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Theorem itg1addlem5 23758
Description: Lemma for itg1add . (Contributed by Mario Carneiro, 27-Jun-2014.)
Hypotheses
Ref Expression
i1fadd.1 (𝜑𝐹 ∈ dom ∫1)
i1fadd.2 (𝜑𝐺 ∈ 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‘(𝐹𝑓 + 𝐺)) = ((∫1𝐹) + (∫1𝐺)))
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.
StepHypRef Expression
1 i1fadd.1 . . . 4 (𝜑𝐹 ∈ dom ∫1)
2 i1frn 23735 . . . 4 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
31, 2syl 17 . . 3 (𝜑 → ran 𝐹 ∈ Fin)
4 i1fadd.2 . . . . . 6 (𝜑𝐺 ∈ dom ∫1)
5 i1frn 23735 . . . . . 6 (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
64, 5syl 17 . . . . 5 (𝜑 → ran 𝐺 ∈ Fin)
76adantr 472 . . . 4 ((𝜑𝑦 ∈ ran 𝐹) → ran 𝐺 ∈ Fin)
8 i1ff 23734 . . . . . . . . . 10 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
91, 8syl 17 . . . . . . . . 9 (𝜑𝐹:ℝ⟶ℝ)
109frnd 6230 . . . . . . . 8 (𝜑 → ran 𝐹 ⊆ ℝ)
1110sselda 3761 . . . . . . 7 ((𝜑𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
1211adantr 472 . . . . . 6 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
1312recnd 10322 . . . . 5 (((𝜑𝑦 ∈ 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})))))
151, 4, 14itg1addlem2 23755 . . . . . . . 8 (𝜑𝐼:(ℝ × ℝ)⟶ℝ)
1615ad2antrr 717 . . . . . . 7 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
17 i1ff 23734 . . . . . . . . . . 11 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
184, 17syl 17 . . . . . . . . . 10 (𝜑𝐺:ℝ⟶ℝ)
1918frnd 6230 . . . . . . . . 9 (𝜑 → ran 𝐺 ⊆ ℝ)
2019sselda 3761 . . . . . . . 8 ((𝜑𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
2120adantlr 706 . . . . . . 7 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
2216, 12, 21fovrnd 7004 . . . . . 6 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℝ)
2322recnd 10322 . . . . 5 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℂ)
2413, 23mulcld 10314 . . . 4 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
257, 24fsumcl 14749 . . 3 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
2621recnd 10322 . . . . 5 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℂ)
2726, 23mulcld 10314 . . . 4 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
287, 27fsumcl 14749 . . 3 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
293, 25, 28fsumadd 14755 . 2 (𝜑 → Σ𝑦 ∈ ran 𝐹𝑧 ∈ 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))
311, 4, 14, 30itg1addlem4 23757 . . 3 (𝜑 → (∫1‘(𝐹𝑓 + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
3213, 26, 23adddird 10319 . . . . . 6 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = ((𝑦 · (𝑦𝐼𝑧)) + (𝑧 · (𝑦𝐼𝑧))))
3332sumeq2dv 14718 . . . . 5 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺((𝑦 · (𝑦𝐼𝑧)) + (𝑧 · (𝑦𝐼𝑧))))
347, 24, 27fsumadd 14755 . . . . 5 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺((𝑦 · (𝑦𝐼𝑧)) + (𝑧 · (𝑦𝐼𝑧))) = (Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
3533, 34eqtrd 2799 . . . 4 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = (Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
3635sumeq2dv 14718 . . 3 (𝜑 → Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
3731, 36eqtrd 2799 . 2 (𝜑 → (∫1‘(𝐹𝑓 + 𝐺)) = Σ𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
38 itg1val 23741 . . . . 5 (𝐹 ∈ dom ∫1 → (∫1𝐹) = Σ𝑦 ∈ (ran 𝐹 ∖ {0})(𝑦 · (vol‘(𝐹 “ {𝑦}))))
391, 38syl 17 . . . 4 (𝜑 → (∫1𝐹) = Σ𝑦 ∈ (ran 𝐹 ∖ {0})(𝑦 · (vol‘(𝐹 “ {𝑦}))))
4018adantr 472 . . . . . . . . 9 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝐺:ℝ⟶ℝ)
416adantr 472 . . . . . . . . 9 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → ran 𝐺 ∈ Fin)
42 inss2 3993 . . . . . . . . . 10 ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧})
