itg1addlem5 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > itg1addlem5		Structured version Visualization version GIF version

Theorem itg1addlem5 23467

Description: Lemma for itg1add . (Contributed by Mario Carneiro, 27-Jun-2014.)

**Hypotheses**
Ref	Expression
i1fadd.1	⊢ (𝜑 → 𝐹 ∈ dom ∫₁)
i1fadd.2	⊢ (𝜑 → 𝐺 ∈ dom ∫₁)
itg1add.3	⊢ 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((^◡𝐹 “ {𝑖}) ∩ (^◡𝐺 “ {𝑗})))))
itg1add.4	⊢ 𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))

**Assertion**
Ref	Expression
itg1addlem5	⊢ (𝜑 → (∫₁‘(𝐹 ∘_𝑓 + 𝐺)) = ((∫₁‘𝐹) + (∫₁‘𝐺)))

Distinct variable groups: 𝑖,𝑗,𝐹 𝑖,𝐺,𝑗 𝜑,𝑖,𝑗 Allowed substitution hints: 𝑃(𝑖,𝑗) 𝐼(𝑖,𝑗)

Proof of Theorem itg1addlem5 Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		i1fadd.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ dom ∫₁)
2		i1frn 23444	. . . 4 ⊢ (𝐹 ∈ dom ∫₁ → ran 𝐹 ∈ Fin)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → ran 𝐹 ∈ Fin)
4		i1fadd.2	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ dom ∫₁)
5		i1frn 23444	. . . . . 6 ⊢ (𝐺 ∈ dom ∫₁ → ran 𝐺 ∈ Fin)
6	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
7	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → ran 𝐺 ∈ Fin)
8		i1ff 23443	. . . . . . . . . 10 ⊢ (𝐹 ∈ dom ∫₁ → 𝐹:ℝ⟶ℝ)
9	1, 8	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
10		frn 6053	. . . . . . . . 9 ⊢ (𝐹:ℝ⟶ℝ → ran 𝐹 ⊆ ℝ)
11	9, 10	syl 17	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
12	11	sselda 3603	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
13	12	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
14	13	recnd 10068	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℂ)
15		itg1add.3	. . . . . . . . 9 ⊢ 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((^◡𝐹 “ {𝑖}) ∩ (^◡𝐺 “ {𝑗})))))
16	1, 4, 15	itg1addlem2 23464	. . . . . . . 8 ⊢ (𝜑 → 𝐼:(ℝ × ℝ)⟶ℝ)
17	16	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
18		i1ff 23443	. . . . . . . . . . 11 ⊢ (𝐺 ∈ dom ∫₁ → 𝐺:ℝ⟶ℝ)
19	4, 18	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
20		frn 6053	. . . . . . . . . 10 ⊢ (𝐺:ℝ⟶ℝ → ran 𝐺 ⊆ ℝ)
21	19, 20	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐺 ⊆ ℝ)
22	21	sselda 3603	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
23	22	adantlr 751	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
24	17, 13, 23	fovrnd 6806	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℝ)
25	24	recnd 10068	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℂ)
26	14, 25	mulcld 10060	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
27	7, 26	fsumcl 14464	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
28	23	recnd 10068	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℂ)
29	28, 25	mulcld 10060	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
30	7, 29	fsumcl 14464	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
31	3, 27, 30	fsumadd 14470	. 2 ⊢ (𝜑 → Σ𝑦 ∈ ran 𝐹(Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))) = (Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
32		itg1add.4	. . . 4 ⊢ 𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))
33	1, 4, 15, 32	itg1addlem4 23466	. . 3 ⊢ (𝜑 → (∫₁‘(𝐹 ∘_𝑓 + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
34	14, 28, 25	adddird 10065	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = ((𝑦 · (𝑦𝐼𝑧)) + (𝑧 · (𝑦𝐼𝑧))))
35	34	sumeq2dv 14433	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺((𝑦 · (𝑦𝐼𝑧)) + (𝑧 · (𝑦𝐼𝑧))))
36	7, 26, 29	fsumadd 14470	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺((𝑦 · (𝑦𝐼𝑧)) + (𝑧 · (𝑦𝐼𝑧))) = (Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
37	35, 36	eqtrd 2656	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = (Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
