Theorem itg1addlem5 25608

Description: Lemma for itg1add 25609. (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 25585	. . . 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 25585	. . . . . 6 ⊢ (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
6	4, 5	syl 17	. . . . 5 ⊢ (𝜑 → ran 𝐺 ∈ Fin)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → ran 𝐺 ∈ Fin)
8		i1ff 25584	. . . . . . . . . 10 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ)
9	1, 8	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)
10	9	frnd 6699	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
11	10	sselda 3949	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
12	11	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
13	12	recnd 11209	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℂ)
14		itg1add.3	. . . . . . . . 9 ⊢ 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))))
15	1, 4, 14	itg1addlem2 25605	. . . . . . . 8 ⊢ (𝜑 → 𝐼:(ℝ × ℝ)⟶ℝ)
16	15	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
17		i1ff 25584	. . . . . . . . . . 11 ⊢ (𝐺 ∈ dom ∫1 → 𝐺:ℝ⟶ℝ)
18	4, 17	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℝ⟶ℝ)
19	18	frnd 6699	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐺 ⊆ ℝ)
20	19	sselda 3949	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
21	20	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
22	16, 12, 21	fovcdmd 7564	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℝ)
23	22	recnd 11209	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℂ)
24	13, 23	mulcld 11201	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
25	7, 24	fsumcl 15706	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
26	21	recnd 11209	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℂ)
27	26, 23	mulcld 11201	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
28	7, 27	fsumcl 15706	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
29	3, 25, 28	fsumadd 15713	. 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))))
30		itg1add.4	. . . 4 ⊢ 𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))
31	1, 4, 14, 30	itg1addlem4 25607	. . 3 ⊢ (𝜑 → (∫1‘(𝐹 ∘f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
32	13, 26, 23	adddird 11206	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = ((𝑦 · (𝑦𝐼𝑧)) + (𝑧 · (𝑦𝐼𝑧))))
33	32	sumeq2dv 15675	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺((𝑦 · (𝑦𝐼𝑧)) + (𝑧 · (𝑦𝐼𝑧))))
34	7, 24, 27	fsumadd 15713	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺((𝑦 · (𝑦𝐼𝑧)) + (𝑧 · (𝑦𝐼𝑧))) = (Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
35	33, 34	eqtrd 2765	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = (Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
36	35	sumeq2dv 15675	. . 3 ⊢ (𝜑 → Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹(Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
37	31, 36	eqtrd 2765	. 2 ⊢ (𝜑 → (∫1‘(𝐹 ∘f + 𝐺)) = Σ𝑦 ∈ ran 𝐹(Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
38		itg1val 25591	. . . . 5 ⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) = Σ𝑦 ∈ (ran 𝐹 ∖ {0})(𝑦 · (vol‘(◡𝐹 “ {𝑦}))))
39	1, 38	syl 17	. . . 4 ⊢ (𝜑 → (∫1‘𝐹) = Σ𝑦 ∈ (ran 𝐹 ∖ {0})(𝑦 · (vol‘(◡𝐹 “ {𝑦}))))
40	18	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝐺:ℝ⟶ℝ)
41	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → ran 𝐺 ∈ Fin)
42		inss2 4204	. . . . . . . . . 10 ⊢ ((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐺 “ {𝑧})
43	42	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐺 “ {𝑧}))
44		i1fima 25586	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ {𝑦}) ∈ dom vol)
