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Theorem itg1addlem5 25081
Description: Lemma for itg1add 25082. (Contributed by Mario Carneiro, 27-Jun-2014.)
Hypotheses
Ref Expression
i1fadd.1 (πœ‘ β†’ 𝐹 ∈ dom ∫1)
i1fadd.2 (πœ‘ β†’ 𝐺 ∈ dom ∫1)
itg1add.3 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (volβ€˜((◑𝐹 β€œ {𝑖}) ∩ (◑𝐺 β€œ {𝑗})))))
itg1add.4 𝑃 = ( + β†Ύ (ran 𝐹 Γ— ran 𝐺))
Assertion
Ref Expression
itg1addlem5 (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = ((∫1β€˜πΉ) + (∫1β€˜πΊ)))
Distinct variable groups:   𝑖,𝑗,𝐹   𝑖,𝐺,𝑗   πœ‘,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 (πœ‘ β†’ 𝐹 ∈ dom ∫1)
2 i1frn 25057 . . . 4 (𝐹 ∈ dom ∫1 β†’ ran 𝐹 ∈ Fin)
31, 2syl 17 . . 3 (πœ‘ β†’ ran 𝐹 ∈ Fin)
4 i1fadd.2 . . . . . 6 (πœ‘ β†’ 𝐺 ∈ dom ∫1)
5 i1frn 25057 . . . . . 6 (𝐺 ∈ dom ∫1 β†’ ran 𝐺 ∈ Fin)
64, 5syl 17 . . . . 5 (πœ‘ β†’ ran 𝐺 ∈ Fin)
76adantr 482 . . . 4 ((πœ‘ ∧ 𝑦 ∈ ran 𝐹) β†’ ran 𝐺 ∈ Fin)
8 i1ff 25056 . . . . . . . . . 10 (𝐹 ∈ dom ∫1 β†’ 𝐹:β„βŸΆβ„)
91, 8syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐹:β„βŸΆβ„)
109frnd 6681 . . . . . . . 8 (πœ‘ β†’ ran 𝐹 βŠ† ℝ)
1110sselda 3949 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ ℝ)
1211adantr 482 . . . . . 6 (((πœ‘ ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑦 ∈ ℝ)
1312recnd 11190 . . . . 5 (((πœ‘ ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑦 ∈ β„‚)
14 itg1add.3 . . . . . . . . 9 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (volβ€˜((◑𝐹 β€œ {𝑖}) ∩ (◑𝐺 β€œ {𝑗})))))
151, 4, 14itg1addlem2 25077 . . . . . . . 8 (πœ‘ β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
1615ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
17 i1ff 25056 . . . . . . . . . . 11 (𝐺 ∈ dom ∫1 β†’ 𝐺:β„βŸΆβ„)
184, 17syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐺:β„βŸΆβ„)
1918frnd 6681 . . . . . . . . 9 (πœ‘ β†’ ran 𝐺 βŠ† ℝ)
2019sselda 3949 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ ℝ)
2120adantlr 714 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ ℝ)
2216, 12, 21fovcdmd 7531 . . . . . 6 (((πœ‘ ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦𝐼𝑧) ∈ ℝ)
2322recnd 11190 . . . . 5 (((πœ‘ ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦𝐼𝑧) ∈ β„‚)
2413, 23mulcld 11182 . . . 4 (((πœ‘ ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦 Β· (𝑦𝐼𝑧)) ∈ β„‚)
257, 24fsumcl 15625 . . 3 ((πœ‘ ∧ 𝑦 ∈ ran 𝐹) β†’ Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)) ∈ β„‚)
2621recnd 11190 . . . . 5 (((πœ‘ ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ β„‚)
2726, 23mulcld 11182 . . . 4 (((πœ‘ ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑧 Β· (𝑦𝐼𝑧)) ∈ β„‚)
287, 27fsumcl 15625 . . 3 ((πœ‘ ∧ 𝑦 ∈ ran 𝐹) β†’ Σ𝑧 ∈ ran 𝐺(𝑧 Β· (𝑦𝐼𝑧)) ∈ β„‚)
293, 25, 28fsumadd 15632 . 2 (πœ‘ β†’ Σ𝑦 ∈ ran 𝐹(Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 Β· (𝑦𝐼𝑧))) = (Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)) + Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 Β· (𝑦𝐼𝑧))))