4342a1i 11 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}))
44 i1fima 23736 . . . . . . . . . . . 12 (𝐹 ∈ dom ∫1 → (𝐹 “ {𝑦}) ∈ dom vol)
451, 44syl 17 . . . . . . . . . . 11 (𝜑 → (𝐹 “ {𝑦}) ∈ dom vol)
4645ad2antrr 717 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐹 “ {𝑦}) ∈ dom vol)
47 i1fima 23736 . . . . . . . . . . . 12 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑧}) ∈ dom vol)
484, 47syl 17 . . . . . . . . . . 11 (𝜑 → (𝐺 “ {𝑧}) ∈ dom vol)
4948ad2antrr 717 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐺 “ {𝑧}) ∈ dom vol)
50 inmbl 23600 . . . . . . . . . 10 (((𝐹 “ {𝑦}) ∈ dom vol ∧ (𝐺 “ {𝑧}) ∈ dom vol) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
5146, 49, 50syl2anc 579 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
5210ssdifssd 3910 . . . . . . . . . . . . 13 (𝜑 → (ran 𝐹 ∖ {0}) ⊆ ℝ)
5352sselda 3761 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝑦 ∈ ℝ)
5453adantr 472 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
5519adantr 472 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → ran 𝐺 ⊆ ℝ)
5655sselda 3761 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
57 eldifsni 4476 . . . . . . . . . . . . 13 (𝑦 ∈ (ran 𝐹 ∖ {0}) → 𝑦 ≠ 0)
5857ad2antlr 718 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ≠ 0)
59 simpl 474 . . . . . . . . . . . . 13 ((𝑦 = 0 ∧ 𝑧 = 0) → 𝑦 = 0)
6059necon3ai 2962 . . . . . . . . . . . 12 (𝑦 ≠ 0 → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
6158, 60syl 17 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
621, 4, 14itg1addlem3 23756 . . . . . . . . . . 11 (((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0)) → (𝑦𝐼𝑧) = (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
6354, 56, 61, 62syl21anc 866 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) = (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
6415ad2antrr 717 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
6564, 54, 56fovrnd 7004 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℝ)
6663, 65eqeltrrd 2845 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
6740, 41, 43, 51, 66itg1addlem1 23750 . . . . . . . 8 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘ 𝑧 ∈ ran 𝐺((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) = Σ𝑧 ∈ ran 𝐺(vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
68 iunin2 4740 . . . . . . . . . 10 𝑧 ∈ ran 𝐺((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ((𝐹 “ {𝑦}) ∩ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}))
691adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝐹 ∈ dom ∫1)
7069, 44syl 17 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑦}) ∈ dom vol)
71 mblss 23589 . . . . . . . . . . . . 13 ((𝐹 “ {𝑦}) ∈ dom vol → (𝐹 “ {𝑦}) ⊆ ℝ)
7270, 71syl 17 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑦}) ⊆ ℝ)
73 iunid 4731 . . . . . . . . . . . . . . 15 𝑧 ∈ ran 𝐺{𝑧} = ran 𝐺
7473imaeq2i 5646 . . . . . . . . . . . . . 14 (𝐺 𝑧 ∈ ran 𝐺{𝑧}) = (𝐺 “ ran 𝐺)
75 imaiun 6695 . . . . . . . . . . . . . 14 (𝐺 𝑧 ∈ ran 𝐺{𝑧}) = 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧})
76 cnvimarndm 5668 . . . . . . . . . . . . . 14 (𝐺 “ ran 𝐺) = dom 𝐺
7774, 75, 763eqtr3i 2795 . . . . . . . . . . . . 13 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}) = dom 𝐺
7840fdmd 6232 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → dom 𝐺 = ℝ)
7977, 78syl5eq 2811 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}) = ℝ)
8072, 79sseqtr4d 3802 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑦}) ⊆ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}))