38	37	sumeq2dv 14433	. . 3 ⊢ (𝜑 → Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹(Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
39	33, 38	eqtrd 2656	. 2 ⊢ (𝜑 → (∫₁‘(𝐹 ∘_𝑓 + 𝐺)) = Σ𝑦 ∈ ran 𝐹(Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
40		itg1val 23450	. . . . 5 ⊢ (𝐹 ∈ dom ∫₁ → (∫₁‘𝐹) = Σ𝑦 ∈ (ran 𝐹 ∖ {0})(𝑦 · (vol‘(^◡𝐹 “ {𝑦}))))
41	1, 40	syl 17	. . . 4 ⊢ (𝜑 → (∫₁‘𝐹) = Σ𝑦 ∈ (ran 𝐹 ∖ {0})(𝑦 · (vol‘(^◡𝐹 “ {𝑦}))))
42	19	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝐺:ℝ⟶ℝ)
43	6	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → ran 𝐺 ∈ Fin)
44		inss2 3834	. . . . . . . . . 10 ⊢ ((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧})) ⊆ (^◡𝐺 “ {𝑧})
45	44	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧})) ⊆ (^◡𝐺 “ {𝑧}))
46		i1fima 23445	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ dom ∫₁ → (^◡𝐹 “ {𝑦}) ∈ dom vol)
47	1, 46	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (^◡𝐹 “ {𝑦}) ∈ dom vol)
48	47	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (^◡𝐹 “ {𝑦}) ∈ dom vol)
49		i1fima 23445	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ dom ∫₁ → (^◡𝐺 “ {𝑧}) ∈ dom vol)
50	4, 49	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (^◡𝐺 “ {𝑧}) ∈ dom vol)
51	50	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (^◡𝐺 “ {𝑧}) ∈ dom vol)
52		inmbl 23310	. . . . . . . . . 10 ⊢ (((^◡𝐹 “ {𝑦}) ∈ dom vol ∧ (^◡𝐺 “ {𝑧}) ∈ dom vol) → ((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧})) ∈ dom vol)
53	48, 51, 52	syl2anc 693	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧})) ∈ dom vol)
54	11	ssdifssd 3748	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ran 𝐹 ∖ {0}) ⊆ ℝ)
55	54	sselda 3603	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝑦 ∈ ℝ)
56	55	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
57	21	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → ran 𝐺 ⊆ ℝ)
58	57	sselda 3603	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
59		eldifsni 4320	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → 𝑦 ≠ 0)
60	59	ad2antlr 763	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ≠ 0)
61		simpl 473	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 0 ∧ 𝑧 = 0) → 𝑦 = 0)
62	61	necon3ai 2819	. . . . . . . . . . . 12 ⊢ (𝑦 ≠ 0 → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
63	60, 62	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
64	1, 4, 15	itg1addlem3 23465	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0)) → (𝑦𝐼𝑧) = (vol‘((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧}))))
65	56, 58, 63, 64	syl21anc 1325	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) = (vol‘((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧}))))
66	16	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
67	66, 56, 58	fovrnd 6806	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℝ)
68	65, 67	eqeltrrd 2702	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol‘((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧}))) ∈ ℝ)
69	42, 43, 45, 53, 68	itg1addlem1 23459	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘∪ 𝑧 ∈ ran 𝐺((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧}))) = Σ𝑧 ∈ ran 𝐺(vol‘((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧}))))
70		iunin2 4584	. . . . . . . . . 10 ⊢ ∪ 𝑧 ∈ ran 𝐺((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧})) = ((^◡𝐹 “ {𝑦}) ∩ ∪ 𝑧 ∈ ran 𝐺(^◡𝐺 “ {𝑧}))
71	1	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝐹 ∈ dom ∫₁)
72	71, 46	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (^◡𝐹 “ {𝑦}) ∈ dom vol)
73		mblss 23299	. . . . . . . . . . . . 13 ⊢ ((^◡𝐹 “ {𝑦}) ∈ dom vol → (^◡𝐹 “ {𝑦}) ⊆ ℝ)
74	72, 73	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (^◡𝐹 “ {𝑦}) ⊆ ℝ)
75		iunid 4575	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑧 ∈ ran 𝐺{𝑧} = ran 𝐺
