 / 𝑗](𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) ↔ ([𝑙 / 𝑗]𝑗 ∈ 𝐴 ∧ [𝑙 / 𝑗]𝑘 ∈ 𝐵))
50		sbcel1v 3841	. . . . . . . . . . . . . . 15 ⊢ ([𝑙 / 𝑗]𝑗 ∈ 𝐴 ↔ 𝑙 ∈ 𝐴)
51		sbcel2 4369	. . . . . . . . . . . . . . 15 ⊢ ([𝑙 / 𝑗]𝑘 ∈ 𝐵 ↔ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵)
52	50, 51	anbi12i 628	. . . . . . . . . . . . . 14 ⊢ (([𝑙 / 𝑗]𝑗 ∈ 𝐴 ∧ [𝑙 / 𝑗]𝑘 ∈ 𝐵) ↔ (𝑙 ∈ 𝐴 ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵))
53	49, 52	bitri 277	. . . . . . . . . . . . 13 ⊢ ([𝑙 / 𝑗](𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) ↔ (𝑙 ∈ 𝐴 ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵))
54		vex 3499	. . . . . . . . . . . . . 14 ⊢ 𝑙 ∈ V
55		sbcel1g 4367	. . . . . . . . . . . . . 14 ⊢ (𝑙 ∈ V → ([𝑙 / 𝑗]𝐶 ∈ (0[,]+∞) ↔ ⦋𝑙 / 𝑗⦌𝐶 ∈ (0[,]+∞)))
56	54, 55	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ([𝑙 / 𝑗]𝐶 ∈ (0[,]+∞) ↔ ⦋𝑙 / 𝑗⦌𝐶 ∈ (0[,]+∞))
57	48, 53, 56	3imtr3g 297	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑙 ∈ 𝐴 ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵) → ⦋𝑙 / 𝑗⦌𝐶 ∈ (0[,]+∞)))
58	57	imp 409	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑙 ∈ 𝐴 ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵)) → ⦋𝑙 / 𝑗⦌𝐶 ∈ (0[,]+∞))
59	43, 44, 45, 58	syl12anc 834	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵) → ⦋𝑙 / 𝑗⦌𝐶 ∈ (0[,]+∞))
60	59	ralrimiva 3184	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ∀𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶 ∈ (0[,]+∞))
61		nfcv 2979	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑙 / 𝑗⦌𝐵
62	61	esumcl 31291	. . . . . . . . 9 ⊢ ((⦋𝑙 / 𝑗⦌𝐵 ∈ 𝑊 ∧ ∀𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶 ∈ (0[,]+∞)) → Σ*𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶 ∈ (0[,]+∞))
63	42, 60, 62	syl2anc 586	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → Σ*𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶 ∈ (0[,]+∞))
64	31, 35, 36, 63	esumsnf 31325	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → Σ*𝑗 ∈ {𝑙}Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶)
65		esum2d.0	. . . . . . . . 9 ⊢ Ⅎ𝑘𝐹
66		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)))
67		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝑧 = ⟨𝑙, 𝑘⟩
68	30	nfeq2 2997	. . . . . . . . . . 11 ⊢ Ⅎ𝑗 𝐹 = ⦋𝑙 / 𝑗⦌𝐶
69	67, 68	nfim 1897	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑧 = ⟨𝑙, 𝑘⟩ → 𝐹 = ⦋𝑙 / 𝑗⦌𝐶)
70		opeq1 4805	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑙 → ⟨𝑗, 𝑘⟩ = ⟨𝑙, 𝑘⟩)
71	70	eqeq2d 2834	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑙 → (𝑧 = ⟨𝑗, 𝑘⟩ ↔ 𝑧 = ⟨𝑙, 𝑘⟩))
72	33	eqeq2d 2834	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑙 → (𝐹 = 𝐶 ↔ 𝐹 = ⦋𝑙 / 𝑗⦌𝐶))
73	71, 72	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑗 = 𝑙 → ((𝑧 = ⟨𝑗, 𝑘⟩ → 𝐹 = 𝐶) ↔ (𝑧 = ⟨𝑙, 𝑘⟩ → 𝐹 = ⦋𝑙 / 𝑗⦌𝐶)))
74		esum2d.1	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑗, 𝑘⟩ → 𝐹 = 𝐶)
75	69, 73, 74	chvarfv 2242	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑙, 𝑘⟩ → 𝐹 = ⦋𝑙 / 𝑗⦌𝐶)
76		vsnid 4604	. . . . . . . . . . . . . . . . 17 ⊢ 𝑗 ∈ {𝑗}
77	76	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝑗 ∈ {𝑗})
78		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ 𝐵)
79	77, 78	opelxpd 5595	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → ⟨𝑗, 𝑘⟩ ∈ ({𝑗} × 𝐵))
80		xp2nd 7724	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (2nd ‘𝑧) ∈ 𝐵)
81		xp1st 7723	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ {𝑗})
82		fvex 6685	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1st ‘𝑧) ∈ V
83	82	elsn 4584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘𝑧) ∈ {𝑗} ↔ (1st ‘𝑧) = 𝑗)
