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Theorem itg1addlem5 25551
Description: Lemma for itg1add 25552. (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 25527 . . . 4 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
31, 2syl 17 . . 3 (𝜑 → ran 𝐹 ∈ Fin)
4 i1fadd.2 . . . . . 6 (𝜑𝐺 ∈ dom ∫1)
5 i1frn 25527 . . . . . 6 (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
64, 5syl 17 . . . . 5 (𝜑 → ran 𝐺 ∈ Fin)
76adantr 480 . . . 4 ((𝜑𝑦 ∈ ran 𝐹) → ran 𝐺 ∈ Fin)
8 i1ff 25526 . . . . . . . . . 10 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
91, 8syl 17 . . . . . . . . 9 (𝜑𝐹:ℝ⟶ℝ)
109frnd 6725 . . . . . . . 8 (𝜑 → ran 𝐹 ⊆ ℝ)
1110sselda 3982 . . . . . . 7 ((𝜑𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
1211adantr 480 . . . . . 6 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
1312recnd 11249 . . . . 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 25547 . . . . . . . 8 (𝜑𝐼:(ℝ × ℝ)⟶ℝ)
1615ad2antrr 723 . . . . . . 7 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
17 i1ff 25526 . . . . . . . . . . 11 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
184, 17syl 17 . . . . . . . . . 10 (𝜑𝐺:ℝ⟶ℝ)
1918frnd 6725 . . . . . . . . 9 (𝜑 → ran 𝐺 ⊆ ℝ)
2019sselda 3982 . . . . . . . 8 ((𝜑𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
2120adantlr 712 . . . . . . 7 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
2216, 12, 21fovcdmd 7583 . . . . . 6 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℝ)
2322recnd 11249 . . . . 5 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℂ)
2413, 23mulcld 11241 . . . 4 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
257, 24fsumcl 15686 . . 3 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
2621recnd 11249 . . . . 5 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℂ)
2726, 23mulcld 11241 . . . 4 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
287, 27fsumcl 15686 . . 3 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
293, 25, 28fsumadd 15693 . 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 25549 . . 3 (𝜑 → (∫1‘(𝐹f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
3213, 26, 23adddird 11246 . . . . . 6 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = ((𝑦 · (𝑦𝐼𝑧)) + (𝑧 · (𝑦𝐼𝑧))))
3332sumeq2dv 15656 . . . . 5 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺((𝑦 · (𝑦𝐼𝑧)) + (𝑧 · (𝑦𝐼𝑧))))
347, 24, 27fsumadd 15693 . . . . 5 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺((𝑦 · (𝑦𝐼𝑧)) + (𝑧 · (𝑦𝐼𝑧))) = (Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
3533, 34eqtrd 2771 . . . 4 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = (Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
3635sumeq2dv 15656 . . 3 (𝜑 → Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
3731, 36eqtrd 2771 . 2 (𝜑 → (∫1‘(𝐹f + 𝐺)) = Σ𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
38 itg1val 25533 . . . . 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 480 . . . . . . . . 9 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝐺:ℝ⟶ℝ)
416adantr 480 . . . . . . . . 9 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → ran 𝐺 ∈ Fin)
42 inss2 4229 . . . . . . . . . 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 25528 . . . . . . . . . . . 12 (𝐹 ∈ dom ∫1 → (𝐹 “ {𝑦}) ∈ dom vol)
451, 44syl 17 . . . . . . . . . . 11 (𝜑 → (𝐹 “ {𝑦}) ∈ dom vol)
4645ad2antrr 723 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐹 “ {𝑦}) ∈ dom vol)
47 i1fima 25528 . . . . . . . . . . . 12 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑧}) ∈ dom vol)
484, 47syl 17 . . . . . . . . . . 11 (𝜑 → (𝐺 “ {𝑧}) ∈ dom vol)
4948ad2antrr 723 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐺 “ {𝑧}) ∈ dom vol)
50 inmbl 25392 . . . . . . . . . 10 (((𝐹 “ {𝑦}) ∈ dom vol ∧ (𝐺 “ {𝑧}) ∈ dom vol) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
5146, 49, 50syl2anc 583 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
5210ssdifssd 4142 . . . . . . . . . . . . 13 (𝜑 → (ran 𝐹 ∖ {0}) ⊆ ℝ)
5352sselda 3982 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝑦 ∈ ℝ)
5453adantr 480 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
5519adantr 480 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → ran 𝐺 ⊆ ℝ)
5655sselda 3982 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