76	75	imaeq2i 5464	. . . . . . . . . . . . . 14 ⊢ (^◡𝐺 “ ∪ 𝑧 ∈ ran 𝐺{𝑧}) = (^◡𝐺 “ ran 𝐺)
77		imaiun 6503	. . . . . . . . . . . . . 14 ⊢ (^◡𝐺 “ ∪ 𝑧 ∈ ran 𝐺{𝑧}) = ∪ 𝑧 ∈ ran 𝐺(^◡𝐺 “ {𝑧})
78		cnvimarndm 5486	. . . . . . . . . . . . . 14 ⊢ (^◡𝐺 “ ran 𝐺) = dom 𝐺
79	76, 77, 78	3eqtr3i 2652	. . . . . . . . . . . . 13 ⊢ ∪ 𝑧 ∈ ran 𝐺(^◡𝐺 “ {𝑧}) = dom 𝐺
80		fdm 6051	. . . . . . . . . . . . . 14 ⊢ (𝐺:ℝ⟶ℝ → dom 𝐺 = ℝ)
81	42, 80	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → dom 𝐺 = ℝ)
82	79, 81	syl5eq 2668	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → ∪ 𝑧 ∈ ran 𝐺(^◡𝐺 “ {𝑧}) = ℝ)
83	74, 82	sseqtr4d 3642	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (^◡𝐹 “ {𝑦}) ⊆ ∪ 𝑧 ∈ ran 𝐺(^◡𝐺 “ {𝑧}))
84		df-ss 3588	. . . . . . . . . . 11 ⊢ ((^◡𝐹 “ {𝑦}) ⊆ ∪ 𝑧 ∈ ran 𝐺(^◡𝐺 “ {𝑧}) ↔ ((^◡𝐹 “ {𝑦}) ∩ ∪ 𝑧 ∈ ran 𝐺(^◡𝐺 “ {𝑧})) = (^◡𝐹 “ {𝑦}))
85	83, 84	sylib 208	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → ((^◡𝐹 “ {𝑦}) ∩ ∪ 𝑧 ∈ ran 𝐺(^◡𝐺 “ {𝑧})) = (^◡𝐹 “ {𝑦}))
86	70, 85	syl5req 2669	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (^◡𝐹 “ {𝑦}) = ∪ 𝑧 ∈ ran 𝐺((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧})))
87	86	fveq2d 6195	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(^◡𝐹 “ {𝑦})) = (vol‘∪ 𝑧 ∈ ran 𝐺((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧}))))
88	65	sumeq2dv 14433	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧) = Σ𝑧 ∈ ran 𝐺(vol‘((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧}))))
89	69, 87, 88	3eqtr4d 2666	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(^◡𝐹 “ {𝑦})) = Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧))
90	89	oveq2d 6666	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝑦 · (vol‘(^◡𝐹 “ {𝑦}))) = (𝑦 · Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧)))
91	55	recnd 10068	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝑦 ∈ ℂ)
92	67	recnd 10068	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℂ)
93	43, 91, 92	fsummulc2 14516	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝑦 · Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
94	90, 93	eqtrd 2656	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝑦 · (vol‘(^◡𝐹 “ {𝑦}))) = Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
95	94	sumeq2dv 14433	. . . 4 ⊢ (𝜑 → Σ𝑦 ∈ (ran 𝐹 ∖ {0})(𝑦 · (vol‘(^◡𝐹 “ {𝑦}))) = Σ𝑦 ∈ (ran 𝐹 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
96		difssd 3738	. . . . 5 ⊢ (𝜑 → (ran 𝐹 ∖ {0}) ⊆ ran 𝐹)
97	56	recnd 10068	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℂ)
98	97, 92	mulcld 10060	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
99	43, 98	fsumcl 14464	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
100		dfin4 3867	. . . . . . . 8 ⊢ (ran 𝐹 ∩ {0}) = (ran 𝐹 ∖ (ran 𝐹 ∖ {0}))
101		inss2 3834	. . . . . . . 8 ⊢ (ran 𝐹 ∩ {0}) ⊆ {0}
102	100, 101	eqsstr3i 3636	. . . . . . 7 ⊢ (ran 𝐹 ∖ (ran 𝐹 ∖ {0})) ⊆ {0}
103	102	sseli 3599	. . . . . 6 ⊢ (𝑦 ∈ (ran 𝐹 ∖ (ran 𝐹 ∖ {0})) → 𝑦 ∈ {0})
104		elsni 4194	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {0} → 𝑦 = 0)
105	104	ad2antlr 763	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 = 0)
106	105	oveq1d 6665	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) = (0 · (𝑦𝐼𝑧)))
107	16	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
108		0re 10040	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
109	105, 108	syl6eqel 2709	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
110	22	adantlr 751	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
111	107, 109, 110	fovrnd 6806	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℝ)
112	111	recnd 10068	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℂ)
113	112	mul02d 10234	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (0 · (𝑦𝐼𝑧)) = 0)