81 df-ss 3746 . . . . . . . . . . 11 ((𝐹 “ {𝑦}) ⊆ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}) ↔ ((𝐹 “ {𝑦}) ∩ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧})) = (𝐹 “ {𝑦}))
8280, 81sylib 209 . . . . . . . . . 10 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → ((𝐹 “ {𝑦}) ∩ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧})) = (𝐹 “ {𝑦}))
8368, 82syl5req 2812 . . . . . . . . 9 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑦}) = 𝑧 ∈ ran 𝐺((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})))
8483fveq2d 6379 . . . . . . . 8 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑦})) = (vol‘ 𝑧 ∈ ran 𝐺((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
8563sumeq2dv 14718 . . . . . . . 8 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧) = Σ𝑧 ∈ ran 𝐺(vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
8667, 84, 853eqtr4d 2809 . . . . . . 7 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑦})) = Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧))
8786oveq2d 6858 . . . . . 6 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝑦 · (vol‘(𝐹 “ {𝑦}))) = (𝑦 · Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧)))
8853recnd 10322 . . . . . . 7 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝑦 ∈ ℂ)
8965recnd 10322 . . . . . . 7 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℂ)
9041, 88, 89fsummulc2 14800 . . . . . 6 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝑦 · Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
9187, 90eqtrd 2799 . . . . 5 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝑦 · (vol‘(𝐹 “ {𝑦}))) = Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
9291sumeq2dv 14718 . . . 4 (𝜑 → Σ𝑦 ∈ (ran 𝐹 ∖ {0})(𝑦 · (vol‘(𝐹 “ {𝑦}))) = Σ𝑦 ∈ (ran 𝐹 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
93 difssd 3900 . . . . 5 (𝜑 → (ran 𝐹 ∖ {0}) ⊆ ran 𝐹)
9454recnd 10322 . . . . . . 7 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℂ)
9594, 89mulcld 10314 . . . . . 6 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
9641, 95fsumcl 14749 . . . . 5 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
97 dfin4 4032 . . . . . . . 8 (ran 𝐹 ∩ {0}) = (ran 𝐹 ∖ (ran 𝐹 ∖ {0}))
98 inss2 3993 . . . . . . . 8 (ran 𝐹 ∩ {0}) ⊆ {0}
9997, 98eqsstr3i 3796 . . . . . . 7 (ran 𝐹 ∖ (ran 𝐹 ∖ {0})) ⊆ {0}
10099sseli 3757 . . . . . 6 (𝑦 ∈ (ran 𝐹 ∖ (ran 𝐹 ∖ {0})) → 𝑦 ∈ {0})
101 elsni 4351 . . . . . . . . . . 11 (𝑦 ∈ {0} → 𝑦 = 0)
102101ad2antlr 718 . . . . . . . . . 10 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 = 0)
103102oveq1d 6857 . . . . . . . . 9 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) = (0 · (𝑦𝐼𝑧)))
10415ad2antrr 717 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
105 0re 10295 . . . . . . . . . . . . 13 0 ∈ ℝ
106102, 105syl6eqel 2852 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
10720adantlr 706 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
108104, 106, 107fovrnd 7004 . . . . . . . . . . 11 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℝ)
109108recnd 10322 . . . . . . . . . 10 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℂ)
110109mul02d 10488 . . . . . . . . 9 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (0 · (𝑦𝐼𝑧)) = 0)
111103, 110eqtrd 2799 . . . . . . . 8 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) = 0)
112111sumeq2dv 14718 . . . . . . 7 ((𝜑𝑦 ∈ {0}) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺0)
1136adantr 472 . . . . . . . . 9 ((𝜑𝑦 ∈ {0}) → ran 𝐺 ∈ Fin)
114113olcd 900 . . . . . . . 8 ((𝜑𝑦 ∈ {0}) → (ran 𝐺 ⊆ (ℤ‘0) ∨ ran 𝐺 ∈ Fin))
115 sumz 14738 . . . . . . . 8 ((ran 𝐺 ⊆ (ℤ‘0) ∨ ran 𝐺 ∈ Fin) → Σ𝑧 ∈ ran 𝐺0 = 0)