45	1, 44	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (◡𝐹 “ {𝑦}) ∈ dom vol)
46	45	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (◡𝐹 “ {𝑦}) ∈ dom vol)
47		i1fima 25586	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ dom ∫1 → (◡𝐺 “ {𝑧}) ∈ dom vol)
48	4, 47	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (◡𝐺 “ {𝑧}) ∈ dom vol)
49	48	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (◡𝐺 “ {𝑧}) ∈ dom vol)
50		inmbl 25450	. . . . . . . . . 10 ⊢ (((◡𝐹 “ {𝑦}) ∈ dom vol ∧ (◡𝐺 “ {𝑧}) ∈ dom vol) → ((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
51	46, 49, 50	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
52	10	ssdifssd 4113	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ran 𝐹 ∖ {0}) ⊆ ℝ)
53	52	sselda 3949	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝑦 ∈ ℝ)
54	53	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
55	19	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → ran 𝐺 ⊆ ℝ)
56	55	sselda 3949	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
57		eldifsni 4757	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (ran 𝐹 ∖ {0}) → 𝑦 ≠ 0)
58	57	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ≠ 0)
59		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 0 ∧ 𝑧 = 0) → 𝑦 = 0)
60	59	necon3ai 2951	. . . . . . . . . . . 12 ⊢ (𝑦 ≠ 0 → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
61	58, 60	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
62	1, 4, 14	itg1addlem3 25606	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0)) → (𝑦𝐼𝑧) = (vol‘((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧}))))
63	54, 56, 61, 62	syl21anc 837	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) = (vol‘((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧}))))
64	15	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
65	64, 54, 56	fovcdmd 7564	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℝ)
66	63, 65	eqeltrrd 2830	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol‘((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
67	40, 41, 43, 51, 66	itg1addlem1 25600	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧}))) = Σ𝑧 ∈ ran 𝐺(vol‘((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧}))))
68		iunin2 5038	. . . . . . . . . 10 ⊢ ∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧})) = ((◡𝐹 “ {𝑦}) ∩ ∪ 𝑧 ∈ ran 𝐺(◡𝐺 “ {𝑧}))
69	1	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝐹 ∈ dom ∫1)
70	69, 44	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑦}) ∈ dom vol)
71		mblss 25439	. . . . . . . . . . . . 13 ⊢ ((◡𝐹 “ {𝑦}) ∈ dom vol → (◡𝐹 “ {𝑦}) ⊆ ℝ)
72	70, 71	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑦}) ⊆ ℝ)
73		iunid 5027	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑧 ∈ ran 𝐺{𝑧} = ran 𝐺
74	73	imaeq2i 6032	. . . . . . . . . . . . . 14 ⊢ (◡𝐺 “ ∪ 𝑧 ∈ ran 𝐺{𝑧}) = (◡𝐺 “ ran 𝐺)
75		imaiun 7222	. . . . . . . . . . . . . 14 ⊢ (◡𝐺 “ ∪ 𝑧 ∈ ran 𝐺{𝑧}) = ∪ 𝑧 ∈ ran 𝐺(◡𝐺 “ {𝑧})
76		cnvimarndm 6057	. . . . . . . . . . . . . 14 ⊢ (◡𝐺 “ ran 𝐺) = dom 𝐺
77	74, 75, 76	3eqtr3i 2761	. . . . . . . . . . . . 13 ⊢ ∪ 𝑧 ∈ ran 𝐺(◡𝐺 “ {𝑧}) = dom 𝐺
78	40	fdmd 6701	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → dom 𝐺 = ℝ)
79	77, 78	eqtrid 2777	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → ∪ 𝑧 ∈ ran 𝐺(◡𝐺 “ {𝑧}) = ℝ)
80	72, 79	sseqtrrd 3987	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑦}) ⊆ ∪ 𝑧 ∈ ran 𝐺(◡𝐺 “ {𝑧}))
81		dfss2 3935	. . . . . . . . . . 11 ⊢ ((◡𝐹 “ {𝑦}) ⊆ ∪ 𝑧 ∈ ran 𝐺(◡𝐺 “ {𝑧}) ↔ ((◡𝐹 “ {𝑦}) ∩ ∪ 𝑧 ∈ ran 𝐺(◡𝐺 “ {𝑧})) = (◡𝐹 “ {𝑦}))
82	80, 81	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → ((◡𝐹 “ {𝑦}) ∩ ∪ 𝑧 ∈ ran 𝐺(◡𝐺 “ {𝑧})) = (◡𝐹 “ {𝑦}))