114	106, 113	eqtrd 2656	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) = 0)
115	114	sumeq2dv 14433	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {0}) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺0)
116	6	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ {0}) → ran 𝐺 ∈ Fin)
117	116	olcd 408	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ {0}) → (ran 𝐺 ⊆ (ℤ_≥‘0) ∨ ran 𝐺 ∈ Fin))
118		sumz 14453	. . . . . . . 8 ⊢ ((ran 𝐺 ⊆ (ℤ_≥‘0) ∨ ran 𝐺 ∈ Fin) → Σ𝑧 ∈ ran 𝐺0 = 0)
119	117, 118	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {0}) → Σ𝑧 ∈ ran 𝐺0 = 0)
120	115, 119	eqtrd 2656	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {0}) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = 0)
121	103, 120	sylan2 491	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ (ran 𝐹 ∖ {0}))) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = 0)
122	96, 99, 121, 3	fsumss 14456	. . . 4 ⊢ (𝜑 → Σ𝑦 ∈ (ran 𝐹 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
123	41, 95, 122	3eqtrd 2660	. . 3 ⊢ (𝜑 → (∫₁‘𝐹) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
124		itg1val 23450	. . . . 5 ⊢ (𝐺 ∈ dom ∫₁ → (∫₁‘𝐺) = Σ𝑧 ∈ (ran 𝐺 ∖ {0})(𝑧 · (vol‘(^◡𝐺 “ {𝑧}))))
125	4, 124	syl 17	. . . 4 ⊢ (𝜑 → (∫₁‘𝐺) = Σ𝑧 ∈ (ran 𝐺 ∖ {0})(𝑧 · (vol‘(^◡𝐺 “ {𝑧}))))
126	9	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝐹:ℝ⟶ℝ)
127	3	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ran 𝐹 ∈ Fin)
128		inss1 3833	. . . . . . . . . 10 ⊢ ((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧})) ⊆ (^◡𝐹 “ {𝑦})
129	128	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → ((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧})) ⊆ (^◡𝐹 “ {𝑦}))
130	47	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (^◡𝐹 “ {𝑦}) ∈ dom vol)
131	50	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (^◡𝐺 “ {𝑧}) ∈ dom vol)
132	130, 131, 52	syl2anc 693	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → ((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧})) ∈ dom vol)
133	11	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ran 𝐹 ⊆ ℝ)
134	133	sselda 3603	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
135	21	ssdifssd 3748	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ran 𝐺 ∖ {0}) ⊆ ℝ)
136	135	sselda 3603	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝑧 ∈ ℝ)
137	136	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
138		eldifsni 4320	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (ran 𝐺 ∖ {0}) → 𝑧 ≠ 0)
139	138	ad2antlr 763	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ≠ 0)
140		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 0 ∧ 𝑧 = 0) → 𝑧 = 0)
141	140	necon3ai 2819	. . . . . . . . . . . 12 ⊢ (𝑧 ≠ 0 → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
142	139, 141	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
143	134, 137, 142, 64	syl21anc 1325	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) = (vol‘((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧}))))
144	16	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
145	144, 134, 137	fovrnd 6806	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
146	143, 145	eqeltrrd 2702	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (vol‘((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧}))) ∈ ℝ)
147	126, 127, 129, 132, 146	itg1addlem1 23459	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘∪ 𝑦 ∈ ran 𝐹((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧}))) = Σ𝑦 ∈ ran 𝐹(vol‘((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧}))))
148		incom 3805	. . . . . . . . . . . . 13 ⊢ ((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧})) = ((^◡𝐺 “ {𝑧}) ∩ (^◡𝐹 “ {𝑦}))
149	148	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ran 𝐹 → ((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧})) = ((^◡𝐺 “ {𝑧}) ∩ (^◡𝐹 “ {𝑦})))