116114, 115syl 17 . . . . . . 7 ((𝜑𝑦 ∈ {0}) → Σ𝑧 ∈ ran 𝐺0 = 0)
117112, 116eqtrd 2799 . . . . . 6 ((𝜑𝑦 ∈ {0}) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = 0)
118100, 117sylan2 586 . . . . 5 ((𝜑𝑦 ∈ (ran 𝐹 ∖ (ran 𝐹 ∖ {0}))) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = 0)
11993, 96, 118, 3fsumss 14741 . . . 4 (𝜑 → Σ𝑦 ∈ (ran 𝐹 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
12039, 92, 1193eqtrd 2803 . . 3 (𝜑 → (∫1𝐹) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
121 itg1val 23741 . . . . 5 (𝐺 ∈ dom ∫1 → (∫1𝐺) = Σ𝑧 ∈ (ran 𝐺 ∖ {0})(𝑧 · (vol‘(𝐺 “ {𝑧}))))
1224, 121syl 17 . . . 4 (𝜑 → (∫1𝐺) = Σ𝑧 ∈ (ran 𝐺 ∖ {0})(𝑧 · (vol‘(𝐺 “ {𝑧}))))
1239adantr 472 . . . . . . . . 9 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝐹:ℝ⟶ℝ)
1243adantr 472 . . . . . . . . 9 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → ran 𝐹 ∈ Fin)
125 inss1 3992 . . . . . . . . . 10 ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {𝑦})
126125a1i 11 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {𝑦}))
12745ad2antrr 717 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝐹 “ {𝑦}) ∈ dom vol)
12848ad2antrr 717 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝐺 “ {𝑧}) ∈ dom vol)
129127, 128, 50syl2anc 579 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
13010adantr 472 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → ran 𝐹 ⊆ ℝ)
131130sselda 3761 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
13219ssdifssd 3910 . . . . . . . . . . . . 13 (𝜑 → (ran 𝐺 ∖ {0}) ⊆ ℝ)
133132sselda 3761 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝑧 ∈ ℝ)
134133adantr 472 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
135 eldifsni 4476 . . . . . . . . . . . . 13 (𝑧 ∈ (ran 𝐺 ∖ {0}) → 𝑧 ≠ 0)
136135ad2antlr 718 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ≠ 0)
137 simpr 477 . . . . . . . . . . . . 13 ((𝑦 = 0 ∧ 𝑧 = 0) → 𝑧 = 0)
138137necon3ai 2962 . . . . . . . . . . . 12 (𝑧 ≠ 0 → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
139136, 138syl 17 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
140131, 134, 139, 62syl21anc 866 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) = (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
14115ad2antrr 717 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
142141, 131, 134fovrnd 7004 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
143140, 142eqeltrrd 2845 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
144123, 124, 126, 129, 143itg1addlem1 23750 . . . . . . . 8 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘ 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) = Σ𝑦 ∈ ran 𝐹(vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
145 incom 3967 . . . . . . . . . . . . 13 ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ((𝐺 “ {𝑧}) ∩ (𝐹 “ {𝑦}))
146145a1i 11 . . . . . . . . . . . 12 (𝑦 ∈ ran 𝐹 → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ((𝐺 “ {𝑧}) ∩ (𝐹 “ {𝑦})))
147146iuneq2i 4695 . . . . . . . . . . 11 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = 𝑦 ∈ ran 𝐹((𝐺 “ {𝑧}) ∩ (𝐹 “ {𝑦}))
148 iunin2 4740 . . . . . . . . . . 11 𝑦 ∈ ran 𝐹((𝐺 “ {𝑧}) ∩ (𝐹 “ {𝑦})) = ((𝐺 “ {𝑧}) ∩ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}))
149147, 148eqtri 2787 . . . . . . . . . 10 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ((𝐺 “ {𝑧}) ∩ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}))
150 cnvimass 5667 . . . . . . . . . . . . 13 (𝐺 “ {𝑧}) ⊆ dom 𝐺
15118fdmd 6232 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐺 = ℝ)
152151adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → dom 𝐺 = ℝ)
153150, 152syl5sseq 3813 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ⊆ ℝ)