84	81, 83	sylib 220	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) = 𝑗)
85		eqop 7733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (𝑧 = ⟨𝑗, 𝑘⟩ ↔ ((1st ‘𝑧) = 𝑗 ∧ (2nd ‘𝑧) = 𝑘)))
86	84, 85	mpbirand 705	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (𝑧 = ⟨𝑗, 𝑘⟩ ↔ (2nd ‘𝑧) = 𝑘))
87		eqcom 2830	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑧) = 𝑘 ↔ 𝑘 = (2nd ‘𝑧))
88	86, 87	syl6bb 289	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (𝑧 = ⟨𝑗, 𝑘⟩ ↔ 𝑘 = (2nd ‘𝑧)))
89	88	ad2antlr 725	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) → (𝑧 = ⟨𝑗, 𝑘⟩ ↔ 𝑘 = (2nd ‘𝑧)))
90	89	ralrimiva 3184	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∀𝑘 ∈ 𝐵 (𝑧 = ⟨𝑗, 𝑘⟩ ↔ 𝑘 = (2nd ‘𝑧)))
91		reu6i 3721	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘𝑧) ∈ 𝐵 ∧ ∀𝑘 ∈ 𝐵 (𝑧 = ⟨𝑗, 𝑘⟩ ↔ 𝑘 = (2nd ‘𝑧))) → ∃!𝑘 ∈ 𝐵 𝑧 = ⟨𝑗, 𝑘⟩)
92	80, 90, 91	syl2an2 684	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∃!𝑘 ∈ 𝐵 𝑧 = ⟨𝑗, 𝑘⟩)
93	79, 92	f1mptrn 30382	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → Fun ◡(𝑘 ∈ 𝐵 ↦ ⟨𝑗, 𝑘⟩))
94	93	ex 415	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑗 ∈ 𝐴 → Fun ◡(𝑘 ∈ 𝐵 ↦ ⟨𝑗, 𝑘⟩)))
95	94	sbcimdv 3845	. . . . . . . . . . . 12 ⊢ (𝜑 → ([𝑙 / 𝑗]𝑗 ∈ 𝐴 → [𝑙 / 𝑗]Fun ◡(𝑘 ∈ 𝐵 ↦ ⟨𝑗, 𝑘⟩)))
96		sbcfung 6381	. . . . . . . . . . . . . 14 ⊢ (𝑙 ∈ V → ([𝑙 / 𝑗]Fun ◡(𝑘 ∈ 𝐵 ↦ ⟨𝑗, 𝑘⟩) ↔ Fun ⦋𝑙 / 𝑗⦌◡(𝑘 ∈ 𝐵 ↦ ⟨𝑗, 𝑘⟩)))
97		csbcnv 5756	. . . . . . . . . . . . . . . 16 ⊢ ◡⦋𝑙 / 𝑗⦌(𝑘 ∈ 𝐵 ↦ ⟨𝑗, 𝑘⟩) = ⦋𝑙 / 𝑗⦌◡(𝑘 ∈ 𝐵 ↦ ⟨𝑗, 𝑘⟩)
98		csbmpt12 5446	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 ∈ V → ⦋𝑙 / 𝑗⦌(𝑘 ∈ 𝐵 ↦ ⟨𝑗, 𝑘⟩) = (𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ ⦋𝑙 / 𝑗⦌⟨𝑗, 𝑘⟩))
99		csbopg 4823	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 ∈ V → ⦋𝑙 / 𝑗⦌⟨𝑗, 𝑘⟩ = ⟨⦋𝑙 / 𝑗⦌𝑗, ⦋𝑙 / 𝑗⦌𝑘⟩)
100		csbvarg 4385	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 ∈ V → ⦋𝑙 / 𝑗⦌𝑗 = 𝑙)
101		csbconstg 3904	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 ∈ V → ⦋𝑙 / 𝑗⦌𝑘 = 𝑘)
102	100, 101	opeq12d 4813	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 ∈ V → ⟨⦋𝑙 / 𝑗⦌𝑗, ⦋𝑙 / 𝑗⦌𝑘⟩ = ⟨𝑙, 𝑘⟩)
103	99, 102	eqtrd 2858	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 ∈ V → ⦋𝑙 / 𝑗⦌⟨𝑗, 𝑘⟩ = ⟨𝑙, 𝑘⟩)
104	103	mpteq2dv 5164	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 ∈ V → (𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ ⦋𝑙 / 𝑗⦌⟨𝑗, 𝑘⟩) = (𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ ⟨𝑙, 𝑘⟩))
105	98, 104	eqtrd 2858	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 ∈ V → ⦋𝑙 / 𝑗⦌(𝑘 ∈ 𝐵 ↦ ⟨𝑗, 𝑘⟩) = (𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ ⟨𝑙, 𝑘⟩))
106	105	cnveqd 5748	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ V → ◡⦋𝑙 / 𝑗⦌(𝑘 ∈ 𝐵 ↦ ⟨𝑗, 𝑘⟩) = ◡(𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ ⟨𝑙, 𝑘⟩))
107	97, 106	syl5eqr 2872	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ V → ⦋𝑙 / 𝑗⦌◡(𝑘 ∈ 𝐵 ↦ ⟨𝑗, 𝑘⟩) = ◡(𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ ⟨𝑙, 𝑘⟩))
108	107	funeqd 6379	. . . . . . . . . . . . . 14 ⊢ (𝑙 ∈ V → (Fun ⦋𝑙 / 𝑗⦌◡(𝑘 ∈ 𝐵 ↦ ⟨𝑗, 𝑘⟩) ↔ Fun ◡(𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ ⟨𝑙, 𝑘⟩)))
109	96, 108	bitrd 281	. . . . . . . . . . . . 13 ⊢ (𝑙 ∈ V → ([𝑙 / 𝑗]Fun ◡(𝑘 ∈ 𝐵 ↦ ⟨𝑗, 𝑘⟩) ↔ Fun ◡(𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ ⟨𝑙, 𝑘⟩)))
110	54, 109	ax-mp 5	. . . . . . . . . . . 12 ⊢ ([𝑙 / 𝑗]Fun ◡(𝑘 ∈ 𝐵 ↦ ⟨𝑗, 𝑘⟩) ↔ Fun ◡(𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ ⟨𝑙, 𝑘⟩))
111	95, 50, 110	3imtr3g 297	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑙 ∈ 𝐴 → Fun ◡(𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ ⟨𝑙, 𝑘⟩)))