83	68, 82	eqtr2id 2778	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑦}) = ∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧})))
84	83	fveq2d 6865	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑦})) = (vol‘∪ 𝑧 ∈ ran 𝐺((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧}))))
85	63	sumeq2dv 15675	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧) = Σ𝑧 ∈ ran 𝐺(vol‘((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧}))))
86	67, 84, 85	3eqtr4d 2775	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑦})) = Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧))
87	86	oveq2d 7406	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝑦 · (vol‘(◡𝐹 “ {𝑦}))) = (𝑦 · Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧)))
88	53	recnd 11209	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝑦 ∈ ℂ)
89	65	recnd 11209	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℂ)
90	41, 88, 89	fsummulc2 15757	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝑦 · Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
91	87, 90	eqtrd 2765	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝑦 · (vol‘(◡𝐹 “ {𝑦}))) = Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
92	91	sumeq2dv 15675	. . . 4 ⊢ (𝜑 → Σ𝑦 ∈ (ran 𝐹 ∖ {0})(𝑦 · (vol‘(◡𝐹 “ {𝑦}))) = Σ𝑦 ∈ (ran 𝐹 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
93		difssd 4103	. . . . 5 ⊢ (𝜑 → (ran 𝐹 ∖ {0}) ⊆ ran 𝐹)
94	54	recnd 11209	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℂ)
95	94, 89	mulcld 11201	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
96	41, 95	fsumcl 15706	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ {0})) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
97		dfin4 4244	. . . . . . . 8 ⊢ (ran 𝐹 ∩ {0}) = (ran 𝐹 ∖ (ran 𝐹 ∖ {0}))
98		inss2 4204	. . . . . . . 8 ⊢ (ran 𝐹 ∩ {0}) ⊆ {0}
99	97, 98	eqsstrri 3997	. . . . . . 7 ⊢ (ran 𝐹 ∖ (ran 𝐹 ∖ {0})) ⊆ {0}
100	99	sseli 3945	. . . . . 6 ⊢ (𝑦 ∈ (ran 𝐹 ∖ (ran 𝐹 ∖ {0})) → 𝑦 ∈ {0})
101		elsni 4609	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {0} → 𝑦 = 0)
102	101	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 = 0)
103	102	oveq1d 7405	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) = (0 · (𝑦𝐼𝑧)))
104	15	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
105		0re 11183	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
106	102, 105	eqeltrdi 2837	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
107	20	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
108	104, 106, 107	fovcdmd 7564	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℝ)
109	108	recnd 11209	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℂ)
110	109	mul02d 11379	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (0 · (𝑦𝐼𝑧)) = 0)
111	103, 110	eqtrd 2765	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) = 0)
112	111	sumeq2dv 15675	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {0}) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺0)
113	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ {0}) → ran 𝐺 ∈ Fin)
114	113	olcd 874	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ {0}) → (ran 𝐺 ⊆ (ℤ≥‘0) ∨ ran 𝐺 ∈ Fin))
115		sumz 15695	. . . . . . . 8 ⊢ ((ran 𝐺 ⊆ (ℤ≥‘0) ∨ ran 𝐺 ∈ Fin) → Σ𝑧 ∈ ran 𝐺0 = 0)
116	114, 115	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ {0}) → Σ𝑧 ∈ ran 𝐺0 = 0)
117	112, 116	eqtrd 2765	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ {0}) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = 0)
118	100, 117	sylan2 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ran 𝐹 ∖ (ran 𝐹 ∖ {0}))) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = 0)