154 iunid 4731 . . . . . . . . . . . . . . 15 𝑦 ∈ ran 𝐹{𝑦} = ran 𝐹
155154imaeq2i 5646 . . . . . . . . . . . . . 14 (𝐹 𝑦 ∈ ran 𝐹{𝑦}) = (𝐹 “ ran 𝐹)
156 imaiun 6695 . . . . . . . . . . . . . 14 (𝐹 𝑦 ∈ ran 𝐹{𝑦}) = 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦})
157 cnvimarndm 5668 . . . . . . . . . . . . . 14 (𝐹 “ ran 𝐹) = dom 𝐹
158155, 156, 1573eqtr3i 2795 . . . . . . . . . . . . 13 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}) = dom 𝐹
1599fdmd 6232 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐹 = ℝ)
160159adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → dom 𝐹 = ℝ)
161158, 160syl5eq 2811 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}) = ℝ)
162153, 161sseqtr4d 3802 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ⊆ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}))
163 df-ss 3746 . . . . . . . . . . 11 ((𝐺 “ {𝑧}) ⊆ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}) ↔ ((𝐺 “ {𝑧}) ∩ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦})) = (𝐺 “ {𝑧}))
164162, 163sylib 209 . . . . . . . . . 10 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → ((𝐺 “ {𝑧}) ∩ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦})) = (𝐺 “ {𝑧}))
165149, 164syl5req 2812 . . . . . . . . 9 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) = 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})))
166165fveq2d 6379 . . . . . . . 8 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) = (vol‘ 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
167140sumeq2dv 14718 . . . . . . . 8 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧) = Σ𝑦 ∈ ran 𝐹(vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
168144, 166, 1673eqtr4d 2809 . . . . . . 7 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) = Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧))
169168oveq2d 6858 . . . . . 6 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝑧 · (vol‘(𝐺 “ {𝑧}))) = (𝑧 · Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧)))
170133recnd 10322 . . . . . . 7 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝑧 ∈ ℂ)
171142recnd 10322 . . . . . . 7 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℂ)
172124, 170, 171fsummulc2 14800 . . . . . 6 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝑧 · Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
173169, 172eqtrd 2799 . . . . 5 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝑧 · (vol‘(𝐺 “ {𝑧}))) = Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
174173sumeq2dv 14718 . . . 4 (𝜑 → Σ𝑧 ∈ (ran 𝐺 ∖ {0})(𝑧 · (vol‘(𝐺 “ {𝑧}))) = Σ𝑧 ∈ (ran 𝐺 ∖ {0})Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
175 difssd 3900 . . . . . 6 (𝜑 → (ran 𝐺 ∖ {0}) ⊆ ran 𝐺)
176170adantr 472 . . . . . . . 8 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
177176, 171mulcld 10314 . . . . . . 7 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
178124, 177fsumcl 14749 . . . . . 6 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
179 dfin4 4032 . . . . . . . . 9 (ran 𝐺 ∩ {0}) = (ran 𝐺 ∖ (ran 𝐺 ∖ {0}))
180 inss2 3993 . . . . . . . . 9 (ran 𝐺 ∩ {0}) ⊆ {0}
181179, 180eqsstr3i 3796 . . . . . . . 8 (ran 𝐺 ∖ (ran 𝐺 ∖ {0})) ⊆ {0}
182181sseli 3757 . . . . . . 7 (𝑧 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {0})) → 𝑧 ∈ {0})
183 elsni 4351 . . . . . . . . . . . 12 (𝑧 ∈ {0} → 𝑧 = 0)
184183ad2antlr 718 . . . . . . . . . . 11 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 = 0)
185184oveq1d 6857 . . . . . . . . . 10 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) = (0 · (𝑦𝐼𝑧)))