57 eldifsni 4793 . . . . . . . . . . . . 13 (𝑦 ∈ (ran 𝐹 ∖ {0}) → 𝑦 ≠ 0)
5857ad2antlr 724 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ≠ 0)
59 simpl 482 . . . . . . . . . . . . 13 ((𝑦 = 0 ∧ 𝑧 = 0) → 𝑦 = 0)
6059necon3ai 2964 . . . . . . . . . . . 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 25548 . . . . . . . . . . 11 (((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0)) → (𝑦𝐼𝑧) = (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
6354, 56, 61, 62syl21anc 835 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) = (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
6415ad2antrr 723 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
6564, 54, 56fovcdmd 7583 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℝ)
6663, 65eqeltrrd 2833 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
6740, 41, 43, 51, 66itg1addlem1 25542 . . . . . . . 8 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘ 𝑧 ∈ ran 𝐺((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) = Σ𝑧 ∈ ran 𝐺(vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
68 iunin2 5074 . . . . . . . . . 10 𝑧 ∈ ran 𝐺((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ((𝐹 “ {𝑦}) ∩ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}))
691adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝐹 ∈ dom ∫1)
7069, 44syl 17 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑦}) ∈ dom vol)
71 mblss 25381 . . . . . . . . . . . . 13 ((𝐹 “ {𝑦}) ∈ dom vol → (𝐹 “ {𝑦}) ⊆ ℝ)
7270, 71syl 17 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑦}) ⊆ ℝ)
73 iunid 5063 . . . . . . . . . . . . . . 15 𝑧 ∈ ran 𝐺{𝑧} = ran 𝐺
7473imaeq2i 6057 . . . . . . . . . . . . . 14 (𝐺 𝑧 ∈ ran 𝐺{𝑧}) = (𝐺 “ ran 𝐺)
75 imaiun 7247 . . . . . . . . . . . . . 14 (𝐺 𝑧 ∈ ran 𝐺{𝑧}) = 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧})
76 cnvimarndm 6081 . . . . . . . . . . . . . 14 (𝐺 “ ran 𝐺) = dom 𝐺
7774, 75, 763eqtr3i 2767 . . . . . . . . . . . . 13 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}) = dom 𝐺
7840fdmd 6728 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → dom 𝐺 = ℝ)
7977, 78eqtrid 2783 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}) = ℝ)
8072, 79sseqtrrd 4023 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑦}) ⊆ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}))
81 df-ss 3965 . . . . . . . . . . 11 ((𝐹 “ {𝑦}) ⊆ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}) ↔ ((𝐹 “ {𝑦}) ∩ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧})) = (𝐹 “ {𝑦}))
8280, 81sylib 217 . . . . . . . . . 10 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → ((𝐹 “ {𝑦}) ∩ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧})) = (𝐹 “ {𝑦}))
8368, 82eqtr2id 2784 . . . . . . . . 9 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑦}) = 𝑧 ∈ ran 𝐺((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})))
8483fveq2d 6895 . . . . . . . 8 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑦})) = (vol‘ 𝑧 ∈ ran 𝐺((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
8563sumeq2dv 15656 . . . . . . . 8 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧) = Σ𝑧 ∈ ran 𝐺(vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
8667, 84, 853eqtr4d 2781 . . . . . . 7 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑦})) = Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧))
8786oveq2d 7428 . . . . . 6 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝑦 · (vol‘(𝐹 “ {𝑦}))) = (𝑦 · Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧)))
8853recnd 11249 . . . . . . 7 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝑦 ∈ ℂ)
8965recnd 11249 . . . . . . 7 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℂ)
9041, 88, 89fsummulc2 15737 . . . . . 6 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝑦 · Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
9187, 90eqtrd 2771 . . . . 5 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝑦 · (vol‘(𝐹 “ {𝑦}))) = Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
9291sumeq2dv 15656 . . . 4 (𝜑 → Σ𝑦 ∈ (ran 𝐹 ∖ {0})(𝑦 · (vol‘(𝐹 “ {𝑦}))) = Σ𝑦 ∈ (ran 𝐹 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
93 difssd 4132 . . . . 5 (𝜑 → (ran 𝐹 ∖ {0}) ⊆ ran 𝐹)
9454recnd 11249 . . . . . . 7 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℂ)