30 itg1add.4 . . . 4 𝑃 = ( + β†Ύ (ran 𝐹 Γ— ran 𝐺))
311, 4, 14, 30itg1addlem4 25079 . . 3 (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)))
3213, 26, 23adddird 11187 . . . . . 6 (((πœ‘ ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = ((𝑦 Β· (𝑦𝐼𝑧)) + (𝑧 Β· (𝑦𝐼𝑧))))
3332sumeq2dv 15595 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ ran 𝐹) β†’ Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺((𝑦 Β· (𝑦𝐼𝑧)) + (𝑧 Β· (𝑦𝐼𝑧))))
347, 24, 27fsumadd 15632 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ ran 𝐹) β†’ Σ𝑧 ∈ ran 𝐺((𝑦 Β· (𝑦𝐼𝑧)) + (𝑧 Β· (𝑦𝐼𝑧))) = (Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 Β· (𝑦𝐼𝑧))))
3533, 34eqtrd 2777 . . . 4 ((πœ‘ ∧ 𝑦 ∈ ran 𝐹) β†’ Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = (Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 Β· (𝑦𝐼𝑧))))
3635sumeq2dv 15595 . . 3 (πœ‘ β†’ Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹(Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 Β· (𝑦𝐼𝑧))))
3731, 36eqtrd 2777 . 2 (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = Σ𝑦 ∈ ran 𝐹(Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 Β· (𝑦𝐼𝑧))))
38 itg1val 25063 . . . . 5 (𝐹 ∈ dom ∫1 β†’ (∫1β€˜πΉ) = Σ𝑦 ∈ (ran 𝐹 βˆ– {0})(𝑦 Β· (volβ€˜(◑𝐹 β€œ {𝑦}))))
391, 38syl 17 . . . 4 (πœ‘ β†’ (∫1β€˜πΉ) = Σ𝑦 ∈ (ran 𝐹 βˆ– {0})(𝑦 Β· (volβ€˜(◑𝐹 β€œ {𝑦}))))
4018adantr 482 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ 𝐺:β„βŸΆβ„)
416adantr 482 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ ran 𝐺 ∈ Fin)
42 inss2 4194 . . . . . . . . . 10 ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† (◑𝐺 β€œ {𝑧})
4342a1i 11 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† (◑𝐺 β€œ {𝑧}))
44 i1fima 25058 . . . . . . . . . . . 12 (𝐹 ∈ dom ∫1 β†’ (◑𝐹 β€œ {𝑦}) ∈ dom vol)
451, 44syl 17 . . . . . . . . . . 11 (πœ‘ β†’ (◑𝐹 β€œ {𝑦}) ∈ dom vol)
4645ad2antrr 725 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (◑𝐹 β€œ {𝑦}) ∈ dom vol)
47 i1fima 25058 . . . . . . . . . . . 12 (𝐺 ∈ dom ∫1 β†’ (◑𝐺 β€œ {𝑧}) ∈ dom vol)
484, 47syl 17 . . . . . . . . . . 11 (πœ‘ β†’ (◑𝐺 β€œ {𝑧}) ∈ dom vol)
4948ad2antrr 725 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (◑𝐺 β€œ {𝑧}) ∈ dom vol)
50 inmbl 24922 . . . . . . . . . 10 (((◑𝐹 β€œ {𝑦}) ∈ dom vol ∧ (◑𝐺 β€œ {𝑧}) ∈ dom vol) β†’ ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) ∈ dom vol)
5146, 49, 50syl2anc 585 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) ∈ dom vol)
5210ssdifssd 4107 . . . . . . . . . . . . 13 (πœ‘ β†’ (ran 𝐹 βˆ– {0}) βŠ† ℝ)
5352sselda 3949 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ 𝑦 ∈ ℝ)
5453adantr 482 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑦 ∈ ℝ)
5519adantr 482 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ ran 𝐺 βŠ† ℝ)
5655sselda 3949 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ ℝ)
57 eldifsni 4755 . . . . . . . . . . . . 13 (𝑦 ∈ (ran 𝐹 βˆ– {0}) β†’ 𝑦 β‰  0)
5857ad2antlr 726 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑦 β‰  0)