119	93, 96, 118, 3	fsumss 15698	. . . 4 ⊢ (𝜑 → Σ𝑦 ∈ (ran 𝐹 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
120	39, 92, 119	3eqtrd 2769	. . 3 ⊢ (𝜑 → (∫1‘𝐹) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
121		itg1val 25591	. . . . 5 ⊢ (𝐺 ∈ dom ∫1 → (∫1‘𝐺) = Σ𝑧 ∈ (ran 𝐺 ∖ {0})(𝑧 · (vol‘(◡𝐺 “ {𝑧}))))
122	4, 121	syl 17	. . . 4 ⊢ (𝜑 → (∫1‘𝐺) = Σ𝑧 ∈ (ran 𝐺 ∖ {0})(𝑧 · (vol‘(◡𝐺 “ {𝑧}))))
123	9	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝐹:ℝ⟶ℝ)
124	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ran 𝐹 ∈ Fin)
125		inss1 4203	. . . . . . . . . 10 ⊢ ((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐹 “ {𝑦})
126	125	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → ((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧})) ⊆ (◡𝐹 “ {𝑦}))
127	45	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (◡𝐹 “ {𝑦}) ∈ dom vol)
128	48	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (◡𝐺 “ {𝑧}) ∈ dom vol)
129	127, 128, 50	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → ((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧})) ∈ dom vol)
130	10	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ran 𝐹 ⊆ ℝ)
131	130	sselda 3949	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
132	19	ssdifssd 4113	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ran 𝐺 ∖ {0}) ⊆ ℝ)
133	132	sselda 3949	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝑧 ∈ ℝ)
134	133	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
135		eldifsni 4757	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (ran 𝐺 ∖ {0}) → 𝑧 ≠ 0)
136	135	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ≠ 0)
137		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 0 ∧ 𝑧 = 0) → 𝑧 = 0)
138	137	necon3ai 2951	. . . . . . . . . . . 12 ⊢ (𝑧 ≠ 0 → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
139	136, 138	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
140	131, 134, 139, 62	syl21anc 837	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) = (vol‘((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧}))))
141	15	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
142	141, 131, 134	fovcdmd 7564	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
143	140, 142	eqeltrrd 2830	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (vol‘((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧}))) ∈ ℝ)
144	123, 124, 126, 129, 143	itg1addlem1 25600	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘∪ 𝑦 ∈ ran 𝐹((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧}))) = Σ𝑦 ∈ ran 𝐹(vol‘((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧}))))
145		incom 4175	. . . . . . . . . . . . 13 ⊢ ((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧})) = ((◡𝐺 “ {𝑧}) ∩ (◡𝐹 “ {𝑦}))
146	145	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ran 𝐹 → ((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧})) = ((◡𝐺 “ {𝑧}) ∩ (◡𝐹 “ {𝑦})))
147	146	iuneq2i 4980	. . . . . . . . . . 11 ⊢ ∪ 𝑦 ∈ ran 𝐹((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧})) = ∪ 𝑦 ∈ ran 𝐹((◡𝐺 “ {𝑧}) ∩ (◡𝐹 “ {𝑦}))
148		iunin2 5038	. . . . . . . . . . 11 ⊢ ∪ 𝑦 ∈ ran 𝐹((◡𝐺 “ {𝑧}) ∩ (◡𝐹 “ {𝑦})) = ((◡𝐺 “ {𝑧}) ∩ ∪ 𝑦 ∈ ran 𝐹(◡𝐹 “ {𝑦}))
149	147, 148	eqtri 2753	. . . . . . . . . 10 ⊢ ∪ 𝑦 ∈ ran 𝐹((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧})) = ((◡𝐺 “ {𝑧}) ∩ ∪ 𝑦 ∈ ran 𝐹(◡𝐹 “ {𝑦}))
150		cnvimass 6056	. . . . . . . . . . . . 13 ⊢ (◡𝐺 “ {𝑧}) ⊆ dom 𝐺
151	18	fdmd 6701	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐺 = ℝ)
152	151	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → dom 𝐺 = ℝ)