9594, 89mulcld 11241 . . . . . 6 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
9641, 95fsumcl 15686 . . . . 5 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
97 dfin4 4267 . . . . . . . 8 (ran 𝐹 ∩ {0}) = (ran 𝐹 ∖ (ran 𝐹 ∖ {0}))
98 inss2 4229 . . . . . . . 8 (ran 𝐹 ∩ {0}) ⊆ {0}
9997, 98eqsstrri 4017 . . . . . . 7 (ran 𝐹 ∖ (ran 𝐹 ∖ {0})) ⊆ {0}
10099sseli 3978 . . . . . 6 (𝑦 ∈ (ran 𝐹 ∖ (ran 𝐹 ∖ {0})) → 𝑦 ∈ {0})
101 elsni 4645 . . . . . . . . . . 11 (𝑦 ∈ {0} → 𝑦 = 0)
102101ad2antlr 724 . . . . . . . . . 10 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 = 0)
103102oveq1d 7427 . . . . . . . . 9 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) = (0 · (𝑦𝐼𝑧)))
10415ad2antrr 723 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
105 0re 11223 . . . . . . . . . . . . 13 0 ∈ ℝ
106102, 105eqeltrdi 2840 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
10720adantlr 712 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
108104, 106, 107fovcdmd 7583 . . . . . . . . . . 11 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℝ)
109108recnd 11249 . . . . . . . . . 10 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℂ)
110109mul02d 11419 . . . . . . . . 9 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (0 · (𝑦𝐼𝑧)) = 0)
111103, 110eqtrd 2771 . . . . . . . 8 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) = 0)
112111sumeq2dv 15656 . . . . . . 7 ((𝜑𝑦 ∈ {0}) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺0)
1136adantr 480 . . . . . . . . 9 ((𝜑𝑦 ∈ {0}) → ran 𝐺 ∈ Fin)
114113olcd 871 . . . . . . . 8 ((𝜑𝑦 ∈ {0}) → (ran 𝐺 ⊆ (ℤ‘0) ∨ ran 𝐺 ∈ Fin))
115 sumz 15675 . . . . . . . 8 ((ran 𝐺 ⊆ (ℤ‘0) ∨ ran 𝐺 ∈ Fin) → Σ𝑧 ∈ ran 𝐺0 = 0)
116114, 115syl 17 . . . . . . 7 ((𝜑𝑦 ∈ {0}) → Σ𝑧 ∈ ran 𝐺0 = 0)
117112, 116eqtrd 2771 . . . . . 6 ((𝜑𝑦 ∈ {0}) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = 0)
118100, 117sylan2 592 . . . . 5 ((𝜑𝑦 ∈ (ran 𝐹 ∖ (ran 𝐹 ∖ {0}))) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = 0)
11993, 96, 118, 3fsumss 15678 . . . 4 (𝜑 → Σ𝑦 ∈ (ran 𝐹 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
12039, 92, 1193eqtrd 2775 . . 3 (𝜑 → (∫1𝐹) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
121 itg1val 25533 . . . . 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 480 . . . . . . . . 9 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝐹:ℝ⟶ℝ)
1243adantr 480 . . . . . . . . 9 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → ran 𝐹 ∈ Fin)
125 inss1 4228 . . . . . . . . . 10 ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {𝑦})
126125a1i 11 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {𝑦}))
12745ad2antrr 723 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝐹 “ {𝑦}) ∈ dom vol)
12848ad2antrr 723 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝐺 “ {𝑧}) ∈ dom vol)
129127, 128, 50syl2anc 583 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
13010adantr 480 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → ran 𝐹 ⊆ ℝ)
131130sselda 3982 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
13219ssdifssd 4142 . . . . . . . . . . . . 13 (𝜑 → (ran 𝐺 ∖ {0}) ⊆ ℝ)
133132sselda 3982 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝑧 ∈ ℝ)
134133adantr 480 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
135 eldifsni 4793 . . . . . . . . . . . . 13 (𝑧 ∈ (ran 𝐺 ∖ {0}) → 𝑧 ≠ 0)
136135ad2antlr 724 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ≠ 0)
137 simpr 484 . . . . . . . . . . . . 13 ((𝑦 = 0 ∧ 𝑧 = 0) → 𝑧 = 0)
138137necon3ai 2964 . . . . . . . . . . . 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 835 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) = (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
14115ad2antrr 723 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
142141, 131, 134fovcdmd 7583 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
143140, 142eqeltrrd 2833 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
144123, 124, 126, 129, 143itg1addlem1 25542 . . . . . . . 8 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘ 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) = Σ𝑦 ∈ ran 𝐹(vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
145 incom 4201 . . . . . . . . . . . . 13 ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ((𝐺 “ {𝑧}) ∩ (𝐹 “ {𝑦}))