59 simpl 484 . . . . . . . . . . . . 13 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ 𝑦 = 0)
6059necon3ai 2969 . . . . . . . . . . . 12 (𝑦 β‰  0 β†’ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))
6158, 60syl 17 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))
621, 4, 14itg1addlem3 25078 . . . . . . . . . . 11 (((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0)) β†’ (𝑦𝐼𝑧) = (volβ€˜((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))))
6354, 56, 61, 62syl21anc 837 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦𝐼𝑧) = (volβ€˜((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))))
6415ad2antrr 725 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
6564, 54, 56fovcdmd 7531 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦𝐼𝑧) ∈ ℝ)
6663, 65eqeltrrd 2839 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (volβ€˜((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))) ∈ ℝ)
6740, 41, 43, 51, 66itg1addlem1 25072 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ (volβ€˜βˆͺ 𝑧 ∈ ran 𝐺((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))) = Σ𝑧 ∈ ran 𝐺(volβ€˜((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))))
68 iunin2 5036 . . . . . . . . . 10 βˆͺ 𝑧 ∈ ran 𝐺((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) = ((◑𝐹 β€œ {𝑦}) ∩ βˆͺ 𝑧 ∈ ran 𝐺(◑𝐺 β€œ {𝑧}))
691adantr 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ 𝐹 ∈ dom ∫1)
7069, 44syl 17 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ (◑𝐹 β€œ {𝑦}) ∈ dom vol)
71 mblss 24911 . . . . . . . . . . . . 13 ((◑𝐹 β€œ {𝑦}) ∈ dom vol β†’ (◑𝐹 β€œ {𝑦}) βŠ† ℝ)
7270, 71syl 17 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ (◑𝐹 β€œ {𝑦}) βŠ† ℝ)
73 iunid 5025 . . . . . . . . . . . . . . 15 βˆͺ 𝑧 ∈ ran 𝐺{𝑧} = ran 𝐺
7473imaeq2i 6016 . . . . . . . . . . . . . 14 (◑𝐺 β€œ βˆͺ 𝑧 ∈ ran 𝐺{𝑧}) = (◑𝐺 β€œ ran 𝐺)
75 imaiun 7197 . . . . . . . . . . . . . 14 (◑𝐺 β€œ βˆͺ 𝑧 ∈ ran 𝐺{𝑧}) = βˆͺ 𝑧 ∈ ran 𝐺(◑𝐺 β€œ {𝑧})
76 cnvimarndm 6039 . . . . . . . . . . . . . 14 (◑𝐺 β€œ ran 𝐺) = dom 𝐺
7774, 75, 763eqtr3i 2773 . . . . . . . . . . . . 13 βˆͺ 𝑧 ∈ ran 𝐺(◑𝐺 β€œ {𝑧}) = dom 𝐺
7840fdmd 6684 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ dom 𝐺 = ℝ)
7977, 78eqtrid 2789 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ βˆͺ 𝑧 ∈ ran 𝐺(◑𝐺 β€œ {𝑧}) = ℝ)
8072, 79sseqtrrd 3990 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ (◑𝐹 β€œ {𝑦}) βŠ† βˆͺ 𝑧 ∈ ran 𝐺(◑𝐺 β€œ {𝑧}))
81 df-ss 3932 . . . . . . . . . . 11 ((◑𝐹 β€œ {𝑦}) βŠ† βˆͺ 𝑧 ∈ ran 𝐺(◑𝐺 β€œ {𝑧}) ↔ ((◑𝐹 β€œ {𝑦}) ∩ βˆͺ 𝑧 ∈ ran 𝐺(◑𝐺 β€œ {𝑧})) = (◑𝐹 β€œ {𝑦}))
8280, 81sylib 217 . . . . . . . . . 10 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ ((◑𝐹 β€œ {𝑦}) ∩ βˆͺ 𝑧 ∈ ran 𝐺(◑𝐺 β€œ {𝑧})) = (◑𝐹 β€œ {𝑦}))
8368, 82eqtr2id 2790 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ (◑𝐹 β€œ {𝑦}) = βˆͺ 𝑧 ∈ ran 𝐺((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})))
8483fveq2d 6851 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ (volβ€˜(◑𝐹 β€œ {𝑦})) = (volβ€˜βˆͺ 𝑧 ∈ ran 𝐺((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))))