112	111	imp 409	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑙 ∈ 𝐴) → Fun ◡(𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ ⟨𝑙, 𝑘⟩))
113	37, 112	syldan 593	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → Fun ◡(𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵 ↦ ⟨𝑙, 𝑘⟩))
114		vsnid 4604	. . . . . . . . . . 11 ⊢ 𝑙 ∈ {𝑙}
115	114	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵) → 𝑙 ∈ {𝑙})
116	115, 45	opelxpd 5595	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵) → ⟨𝑙, 𝑘⟩ ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵))
117	65, 66, 61, 75, 42, 113, 59, 116	esumc 31312	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → Σ*𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶 = Σ*𝑧 ∈ {𝑡 ∣ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = ⟨𝑙, 𝑘⟩}𝐹)
118		nfab1 2981	. . . . . . . . . 10 ⊢ Ⅎ𝑡{𝑡 ∣ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = ⟨𝑙, 𝑘⟩}
119		nfcv 2979	. . . . . . . . . 10 ⊢ Ⅎ𝑡({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)
120		opeq1 4805	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑙 → ⟨𝑖, 𝑘⟩ = ⟨𝑙, 𝑘⟩)
121	120	eqeq2d 2834	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑙 → (𝑡 = ⟨𝑖, 𝑘⟩ ↔ 𝑡 = ⟨𝑙, 𝑘⟩))
122	121	rexbidv 3299	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑙 → (∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = ⟨𝑖, 𝑘⟩ ↔ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = ⟨𝑙, 𝑘⟩))
123	54, 122	rexsn 4622	. . . . . . . . . . 11 ⊢ (∃𝑖 ∈ {𝑙}∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = ⟨𝑖, 𝑘⟩ ↔ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = ⟨𝑙, 𝑘⟩)
124		elxp2 5581	. . . . . . . . . . 11 ⊢ (𝑡 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵) ↔ ∃𝑖 ∈ {𝑙}∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = ⟨𝑖, 𝑘⟩)
125		abid 2805	. . . . . . . . . . 11 ⊢ (𝑡 ∈ {𝑡 ∣ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = ⟨𝑙, 𝑘⟩} ↔ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = ⟨𝑙, 𝑘⟩)
126	123, 124, 125	3bitr4ri 306	. . . . . . . . . 10 ⊢ (𝑡 ∈ {𝑡 ∣ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = ⟨𝑙, 𝑘⟩} ↔ 𝑡 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵))
127	118, 119, 126	eqri 3989	. . . . . . . . 9 ⊢ {𝑡 ∣ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = ⟨𝑙, 𝑘⟩} = ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)
128		esumeq1 31295	. . . . . . . . 9 ⊢ ({𝑡 ∣ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = ⟨𝑙, 𝑘⟩} = ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵) → Σ*𝑧 ∈ {𝑡 ∣ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = ⟨𝑙, 𝑘⟩}𝐹 = Σ*𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)𝐹)
129	127, 128	ax-mp 5	. . . . . . . 8 ⊢ Σ*𝑧 ∈ {𝑡 ∣ ∃𝑘 ∈ ⦋ 𝑙 / 𝑗⦌𝐵𝑡 = ⟨𝑙, 𝑘⟩}𝐹 = Σ*𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)𝐹
130	117, 129	syl6eq 2874	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → Σ*𝑘 ∈ ⦋𝑙 / 𝑗⦌𝐵⦋𝑙 / 𝑗⦌𝐶 = Σ*𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)𝐹)
131	64, 130	eqtrd 2858	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → Σ*𝑗 ∈ {𝑙}Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)𝐹)
132	131	adantr 483	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹) → Σ*𝑗 ∈ {𝑙}Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)𝐹)
133	28, 132	oveq12d 7176	. . . 4 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹) → (Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 +𝑒 Σ*𝑗 ∈ {𝑙}Σ*𝑘 ∈ 𝐵𝐶) = (Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹 +𝑒 Σ*𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)𝐹))