153	150, 152	sseqtrid 3992	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) ⊆ ℝ)
154		iunid 5027	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑦 ∈ ran 𝐹{𝑦} = ran 𝐹
155	154	imaeq2i 6032	. . . . . . . . . . . . . 14 ⊢ (◡𝐹 “ ∪ 𝑦 ∈ ran 𝐹{𝑦}) = (◡𝐹 “ ran 𝐹)
156		imaiun 7222	. . . . . . . . . . . . . 14 ⊢ (◡𝐹 “ ∪ 𝑦 ∈ ran 𝐹{𝑦}) = ∪ 𝑦 ∈ ran 𝐹(◡𝐹 “ {𝑦})
157		cnvimarndm 6057	. . . . . . . . . . . . . 14 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
158	155, 156, 157	3eqtr3i 2761	. . . . . . . . . . . . 13 ⊢ ∪ 𝑦 ∈ ran 𝐹(◡𝐹 “ {𝑦}) = dom 𝐹
159	9	fdmd 6701	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐹 = ℝ)
160	159	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → dom 𝐹 = ℝ)
161	158, 160	eqtrid 2777	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ∪ 𝑦 ∈ ran 𝐹(◡𝐹 “ {𝑦}) = ℝ)
162	153, 161	sseqtrrd 3987	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) ⊆ ∪ 𝑦 ∈ ran 𝐹(◡𝐹 “ {𝑦}))
163		dfss2 3935	. . . . . . . . . . 11 ⊢ ((◡𝐺 “ {𝑧}) ⊆ ∪ 𝑦 ∈ ran 𝐹(◡𝐹 “ {𝑦}) ↔ ((◡𝐺 “ {𝑧}) ∩ ∪ 𝑦 ∈ ran 𝐹(◡𝐹 “ {𝑦})) = (◡𝐺 “ {𝑧}))
164	162, 163	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ((◡𝐺 “ {𝑧}) ∩ ∪ 𝑦 ∈ ran 𝐹(◡𝐹 “ {𝑦})) = (◡𝐺 “ {𝑧}))
165	149, 164	eqtr2id 2778	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (◡𝐺 “ {𝑧}) = ∪ 𝑦 ∈ ran 𝐹((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧})))
166	165	fveq2d 6865	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) = (vol‘∪ 𝑦 ∈ ran 𝐹((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧}))))
167	140	sumeq2dv 15675	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧) = Σ𝑦 ∈ ran 𝐹(vol‘((◡𝐹 “ {𝑦}) ∩ (◡𝐺 “ {𝑧}))))
168	144, 166, 167	3eqtr4d 2775	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(◡𝐺 “ {𝑧})) = Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧))
169	168	oveq2d 7406	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝑧 · (vol‘(◡𝐺 “ {𝑧}))) = (𝑧 · Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧)))
170	133	recnd 11209	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝑧 ∈ ℂ)
171	142	recnd 11209	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℂ)
172	124, 170, 171	fsummulc2 15757	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝑧 · Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
173	169, 172	eqtrd 2765	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝑧 · (vol‘(◡𝐺 “ {𝑧}))) = Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
174	173	sumeq2dv 15675	. . . 4 ⊢ (𝜑 → Σ𝑧 ∈ (ran 𝐺 ∖ {0})(𝑧 · (vol‘(◡𝐺 “ {𝑧}))) = Σ𝑧 ∈ (ran 𝐺 ∖ {0})Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
175		difssd 4103	. . . . . 6 ⊢ (𝜑 → (ran 𝐺 ∖ {0}) ⊆ ran 𝐺)
176	170	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
177	176, 171	mulcld 11201	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
178	124, 177	fsumcl 15706	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
179		dfin4 4244	. . . . . . . . 9 ⊢ (ran 𝐺 ∩ {0}) = (ran 𝐺 ∖ (ran 𝐺 ∖ {0}))
180		inss2 4204	. . . . . . . . 9 ⊢ (ran 𝐺 ∩ {0}) ⊆ {0}
181	179, 180	eqsstrri 3997	. . . . . . . 8 ⊢ (ran 𝐺 ∖ (ran 𝐺 ∖ {0})) ⊆ {0}
182	181	sseli 3945	. . . . . . 7 ⊢ (𝑧 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {0})) → 𝑧 ∈ {0})
183		elsni 4609	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {0} → 𝑧 = 0)
184	183	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 = 0)
185	184	oveq1d 7405	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) = (0 · (𝑦𝐼𝑧)))
186	15	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
187	11	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
188	184, 105	eqeltrdi 2837	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