8563sumeq2dv 15595 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧) = Σ𝑧 ∈ ran 𝐺(volβ€˜((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))))
8667, 84, 853eqtr4d 2787 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ (volβ€˜(◑𝐹 β€œ {𝑦})) = Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧))
8786oveq2d 7378 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ (𝑦 Β· (volβ€˜(◑𝐹 β€œ {𝑦}))) = (𝑦 Β· Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧)))
8853recnd 11190 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ 𝑦 ∈ β„‚)
8965recnd 11190 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦𝐼𝑧) ∈ β„‚)
9041, 88, 89fsummulc2 15676 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ (𝑦 Β· Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)))
9187, 90eqtrd 2777 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ (𝑦 Β· (volβ€˜(◑𝐹 β€œ {𝑦}))) = Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)))
9291sumeq2dv 15595 . . . 4 (πœ‘ β†’ Σ𝑦 ∈ (ran 𝐹 βˆ– {0})(𝑦 Β· (volβ€˜(◑𝐹 β€œ {𝑦}))) = Σ𝑦 ∈ (ran 𝐹 βˆ– {0})Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)))
93 difssd 4097 . . . . 5 (πœ‘ β†’ (ran 𝐹 βˆ– {0}) βŠ† ran 𝐹)
9454recnd 11190 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑦 ∈ β„‚)
9594, 89mulcld 11182 . . . . . 6 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦 Β· (𝑦𝐼𝑧)) ∈ β„‚)
9641, 95fsumcl 15625 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)) ∈ β„‚)
97 dfin4 4232 . . . . . . . 8 (ran 𝐹 ∩ {0}) = (ran 𝐹 βˆ– (ran 𝐹 βˆ– {0}))
98 inss2 4194 . . . . . . . 8 (ran 𝐹 ∩ {0}) βŠ† {0}
9997, 98eqsstrri 3984 . . . . . . 7 (ran 𝐹 βˆ– (ran 𝐹 βˆ– {0})) βŠ† {0}
10099sseli 3945 . . . . . 6 (𝑦 ∈ (ran 𝐹 βˆ– (ran 𝐹 βˆ– {0})) β†’ 𝑦 ∈ {0})
101 elsni 4608 . . . . . . . . . . 11 (𝑦 ∈ {0} β†’ 𝑦 = 0)
102101ad2antlr 726 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑦 = 0)
103102oveq1d 7377 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦 Β· (𝑦𝐼𝑧)) = (0 Β· (𝑦𝐼𝑧)))
10415ad2antrr 725 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
105 0re 11164 . . . . . . . . . . . . 13 0 ∈ ℝ
106102, 105eqeltrdi 2846 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑦 ∈ ℝ)
10720adantlr 714 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ ℝ)
108104, 106, 107fovcdmd 7531 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦𝐼𝑧) ∈ ℝ)
109108recnd 11190 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦𝐼𝑧) ∈ β„‚)
110109mul02d 11360 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) β†’ (0 Β· (𝑦𝐼𝑧)) = 0)
111103, 110eqtrd 2777 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦 Β· (𝑦𝐼𝑧)) = 0)
112111sumeq2dv 15595 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ {0}) β†’ Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺0)
1136adantr 482 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ {0}) β†’ ran 𝐺 ∈ Fin)