150	149	iuneq2i 4539	. . . . . . . . . . 11 ⊢ ∪ 𝑦 ∈ ran 𝐹((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧})) = ∪ 𝑦 ∈ ran 𝐹((^◡𝐺 “ {𝑧}) ∩ (^◡𝐹 “ {𝑦}))
151		iunin2 4584	. . . . . . . . . . 11 ⊢ ∪ 𝑦 ∈ ran 𝐹((^◡𝐺 “ {𝑧}) ∩ (^◡𝐹 “ {𝑦})) = ((^◡𝐺 “ {𝑧}) ∩ ∪ 𝑦 ∈ ran 𝐹(^◡𝐹 “ {𝑦}))
152	150, 151	eqtri 2644	. . . . . . . . . 10 ⊢ ∪ 𝑦 ∈ ran 𝐹((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧})) = ((^◡𝐺 “ {𝑧}) ∩ ∪ 𝑦 ∈ ran 𝐹(^◡𝐹 “ {𝑦}))
153		cnvimass 5485	. . . . . . . . . . . . 13 ⊢ (^◡𝐺 “ {𝑧}) ⊆ dom 𝐺
154	19, 80	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐺 = ℝ)
155	154	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → dom 𝐺 = ℝ)
156	153, 155	syl5sseq 3653	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (^◡𝐺 “ {𝑧}) ⊆ ℝ)
157		iunid 4575	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑦 ∈ ran 𝐹{𝑦} = ran 𝐹
158	157	imaeq2i 5464	. . . . . . . . . . . . . 14 ⊢ (^◡𝐹 “ ∪ 𝑦 ∈ ran 𝐹{𝑦}) = (^◡𝐹 “ ran 𝐹)
159		imaiun 6503	. . . . . . . . . . . . . 14 ⊢ (^◡𝐹 “ ∪ 𝑦 ∈ ran 𝐹{𝑦}) = ∪ 𝑦 ∈ ran 𝐹(^◡𝐹 “ {𝑦})
160		cnvimarndm 5486	. . . . . . . . . . . . . 14 ⊢ (^◡𝐹 “ ran 𝐹) = dom 𝐹
161	158, 159, 160	3eqtr3i 2652	. . . . . . . . . . . . 13 ⊢ ∪ 𝑦 ∈ ran 𝐹(^◡𝐹 “ {𝑦}) = dom 𝐹
162		fdm 6051	. . . . . . . . . . . . . . 15 ⊢ (𝐹:ℝ⟶ℝ → dom 𝐹 = ℝ)
163	9, 162	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐹 = ℝ)
164	163	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → dom 𝐹 = ℝ)
165	161, 164	syl5eq 2668	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ∪ 𝑦 ∈ ran 𝐹(^◡𝐹 “ {𝑦}) = ℝ)
166	156, 165	sseqtr4d 3642	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (^◡𝐺 “ {𝑧}) ⊆ ∪ 𝑦 ∈ ran 𝐹(^◡𝐹 “ {𝑦}))
167		df-ss 3588	. . . . . . . . . . 11 ⊢ ((^◡𝐺 “ {𝑧}) ⊆ ∪ 𝑦 ∈ ran 𝐹(^◡𝐹 “ {𝑦}) ↔ ((^◡𝐺 “ {𝑧}) ∩ ∪ 𝑦 ∈ ran 𝐹(^◡𝐹 “ {𝑦})) = (^◡𝐺 “ {𝑧}))
168	166, 167	sylib 208	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ((^◡𝐺 “ {𝑧}) ∩ ∪ 𝑦 ∈ ran 𝐹(^◡𝐹 “ {𝑦})) = (^◡𝐺 “ {𝑧}))
169	152, 168	syl5req 2669	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (^◡𝐺 “ {𝑧}) = ∪ 𝑦 ∈ ran 𝐹((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧})))
170	169	fveq2d 6195	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(^◡𝐺 “ {𝑧})) = (vol‘∪ 𝑦 ∈ ran 𝐹((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧}))))
171	143	sumeq2dv 14433	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧) = Σ𝑦 ∈ ran 𝐹(vol‘((^◡𝐹 “ {𝑦}) ∩ (^◡𝐺 “ {𝑧}))))
172	147, 170, 171	3eqtr4d 2666	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(^◡𝐺 “ {𝑧})) = Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧))
173	172	oveq2d 6666	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝑧 · (vol‘(^◡𝐺 “ {𝑧}))) = (𝑧 · Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧)))
174	136	recnd 10068	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝑧 ∈ ℂ)
175	145	recnd 10068	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℂ)
176	127, 174, 175	fsummulc2 14516	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝑧 · Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
177	173, 176	eqtrd 2656	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝑧 · (vol‘(^◡𝐺 “ {𝑧}))) = Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
178	177	sumeq2dv 14433	. . . 4 ⊢ (𝜑 → Σ𝑧 ∈ (ran 𝐺 ∖ {0})(𝑧 · (vol‘(^◡𝐺 “ {𝑧}))) = Σ𝑧 ∈ (ran 𝐺 ∖ {0})Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
179		difssd 3738	. . . . . 6 ⊢ (𝜑 → (ran 𝐺 ∖ {0}) ⊆ ran 𝐺)
180	174	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
181	180, 175	mulcld 10060	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
182	127, 181	fsumcl 14464	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
183		dfin4 3867	. . . . . . . . 9 ⊢ (ran 𝐺 ∩ {0}) = (ran 𝐺 ∖ (ran 𝐺 ∖ {0}))
184		inss2 3834	. . . . . . . . 9 ⊢ (ran 𝐺 ∩ {0}) ⊆ {0}