189	186, 187, 188	fovcdmd 7564	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
190	189	recnd 11209	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℂ)
191	190	mul02d 11379	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (0 · (𝑦𝐼𝑧)) = 0)
192	185, 191	eqtrd 2765	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) = 0)
193	192	sumeq2dv 15675	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {0}) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹0)
194	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ {0}) → ran 𝐹 ∈ Fin)
195	194	olcd 874	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ {0}) → (ran 𝐹 ⊆ (ℤ≥‘0) ∨ ran 𝐹 ∈ Fin))
196		sumz 15695	. . . . . . . . 9 ⊢ ((ran 𝐹 ⊆ (ℤ≥‘0) ∨ ran 𝐹 ∈ Fin) → Σ𝑦 ∈ ran 𝐹0 = 0)
197	195, 196	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ {0}) → Σ𝑦 ∈ ran 𝐹0 = 0)
198	193, 197	eqtrd 2765	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ {0}) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = 0)
199	182, 198	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {0}))) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = 0)
200	175, 178, 199, 6	fsumss 15698	. . . . 5 ⊢ (𝜑 → Σ𝑧 ∈ (ran 𝐺 ∖ {0})Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
201	20	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
202	201	recnd 11209	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
203	15	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
204	10	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ran 𝐺) → ran 𝐹 ⊆ ℝ)
205	204	sselda 3949	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
206	203, 205, 201	fovcdmd 7564	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
207	206	recnd 11209	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℂ)
208	202, 207	mulcld 11201	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
209	208	anasss 466	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐹)) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
210	6, 3, 209	fsumcom 15748	. . . . 5 ⊢ (𝜑 → Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)))
211	200, 210	eqtrd 2765	. . . 4 ⊢ (𝜑 → Σ𝑧 ∈ (ran 𝐺 ∖ {0})Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)))
212	122, 174, 211	3eqtrd 2769	. . 3 ⊢ (𝜑 → (∫1‘𝐺) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)))
213	120, 212	oveq12d 7408	. 2 ⊢ (𝜑 → ((∫1‘𝐹) + (∫1‘𝐺)) = (Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
214	29, 37, 213	3eqtr4d 2775	1 ⊢ (𝜑 → (∫1‘(𝐹 ∘f + 𝐺)) = ((∫1‘𝐹) + (∫1‘𝐺)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2926   ∖ cdif 3914   ∩ cin 3916   ⊆ wss 3917  ifcif 4491  {csn 4592  ∪ ciun 4958   × cxp 5639  ^◡ccnv 5640  dom cdm 5641  ran crn 5642   ↾ cres 5643   “ cima 5644  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392   ∘_f cof 7654  Fincfn 8921  ℂcc 11073  ℝcr 11074  0cc0 11075   + caddc 11078   · cmul 11080  ℤ_≥cuz 12800  Σcsu 15659  volcvol 25371  ∫₁citg1 25523

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153  ax-addf 11154

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-disj 5078  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-oi 9470  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-z 12537  df-uz 12801  df-q 12915  df-rp 12959  df-xadd 13080  df-ioo 13317  df-ico 13319  df-icc 13320  df-fz 13476  df-fzo 13623  df-fl 13761  df-seq 13974  df-exp 14034  df-hash 14303  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15461  df-sum 15660  df-xmet 21264  df-met 21265  df-ovol 25372  df-vol 25373  df-mbf 25527  df-itg1 25528

This theorem is referenced by:  itg1add  25609