114113olcd 873 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ {0}) β†’ (ran 𝐺 βŠ† (β„€β‰₯β€˜0) ∨ ran 𝐺 ∈ Fin))
115 sumz 15614 . . . . . . . 8 ((ran 𝐺 βŠ† (β„€β‰₯β€˜0) ∨ ran 𝐺 ∈ Fin) β†’ Σ𝑧 ∈ ran 𝐺0 = 0)
116114, 115syl 17 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ {0}) β†’ Σ𝑧 ∈ ran 𝐺0 = 0)
117112, 116eqtrd 2777 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ {0}) β†’ Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)) = 0)
118100, 117sylan2 594 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– (ran 𝐹 βˆ– {0}))) β†’ Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)) = 0)
11993, 96, 118, 3fsumss 15617 . . . 4 (πœ‘ β†’ Σ𝑦 ∈ (ran 𝐹 βˆ– {0})Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)))
12039, 92, 1193eqtrd 2781 . . 3 (πœ‘ β†’ (∫1β€˜πΉ) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)))
121 itg1val 25063 . . . . 5 (𝐺 ∈ dom ∫1 β†’ (∫1β€˜πΊ) = Σ𝑧 ∈ (ran 𝐺 βˆ– {0})(𝑧 Β· (volβ€˜(◑𝐺 β€œ {𝑧}))))
1224, 121syl 17 . . . 4 (πœ‘ β†’ (∫1β€˜πΊ) = Σ𝑧 ∈ (ran 𝐺 βˆ– {0})(𝑧 Β· (volβ€˜(◑𝐺 β€œ {𝑧}))))
1239adantr 482 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ 𝐹:β„βŸΆβ„)
1243adantr 482 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ ran 𝐹 ∈ Fin)
125 inss1 4193 . . . . . . . . . 10 ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† (◑𝐹 β€œ {𝑦})
126125a1i 11 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† (◑𝐹 β€œ {𝑦}))
12745ad2antrr 725 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ (◑𝐹 β€œ {𝑦}) ∈ dom vol)
12848ad2antrr 725 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ (◑𝐺 β€œ {𝑧}) ∈ dom vol)
129127, 128, 50syl2anc 585 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) ∈ dom vol)
13010adantr 482 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ ran 𝐹 βŠ† ℝ)
131130sselda 3949 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ ℝ)
13219ssdifssd 4107 . . . . . . . . . . . . 13 (πœ‘ β†’ (ran 𝐺 βˆ– {0}) βŠ† ℝ)
133132sselda 3949 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ 𝑧 ∈ ℝ)
134133adantr 482 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ ℝ)
135 eldifsni 4755 . . . . . . . . . . . . 13 (𝑧 ∈ (ran 𝐺 βˆ– {0}) β†’ 𝑧 β‰  0)
136135ad2antlr 726 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 β‰  0)
137 simpr 486 . . . . . . . . . . . . 13 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ 𝑧 = 0)
138137necon3ai 2969 . . . . . . . . . . . 12 (𝑧 β‰  0 β†’ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))
139136, 138syl 17 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))
140131, 134, 139, 62syl21anc 837 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦𝐼𝑧) = (volβ€˜((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))))
14115ad2antrr 725 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
142141, 131, 134fovcdmd 7531 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦𝐼𝑧) ∈ ℝ)
143140, 142eqeltrrd 2839 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ (volβ€˜((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))) ∈ ℝ)