134		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑗(𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)))
135		nfcv 2979	. . . . . 6 ⊢ Ⅎ𝑗𝑏
136		nfcv 2979	. . . . . 6 ⊢ Ⅎ𝑗{𝑙}
137		vex 3499	. . . . . . 7 ⊢ 𝑏 ∈ V
138	137	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → 𝑏 ∈ V)
139		snex 5334	. . . . . . 7 ⊢ {𝑙} ∈ V
140	139	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → {𝑙} ∈ V)
141	36	eldifbd 3951	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ¬ 𝑙 ∈ 𝑏)
142		disjsn 4649	. . . . . . 7 ⊢ ((𝑏 ∩ {𝑙}) = ∅ ↔ ¬ 𝑙 ∈ 𝑏)
143	141, 142	sylibr 236	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → (𝑏 ∩ {𝑙}) = ∅)
144		simpll 765	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝑏) → 𝜑)
145		simprl 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → 𝑏 ⊆ 𝐴)
146	145	sselda 3969	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝑏) → 𝑗 ∈ 𝐴)
147	46	anassrs 470	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ (0[,]+∞))
148	147	ralrimiva 3184	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞))
149		nfcv 2979	. . . . . . . . 9 ⊢ Ⅎ𝑘𝐵
150	149	esumcl 31291	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑊 ∧ ∀𝑘 ∈ 𝐵 𝐶 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
151	38, 148, 150	syl2anc 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → Σ*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
152	144, 146, 151	syl2anc 586	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝑏) → Σ*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
153		simpll 765	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ {𝑙}) → 𝜑)
154	37	snssd 4744	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → {𝑙} ⊆ 𝐴)
155	154	sselda 3969	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ {𝑙}) → 𝑗 ∈ 𝐴)
156	153, 155, 151	syl2anc 586	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ {𝑙}) → Σ*𝑘 ∈ 𝐵𝐶 ∈ (0[,]+∞))
157	134, 135, 136, 138, 140, 143, 152, 156	esumsplit 31314	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → Σ*𝑗 ∈ (𝑏 ∪ {𝑙})Σ*𝑘 ∈ 𝐵𝐶 = (Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 +𝑒 Σ*𝑗 ∈ {𝑙}Σ*𝑘 ∈ 𝐵𝐶))
158	157	adantr 483	. . . 4 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹) → Σ*𝑗 ∈ (𝑏 ∪ {𝑙})Σ*𝑘 ∈ 𝐵𝐶 = (Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 +𝑒 Σ*𝑗 ∈ {𝑙}Σ*𝑘 ∈ 𝐵𝐶))
159		iunxun 5018	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵) = (∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∪ ∪ 𝑗 ∈ {𝑙} ({𝑗} × 𝐵))
160	136, 29	nfxp 5590	. . . . . . . . . . 11 ⊢ Ⅎ𝑗({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)
161		sneq 4579	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑙 → {𝑗} = {𝑙})
162	161, 32	xpeq12d 5588	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑙 → ({𝑗} × 𝐵) = ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵))
163	160, 162	iunxsngf 5016	. . . . . . . . . 10 ⊢ (𝑙 ∈ V → ∪ 𝑗 ∈ {𝑙} ({𝑗} × 𝐵) = ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵))
164	54, 163	ax-mp 5	. . . . . . . . 9 ⊢ ∪ 𝑗 ∈ {𝑙} ({𝑗} × 𝐵) = ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)
165	164	uneq2i 4138	. . . . . . . 8 ⊢ (∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∪ ∪ 𝑗 ∈ {𝑙} ({𝑗} × 𝐵)) = (∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∪ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵))
166	159, 165	eqtri 2846	. . . . . . 7 ⊢ ∪ 𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵) = (∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∪ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵))