146145a1i 11 . . . . . . . . . . . 12 (𝑦 ∈ ran 𝐹 → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ((𝐺 “ {𝑧}) ∩ (𝐹 “ {𝑦})))
147146iuneq2i 5018 . . . . . . . . . . 11 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = 𝑦 ∈ ran 𝐹((𝐺 “ {𝑧}) ∩ (𝐹 “ {𝑦}))
148 iunin2 5074 . . . . . . . . . . 11 𝑦 ∈ ran 𝐹((𝐺 “ {𝑧}) ∩ (𝐹 “ {𝑦})) = ((𝐺 “ {𝑧}) ∩ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}))
149147, 148eqtri 2759 . . . . . . . . . 10 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ((𝐺 “ {𝑧}) ∩ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}))
150 cnvimass 6080 . . . . . . . . . . . . 13 (𝐺 “ {𝑧}) ⊆ dom 𝐺
15118fdmd 6728 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐺 = ℝ)
152151adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → dom 𝐺 = ℝ)
153150, 152sseqtrid 4034 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ⊆ ℝ)
154 iunid 5063 . . . . . . . . . . . . . . 15 𝑦 ∈ ran 𝐹{𝑦} = ran 𝐹
155154imaeq2i 6057 . . . . . . . . . . . . . 14 (𝐹 𝑦 ∈ ran 𝐹{𝑦}) = (𝐹 “ ran 𝐹)
156 imaiun 7247 . . . . . . . . . . . . . 14 (𝐹 𝑦 ∈ ran 𝐹{𝑦}) = 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦})
157 cnvimarndm 6081 . . . . . . . . . . . . . 14 (𝐹 “ ran 𝐹) = dom 𝐹
158155, 156, 1573eqtr3i 2767 . . . . . . . . . . . . 13 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}) = dom 𝐹
1599fdmd 6728 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐹 = ℝ)
160159adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → dom 𝐹 = ℝ)
161158, 160eqtrid 2783 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}) = ℝ)
162153, 161sseqtrrd 4023 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ⊆ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}))
163 df-ss 3965 . . . . . . . . . . 11 ((𝐺 “ {𝑧}) ⊆ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}) ↔ ((𝐺 “ {𝑧}) ∩ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦})) = (𝐺 “ {𝑧}))
164162, 163sylib 217 . . . . . . . . . 10 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → ((𝐺 “ {𝑧}) ∩ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦})) = (𝐺 “ {𝑧}))
165149, 164eqtr2id 2784 . . . . . . . . 9 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) = 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})))
166165fveq2d 6895 . . . . . . . 8 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) = (vol‘ 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
167140sumeq2dv 15656 . . . . . . . 8 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧) = Σ𝑦 ∈ ran 𝐹(vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
168144, 166, 1673eqtr4d 2781 . . . . . . 7 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) = Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧))
169168oveq2d 7428 . . . . . 6 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝑧 · (vol‘(𝐺 “ {𝑧}))) = (𝑧 · Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧)))
170133recnd 11249 . . . . . . 7 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝑧 ∈ ℂ)
171142recnd 11249 . . . . . . 7 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℂ)
172124, 170, 171fsummulc2 15737 . . . . . 6 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝑧 · Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
173169, 172eqtrd 2771 . . . . 5 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝑧 · (vol‘(𝐺 “ {𝑧}))) = Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
174173sumeq2dv 15656 . . . 4 (𝜑 → Σ𝑧 ∈ (ran 𝐺 ∖ {0})(𝑧 · (vol‘(𝐺 “ {𝑧}))) = Σ𝑧 ∈ (ran 𝐺 ∖ {0})Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
175 difssd 4132 . . . . . 6 (𝜑 → (ran 𝐺 ∖ {0}) ⊆ ran 𝐺)
176170adantr 480 . . . . . . . 8 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
177176, 171mulcld 11241 . . . . . . 7 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
178124, 177fsumcl 15686 . . . . . 6 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
179 dfin4 4267 . . . . . . . . 9 (ran 𝐺 ∩ {0}) = (ran 𝐺 ∖ (ran 𝐺 ∖ {0}))
180 inss2 4229 . . . . . . . . 9 (ran 𝐺 ∩ {0}) ⊆ {0}
181179, 180eqsstrri 4017 . . . . . . . 8 (ran 𝐺 ∖ (ran 𝐺 ∖ {0})) ⊆ {0}
182181sseli 3978 . . . . . . 7 (𝑧 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {0})) → 𝑧 ∈ {0})
183 elsni 4645 . . . . . . . . . . . 12 (𝑧 ∈ {0} → 𝑧 = 0)