144123, 124, 126, 129, 143itg1addlem1 25072 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (volβ€˜βˆͺ 𝑦 ∈ ran 𝐹((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))) = Σ𝑦 ∈ ran 𝐹(volβ€˜((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))))
145 incom 4166 . . . . . . . . . . . . 13 ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) = ((◑𝐺 β€œ {𝑧}) ∩ (◑𝐹 β€œ {𝑦}))
146145a1i 11 . . . . . . . . . . . 12 (𝑦 ∈ ran 𝐹 β†’ ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) = ((◑𝐺 β€œ {𝑧}) ∩ (◑𝐹 β€œ {𝑦})))
147146iuneq2i 4980 . . . . . . . . . . 11 βˆͺ 𝑦 ∈ ran 𝐹((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) = βˆͺ 𝑦 ∈ ran 𝐹((◑𝐺 β€œ {𝑧}) ∩ (◑𝐹 β€œ {𝑦}))
148 iunin2 5036 . . . . . . . . . . 11 βˆͺ 𝑦 ∈ ran 𝐹((◑𝐺 β€œ {𝑧}) ∩ (◑𝐹 β€œ {𝑦})) = ((◑𝐺 β€œ {𝑧}) ∩ βˆͺ 𝑦 ∈ ran 𝐹(◑𝐹 β€œ {𝑦}))
149147, 148eqtri 2765 . . . . . . . . . 10 βˆͺ 𝑦 ∈ ran 𝐹((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) = ((◑𝐺 β€œ {𝑧}) ∩ βˆͺ 𝑦 ∈ ran 𝐹(◑𝐹 β€œ {𝑦}))
150 cnvimass 6038 . . . . . . . . . . . . 13 (◑𝐺 β€œ {𝑧}) βŠ† dom 𝐺
15118fdmd 6684 . . . . . . . . . . . . . 14 (πœ‘ β†’ dom 𝐺 = ℝ)
152151adantr 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ dom 𝐺 = ℝ)
153150, 152sseqtrid 4001 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (◑𝐺 β€œ {𝑧}) βŠ† ℝ)
154 iunid 5025 . . . . . . . . . . . . . . 15 βˆͺ 𝑦 ∈ ran 𝐹{𝑦} = ran 𝐹
155154imaeq2i 6016 . . . . . . . . . . . . . 14 (◑𝐹 β€œ βˆͺ 𝑦 ∈ ran 𝐹{𝑦}) = (◑𝐹 β€œ ran 𝐹)
156 imaiun 7197 . . . . . . . . . . . . . 14 (◑𝐹 β€œ βˆͺ 𝑦 ∈ ran 𝐹{𝑦}) = βˆͺ 𝑦 ∈ ran 𝐹(◑𝐹 β€œ {𝑦})
157 cnvimarndm 6039 . . . . . . . . . . . . . 14 (◑𝐹 β€œ ran 𝐹) = dom 𝐹
158155, 156, 1573eqtr3i 2773 . . . . . . . . . . . . 13 βˆͺ 𝑦 ∈ ran 𝐹(◑𝐹 β€œ {𝑦}) = dom 𝐹
1599fdmd 6684 . . . . . . . . . . . . . 14 (πœ‘ β†’ dom 𝐹 = ℝ)
160159adantr 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ dom 𝐹 = ℝ)
161158, 160eqtrid 2789 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ βˆͺ 𝑦 ∈ ran 𝐹(◑𝐹 β€œ {𝑦}) = ℝ)
162153, 161sseqtrrd 3990 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (◑𝐺 β€œ {𝑧}) βŠ† βˆͺ 𝑦 ∈ ran 𝐹(◑𝐹 β€œ {𝑦}))
163 df-ss 3932 . . . . . . . . . . 11 ((◑𝐺 β€œ {𝑧}) βŠ† βˆͺ 𝑦 ∈ ran 𝐹(◑𝐹 β€œ {𝑦}) ↔ ((◑𝐺 β€œ {𝑧}) ∩ βˆͺ 𝑦 ∈ ran 𝐹(◑𝐹 β€œ {𝑦})) = (◑𝐺 β€œ {𝑧}))
164162, 163sylib 217 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ ((◑𝐺 β€œ {𝑧}) ∩ βˆͺ 𝑦 ∈ ran 𝐹(◑𝐹 β€œ {𝑦})) = (◑𝐺 β€œ {𝑧}))
165149, 164eqtr2id 2790 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (◑𝐺 β€œ {𝑧}) = βˆͺ 𝑦 ∈ ran 𝐹((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})))
166165fveq2d 6851 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (volβ€˜(◑𝐺 β€œ {𝑧})) = (volβ€˜βˆͺ 𝑦 ∈ ran 𝐹((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))))
167140sumeq2dv 15595 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧) = Σ𝑦 ∈ ran 𝐹(volβ€˜((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))))
168144, 166, 1673eqtr4d 2787 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (volβ€˜(◑𝐺 β€œ {𝑧})) = Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧))