167		esumeq1 31295	. . . . . . 7 ⊢ (∪ 𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵) = (∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∪ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) → Σ*𝑧 ∈ ∪ 𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵)𝐹 = Σ*𝑧 ∈ (∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∪ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵))𝐹)
168	166, 167	ax-mp 5	. . . . . 6 ⊢ Σ*𝑧 ∈ ∪ 𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵)𝐹 = Σ*𝑧 ∈ (∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∪ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵))𝐹
169		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑧(𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏)))
170		nfcv 2979	. . . . . . 7 ⊢ Ⅎ𝑧∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵)
171		nfcv 2979	. . . . . . 7 ⊢ Ⅎ𝑧({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)
172		snex 5334	. . . . . . . . . 10 ⊢ {𝑗} ∈ V
173	146, 39	syldan 593	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝑏) → 𝐵 ∈ 𝑊)
174		xpexg 7475	. . . . . . . . . 10 ⊢ (({𝑗} ∈ V ∧ 𝐵 ∈ 𝑊) → ({𝑗} × 𝐵) ∈ V)
175	172, 173, 174	sylancr 589	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝑏) → ({𝑗} × 𝐵) ∈ V)
176	175	ralrimiva 3184	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ∀𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∈ V)
177		iunexg 7666	. . . . . . . 8 ⊢ ((𝑏 ∈ V ∧ ∀𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∈ V) → ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∈ V)
178	137, 176, 177	sylancr 589	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∈ V)
179		xpexg 7475	. . . . . . . 8 ⊢ (({𝑙} ∈ V ∧ ⦋𝑙 / 𝑗⦌𝐵 ∈ 𝑊) → ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵) ∈ V)
180	139, 42, 179	sylancr 589	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵) ∈ V)
181		simpr 487	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝑏) → 𝑗 ∈ 𝑏)
182	141	adantr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝑏) → ¬ 𝑙 ∈ 𝑏)
183		nelne2 3117	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ 𝑏 ∧ ¬ 𝑙 ∈ 𝑏) → 𝑗 ≠ 𝑙)
184	181, 182, 183	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝑏) → 𝑗 ≠ 𝑙)
185		disjsn2 4650	. . . . . . . . . 10 ⊢ (𝑗 ≠ 𝑙 → ({𝑗} ∩ {𝑙}) = ∅)
186		xpdisj1 6020	. . . . . . . . . 10 ⊢ (({𝑗} ∩ {𝑙}) = ∅ → (({𝑗} × 𝐵) ∩ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) = ∅)
187	184, 185, 186	3syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑗 ∈ 𝑏) → (({𝑗} × 𝐵) ∩ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) = ∅)
188	187	iuneq2dv 4945	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ∪ 𝑗 ∈ 𝑏 (({𝑗} × 𝐵) ∩ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) = ∪ 𝑗 ∈ 𝑏 ∅)
189	160	iunin1f 30311	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ 𝑏 (({𝑗} × 𝐵) ∩ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) = (∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∩ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵))
190		iun0 4987	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ 𝑏 ∅ = ∅
191	188, 189, 190	3eqtr3g 2881	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → (∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∩ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) = ∅)
192		simpll 765	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵)) → 𝜑)
193		iunss1 4935	. . . . . . . . . 10 ⊢ (𝑏 ⊆ 𝐴 → ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
194	145, 193	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
195	194	sselda 3969	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵)) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
196		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝜑
197		nfiu1 4955	. . . . . . . . . . 11 ⊢ Ⅎ𝑗∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
198	197	nfcri 2973	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)
199	196, 198	nfan 1900	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
200		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵))
201		nfcv 2979	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(0[,]+∞)
202	65, 201	nfel 2994	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝐹 ∈ (0[,]+∞)
203	74	adantl 484	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝐹 = 𝐶)
204		simp-5l 783	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝜑)
205		simp-4r 782	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝑗 ∈ 𝐴)
206		simplr 767	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝑘 ∈ 𝐵)
207	204, 205, 206, 46	syl12anc 834	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝐶 ∈ (0[,]+∞))
208	203, 207	eqeltrd 2915	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) ∧ 𝑘 ∈ 𝐵) ∧ 𝑧 = ⟨𝑗, 𝑘⟩) → 𝐹 ∈ (0[,]+∞))
209		elsnxp 6144	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ 𝐴 → (𝑧 ∈ ({𝑗} × 𝐵) ↔ ∃𝑘 ∈ 𝐵 𝑧 = ⟨𝑗, 𝑘⟩))
210	209	biimpa 479	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∃𝑘 ∈ 𝐵 𝑧 = ⟨𝑗, 𝑘⟩)
211	210	adantll 712	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → ∃𝑘 ∈ 𝐵 𝑧 = ⟨𝑗, 𝑘⟩)
212	200, 202, 208, 211	r19.29af2 3332	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) ∧ 𝑗 ∈ 𝐴) ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
213		simpr 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
214		eliun 4925	. . . . . . . . . 10 ⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵))
215	213, 214	sylib 220	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → ∃𝑗 ∈ 𝐴 𝑧 ∈ ({𝑗} × 𝐵))
216	199, 212, 215	r19.29af 3333	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
217	192, 195, 216	syl2anc 586	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑧 ∈ ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵)) → 𝐹 ∈ (0[,]+∞))
218		simpll 765	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) → 𝜑)
219		nfcv 2979	. . . . . . . . . . 11 ⊢ Ⅎ𝑗𝐴
220		nfcv 2979	. . . . . . . . . . 11 ⊢ Ⅎ𝑗𝑙
221	219, 220, 160, 162	ssiun2sf 30313	. . . . . . . . . 10 ⊢ (𝑙 ∈ 𝐴 → ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵) ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
222	37, 221	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵) ⊆ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
223	222	sselda 3969	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) → 𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵))
224	218, 223, 216	syl2anc 586	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ 𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)) → 𝐹 ∈ (0[,]+∞))
225	169, 170, 171, 178, 180, 191, 217, 224	esumsplit 31314	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → Σ*𝑧 ∈ (∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵) ∪ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵))𝐹 = (Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹 +𝑒 Σ*𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)𝐹))
226	168, 225	syl5eq 2870	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → Σ*𝑧 ∈ ∪ 𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵)𝐹 = (Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹 +𝑒 Σ*𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)𝐹))
227	226	adantr 483	. . . 4 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹) → Σ*𝑧 ∈ ∪ 𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵)𝐹 = (Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹 +𝑒 Σ*𝑧 ∈ ({𝑙} × ⦋𝑙 / 𝑗⦌𝐵)𝐹))