185	183, 184	eqsstr3i 3636	. . . . . . . 8 ⊢ (ran 𝐺 ∖ (ran 𝐺 ∖ {0})) ⊆ {0}
186	185	sseli 3599	. . . . . . 7 ⊢ (𝑧 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {0})) → 𝑧 ∈ {0})
187		elsni 4194	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {0} → 𝑧 = 0)
188	187	ad2antlr 763	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 = 0)
189	188	oveq1d 6665	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) = (0 · (𝑦𝐼𝑧)))
190	16	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
191	12	adantlr 751	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
192	188, 108	syl6eqel 2709	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
193	190, 191, 192	fovrnd 6806	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
194	193	recnd 10068	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℂ)
195	194	mul02d 10234	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (0 · (𝑦𝐼𝑧)) = 0)
196	189, 195	eqtrd 2656	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) = 0)
197	196	sumeq2dv 14433	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {0}) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹0)
198	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ {0}) → ran 𝐹 ∈ Fin)
199	198	olcd 408	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ {0}) → (ran 𝐹 ⊆ (ℤ_≥‘0) ∨ ran 𝐹 ∈ Fin))
200		sumz 14453	. . . . . . . . 9 ⊢ ((ran 𝐹 ⊆ (ℤ_≥‘0) ∨ ran 𝐹 ∈ Fin) → Σ𝑦 ∈ ran 𝐹0 = 0)
201	199, 200	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {0}) → Σ𝑦 ∈ ran 𝐹0 = 0)
202	197, 201	eqtrd 2656	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {0}) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = 0)
203	186, 202	sylan2 491	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {0}))) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = 0)
204	179, 182, 203, 6	fsumss 14456	. . . . 5 ⊢ (𝜑 → Σ𝑧 ∈ (ran 𝐺 ∖ {0})Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
205	22	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
206	205	recnd 10068	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
207	16	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
208	11	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → ran 𝐹 ⊆ ℝ)
209	208	sselda 3603	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
210	207, 209, 205	fovrnd 6806	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
211	210	recnd 10068	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℂ)
212	206, 211	mulcld 10060	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
213	212	anasss 679	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐹)) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
214	6, 3, 213	fsumcom 14507	. . . . 5 ⊢ (𝜑 → Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)))
215	204, 214	eqtrd 2656	. . . 4 ⊢ (𝜑 → Σ𝑧 ∈ (ran 𝐺 ∖ {0})Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)))
216	125, 178, 215	3eqtrd 2660	. . 3 ⊢ (𝜑 → (∫₁‘𝐺) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)))
217	123, 216	oveq12d 6668	. 2 ⊢ (𝜑 → ((∫₁‘𝐹) + (∫₁‘𝐺)) = (Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
218	31, 39, 217	3eqtr4d 2666	1 ⊢ (𝜑 → (∫₁‘(𝐹 ∘_𝑓 + 𝐺)) = ((∫₁‘𝐹) + (∫₁‘𝐺)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 ifcif 4086 {csn 4177 ∪ ciun 4520 × cxp 5112 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 ↾ cres 5116 “ cima 5117 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ∘_𝑓 cof 6895 Fincfn 7955 ℂcc 9934 ℝcr 9935 0cc0 9936 + caddc 9939 · cmul 9941 ℤ_≥cuz 11687 Σcsu 14416 volcvol 23232 ∫₁citg1 23384

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xadd 11947 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-xmet 19739 df-met 19740 df-ovol 23233 df-vol 23234 df-mbf 23388 df-itg1 23389

This theorem is referenced by: itg1add 23468

W3C validator