169168oveq2d 7378 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (𝑧 Β· (volβ€˜(◑𝐺 β€œ {𝑧}))) = (𝑧 Β· Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧)))
170133recnd 11190 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ 𝑧 ∈ β„‚)
171142recnd 11190 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦𝐼𝑧) ∈ β„‚)
172124, 170, 171fsummulc2 15676 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (𝑧 Β· Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)))
173169, 172eqtrd 2777 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (𝑧 Β· (volβ€˜(◑𝐺 β€œ {𝑧}))) = Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)))
174173sumeq2dv 15595 . . . 4 (πœ‘ β†’ Σ𝑧 ∈ (ran 𝐺 βˆ– {0})(𝑧 Β· (volβ€˜(◑𝐺 β€œ {𝑧}))) = Σ𝑧 ∈ (ran 𝐺 βˆ– {0})Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)))
175 difssd 4097 . . . . . 6 (πœ‘ β†’ (ran 𝐺 βˆ– {0}) βŠ† ran 𝐺)
176170adantr 482 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ β„‚)
177176, 171mulcld 11182 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑧 Β· (𝑦𝐼𝑧)) ∈ β„‚)
178124, 177fsumcl 15625 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)) ∈ β„‚)
179 dfin4 4232 . . . . . . . . 9 (ran 𝐺 ∩ {0}) = (ran 𝐺 βˆ– (ran 𝐺 βˆ– {0}))
180 inss2 4194 . . . . . . . . 9 (ran 𝐺 ∩ {0}) βŠ† {0}
181179, 180eqsstrri 3984 . . . . . . . 8 (ran 𝐺 βˆ– (ran 𝐺 βˆ– {0})) βŠ† {0}
182181sseli 3945 . . . . . . 7 (𝑧 ∈ (ran 𝐺 βˆ– (ran 𝐺 βˆ– {0})) β†’ 𝑧 ∈ {0})
183 elsni 4608 . . . . . . . . . . . 12 (𝑧 ∈ {0} β†’ 𝑧 = 0)
184183ad2antlr 726 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 = 0)
185184oveq1d 7377 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑧 Β· (𝑦𝐼𝑧)) = (0 Β· (𝑦𝐼𝑧)))
18615ad2antrr 725 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
18711adantlr 714 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ ℝ)
188184, 105eqeltrdi 2846 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ ℝ)
189186, 187, 188fovcdmd 7531 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦𝐼𝑧) ∈ ℝ)
190189recnd 11190 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦𝐼𝑧) ∈ β„‚)
191190mul02d 11360 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) β†’ (0 Β· (𝑦𝐼𝑧)) = 0)
192185, 191eqtrd 2777 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑧 Β· (𝑦𝐼𝑧)) = 0)
193192sumeq2dv 15595 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ {0}) β†’ Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹0)
1943adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ {0}) β†’ ran 𝐹 ∈ Fin)
195194olcd 873 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ {0}) β†’ (ran 𝐹 βŠ† (β„€β‰₯β€˜0) ∨ ran 𝐹 ∈ Fin))
196 sumz 15614 . . . . . . . . 9 ((ran 𝐹 βŠ† (β„€β‰₯β€˜0) ∨ ran 𝐹 ∈ Fin) β†’ Σ𝑦 ∈ ran 𝐹0 = 0)
197195, 196syl 17 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ {0}) β†’ Σ𝑦 ∈ ran 𝐹0 = 0)
198193, 197eqtrd 2777 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ {0}) β†’ Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)) = 0)