18615ad2antrr 717 . . . . . . . . . . . . 13 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
18711adantlr 706 . . . . . . . . . . . . 13 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
188184, 105syl6eqel 2852 . . . . . . . . . . . . 13 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
189186, 187, 188fovrnd 7004 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
190189recnd 10322 . . . . . . . . . . 11 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℂ)
191190mul02d 10488 . . . . . . . . . 10 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (0 · (𝑦𝐼𝑧)) = 0)
192185, 191eqtrd 2799 . . . . . . . . 9 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) = 0)
193192sumeq2dv 14718 . . . . . . . 8 ((𝜑𝑧 ∈ {0}) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹0)
1943adantr 472 . . . . . . . . . 10 ((𝜑𝑧 ∈ {0}) → ran 𝐹 ∈ Fin)
195194olcd 900 . . . . . . . . 9 ((𝜑𝑧 ∈ {0}) → (ran 𝐹 ⊆ (ℤ‘0) ∨ ran 𝐹 ∈ Fin))
196 sumz 14738 . . . . . . . . 9 ((ran 𝐹 ⊆ (ℤ‘0) ∨ ran 𝐹 ∈ Fin) → Σ𝑦 ∈ ran 𝐹0 = 0)
197195, 196syl 17 . . . . . . . 8 ((𝜑𝑧 ∈ {0}) → Σ𝑦 ∈ ran 𝐹0 = 0)
198193, 197eqtrd 2799 . . . . . . 7 ((𝜑𝑧 ∈ {0}) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = 0)
199182, 198sylan2 586 . . . . . 6 ((𝜑𝑧 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {0}))) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = 0)
200175, 178, 199, 6fsumss 14741 . . . . 5 (𝜑 → Σ𝑧 ∈ (ran 𝐺 ∖ {0})Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
20120adantr 472 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
202201recnd 10322 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
20315ad2antrr 717 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
20410adantr 472 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ran 𝐺) → ran 𝐹 ⊆ ℝ)
205204sselda 3761 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
206203, 205, 201fovrnd 7004 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
207206recnd 10322 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℂ)
208202, 207mulcld 10314 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
209208anasss 458 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ran 𝐺𝑦 ∈ ran 𝐹)) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
2106, 3, 209fsumcom 14791 . . . . 5 (𝜑 → Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)))
211200, 210eqtrd 2799 . . . 4 (𝜑 → Σ𝑧 ∈ (ran 𝐺 ∖ {0})Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)))
212122, 174, 2113eqtrd 2803 . . 3 (𝜑 → (∫1𝐺) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)))
213120, 212oveq12d 6860 . 2 (𝜑 → ((∫1𝐹) + (∫1𝐺)) = (Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
21429, 37, 2133eqtr4d 2809 1 (𝜑 → (∫1‘(𝐹𝑓 + 𝐺)) = ((∫1𝐹) + (∫1𝐺)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wo 873   = wceq 1652  wcel 2155  wne 2937  cdif 3729  cin 3731  wss 3732  ifcif 4243  {csn 4334   ciun 4676   × cxp 5275  ccnv 5276  dom cdm 5277  ran crn 5278  cres 5279  cima 5280  wf 6064  cfv 6068  (class class class)co 6842  cmpt2 6844  𝑓 cof 7093  Fincfn 8160  cc 10187  cr 10188  0cc0 10189   + caddc 10192   · cmul 10194  cuz 11886  Σcsu 14701  volcvol 23521  1citg1 23673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267  ax-addf 10268
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-disj 4778  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-q 11990  df-rp 12029  df-xadd 12147  df-ioo 12381  df-ico 12383  df-icc 12384  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-clim 14504  df-sum 14702  df-xmet 20012  df-met 20013  df-ovol 23522  df-vol 23523  df-mbf 23677  df-itg1 23678
This theorem is referenced by:  itg1add  23759
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