184183ad2antlr 724 . . . . . . . . . . 11 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 = 0)
185184oveq1d 7427 . . . . . . . . . 10 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) = (0 · (𝑦𝐼𝑧)))
18615ad2antrr 723 . . . . . . . . . . . . 13 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
18711adantlr 712 . . . . . . . . . . . . 13 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
188184, 105eqeltrdi 2840 . . . . . . . . . . . . 13 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
189186, 187, 188fovcdmd 7583 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
190189recnd 11249 . . . . . . . . . . 11 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℂ)
191190mul02d 11419 . . . . . . . . . 10 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (0 · (𝑦𝐼𝑧)) = 0)
192185, 191eqtrd 2771 . . . . . . . . 9 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) = 0)
193192sumeq2dv 15656 . . . . . . . 8 ((𝜑𝑧 ∈ {0}) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹0)
1943adantr 480 . . . . . . . . . 10 ((𝜑𝑧 ∈ {0}) → ran 𝐹 ∈ Fin)
195194olcd 871 . . . . . . . . 9 ((𝜑𝑧 ∈ {0}) → (ran 𝐹 ⊆ (ℤ‘0) ∨ ran 𝐹 ∈ Fin))
196 sumz 15675 . . . . . . . . 9 ((ran 𝐹 ⊆ (ℤ‘0) ∨ ran 𝐹 ∈ Fin) → Σ𝑦 ∈ ran 𝐹0 = 0)
197195, 196syl 17 . . . . . . . 8 ((𝜑𝑧 ∈ {0}) → Σ𝑦 ∈ ran 𝐹0 = 0)
198193, 197eqtrd 2771 . . . . . . 7 ((𝜑𝑧 ∈ {0}) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = 0)
199182, 198sylan2 592 . . . . . 6 ((𝜑𝑧 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {0}))) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = 0)
200175, 178, 199, 6fsumss 15678 . . . . 5 (𝜑 → Σ𝑧 ∈ (ran 𝐺 ∖ {0})Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
20120adantr 480 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
202201recnd 11249 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
20315ad2antrr 723 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
20410adantr 480 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ran 𝐺) → ran 𝐹 ⊆ ℝ)
205204sselda 3982 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
206203, 205, 201fovcdmd 7583 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
207206recnd 11249 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℂ)
208202, 207mulcld 11241 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
209208anasss 466 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ran 𝐺𝑦 ∈ ran 𝐹)) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
2106, 3, 209fsumcom 15728 . . . . 5 (𝜑 → Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)))
211200, 210eqtrd 2771 . . . 4 (𝜑 → Σ𝑧 ∈ (ran 𝐺 ∖ {0})Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)))
212122, 174, 2113eqtrd 2775 . . 3 (𝜑 → (∫1𝐺) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)))
213120, 212oveq12d 7430 . 2 (𝜑 → ((∫1𝐹) + (∫1𝐺)) = (Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
21429, 37, 2133eqtr4d 2781 1 (𝜑 → (∫1‘(𝐹f + 𝐺)) = ((∫1𝐹) + (∫1𝐺)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 844   = wceq 1540  wcel 2105  wne 2939  cdif 3945  cin 3947  wss 3948  ifcif 4528  {csn 4628   ciun 4997   × cxp 5674  ccnv 5675  dom cdm 5676  ran crn 5677  cres 5678  cima 5679  wf 6539  cfv 6543  (class class class)co 7412  cmpo 7414  f cof 7672  Fincfn 8945  cc 11114  cr 11115  0cc0 11116   + caddc 11119   · cmul 11121  cuz 12829  Σcsu 15639  volcvol 25313  1citg1 25465
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9642  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194  ax-addf 11195
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-2o 8473  df-er 8709  df-map 8828  df-pm 8829  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9443  df-inf 9444  df-oi 9511  df-dju 9902  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-div 11879  df-nn 12220  df-2 12282  df-3 12283  df-n0 12480  df-z 12566  df-uz 12830  df-q 12940  df-rp 12982  df-xadd 13100  df-ioo 13335  df-ico 13337  df-icc 13338  df-fz 13492  df-fzo 13635  df-fl 13764  df-seq 13974  df-exp 14035  df-hash 14298  df-cj 15053  df-re 15054  df-im 15055  df-sqrt 15189  df-abs 15190  df-clim 15439  df-sum 15640  df-xmet 21227  df-met 21228  df-ovol 25314  df-vol 25315  df-mbf 25469  df-itg1 25470
This theorem is referenced by:  itg1add  25552
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