199182, 198sylan2 594 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– (ran 𝐺 βˆ– {0}))) β†’ Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)) = 0)
200175, 178, 199, 6fsumss 15617 . . . . 5 (πœ‘ β†’ Σ𝑧 ∈ (ran 𝐺 βˆ– {0})Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)))
20120adantr 482 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ ℝ)
202201recnd 11190 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ β„‚)
20315ad2antrr 725 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
20410adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran 𝐹 βŠ† ℝ)
205204sselda 3949 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ ℝ)
206203, 205, 201fovcdmd 7531 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦𝐼𝑧) ∈ ℝ)
207206recnd 11190 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦𝐼𝑧) ∈ β„‚)
208202, 207mulcld 11182 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑧 Β· (𝑦𝐼𝑧)) ∈ β„‚)
209208anasss 468 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐹)) β†’ (𝑧 Β· (𝑦𝐼𝑧)) ∈ β„‚)
2106, 3, 209fsumcom 15667 . . . . 5 (πœ‘ β†’ Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 Β· (𝑦𝐼𝑧)))
211200, 210eqtrd 2777 . . . 4 (πœ‘ β†’ Σ𝑧 ∈ (ran 𝐺 βˆ– {0})Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 Β· (𝑦𝐼𝑧)))
212122, 174, 2113eqtrd 2781 . . 3 (πœ‘ β†’ (∫1β€˜πΊ) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 Β· (𝑦𝐼𝑧)))
213120, 212oveq12d 7380 . 2 (πœ‘ β†’ ((∫1β€˜πΉ) + (∫1β€˜πΊ)) = (Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)) + Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 Β· (𝑦𝐼𝑧))))
21429, 37, 2133eqtr4d 2787 1 (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = ((∫1β€˜πΉ) + (∫1β€˜πΊ)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2944   βˆ– cdif 3912   ∩ cin 3914   βŠ† wss 3915  ifcif 4491  {csn 4591  βˆͺ ciun 4959   Γ— cxp 5636  β—‘ccnv 5637  dom cdm 5638  ran crn 5639   β†Ύ cres 5640   β€œ cima 5641  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362   ∈ cmpo 7364   ∘f cof 7620  Fincfn 8890  β„‚cc 11056  β„cr 11057  0cc0 11058   + caddc 11061   Β· cmul 11063  β„€β‰₯cuz 12770  Ξ£csu 15577  volcvol 24843  βˆ«1citg1 24995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136  ax-addf 11137
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-disj 5076  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-er 8655  df-map 8774  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9385  df-inf 9386  df-oi 9453  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-z 12507  df-uz 12771  df-q 12881  df-rp 12923  df-xadd 13041  df-ioo 13275  df-ico 13277  df-icc 13278  df-fz 13432  df-fzo 13575  df-fl 13704  df-seq 13914  df-exp 13975  df-hash 14238  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-clim 15377  df-sum 15578  df-xmet 20805  df-met 20806  df-ovol 24844  df-vol 24845  df-mbf 24999  df-itg1 25000
This theorem is referenced by:  itg1add  25082
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