228	133, 158, 227	3eqtr4d 2868	. . 3 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) ∧ Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹) → Σ*𝑗 ∈ (𝑏 ∪ {𝑙})Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵)𝐹)
229	228	ex 415	. 2 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑙 ∈ (𝐴 ∖ 𝑏))) → (Σ*𝑗 ∈ 𝑏Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝑏 ({𝑗} × 𝐵)𝐹 → Σ*𝑗 ∈ (𝑏 ∪ {𝑙})Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ (𝑏 ∪ {𝑙})({𝑗} × 𝐵)𝐹))
230		esum2dlem.e	. 2 ⊢ (𝜑 → 𝐴 ∈ Fin)
231	5, 10, 15, 20, 27, 229, 230	findcard2d 8762	1 ⊢ (𝜑 → Σ*𝑗 ∈ 𝐴Σ*𝑘 ∈ 𝐵𝐶 = Σ*𝑧 ∈ ∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵)𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {cab 2801  Ⅎwnfc 2963   ≠ wne 3018  ∀wral 3140  ∃wrex 3141  ∃!wreu 3142  Vcvv 3496  [wsbc 3774  ⦋csb 3885   ∖ cdif 3935   ∪ cun 3936   ∩ cin 3937   ⊆ wss 3938  ∅c0 4293  {csn 4569  ⟨cop 4575  ∪ ciun 4921   ↦ cmpt 5148   × cxp 5555  ^◡ccnv 5556  Fun wfun 6351  ‘cfv 6357  (class class class)co 7158  1^st c1st 7689  2^nd c2nd 7690  Fincfn 8511  0cc0 10539  +∞cpnf 10674   +_𝑒 cxad 12508  [,]cicc 12744  Σ^*cesum 31288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617  ax-addf 10618  ax-mulf 10619

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-of 7411  df-om 7583  df-1st 7691  df-2nd 7692  df-supp 7833  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-map 8410  df-pm 8411  df-ixp 8464  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-fsupp 8836  df-fi 8877  df-sup 8908  df-inf 8909  df-oi 8976  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-q 12352  df-rp 12393  df-xneg 12510  df-xadd 12511  df-xmul 12512  df-ioo 12745  df-ioc 12746  df-ico 12747  df-icc 12748  df-fz 12896  df-fzo 13037  df-fl 13165  df-mod 13241  df-seq 13373  df-exp 13433  df-fac 13637  df-bc 13666  df-hash 13694  df-shft 14428  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-limsup 14830  df-clim 14847  df-rlim 14848  df-sum 15045  df-ef 15423  df-sin 15425  df-cos 15426  df-pi 15428  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-mulr 16581  df-starv 16582  df-sca 16583  df-vsca 16584  df-ip 16585  df-tset 16586  df-ple 16587  df-ds 16589  df-unif 16590  df-hom 16591  df-cco 16592  df-rest 16698  df-topn 16699  df-0g 16717  df-gsum 16718  df-topgen 16719  df-pt 16720  df-prds 16723  df-ordt 16776  df-xrs 16777  df-qtop 16782  df-imas 16783  df-xps 16785  df-mre 16859  df-mrc 16860  df-acs 16862  df-ps 17812  df-tsr 17813  df-plusf 17853  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-mhm 17958  df-submnd 17959  df-grp 18108  df-minusg 18109  df-sbg 18110  df-mulg 18227  df-subg 18278  df-cntz 18449  df-cmn 18910  df-abl 18911  df-mgp 19242  df-ur 19254  df-ring 19301  df-cring 19302  df-subrg 19535  df-abv 19590  df-lmod 19638  df-scaf 19639  df-sra 19946  df-rgmod 19947  df-psmet 20539  df-xmet 20540  df-met 20541  df-bl 20542  df-mopn 20543  df-fbas 20544  df-fg 20545  df-cnfld 20548  df-top 21504  df-topon 21521  df-topsp 21543  df-bases 21556  df-cld 21629  df-ntr 21630  df-cls 21631  df-nei 21708  df-lp 21746  df-perf 21747  df-cn 21837  df-cnp 21838  df-haus 21925  df-tx 22172  df-hmeo 22365  df-fil 22456  df-fm 22548  df-flim 22549  df-flf 22550  df-tmd 22682  df-tgp 22683  df-tsms 22737  df-trg 22770  df-xms 22932  df-ms 22933  df-tms 22934  df-nm 23194  df-ngp 23195  df-nrg 23197  df-nlm 23198  df-ii 23487  df-cncf 23488  df-limc 24466  df-dv 24467  df-log 25142  df-esum 31289

This theorem is referenced by:  esum2d  31354