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Theorem itg1addlem5 25646
Description: Lemma for itg1add 25647. (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 25622 . . . 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 25622 . . . . . 6 (𝐺 ∈ dom ∫1 β†’ ran 𝐺 ∈ Fin)
64, 5syl 17 . . . . 5 (πœ‘ β†’ ran 𝐺 ∈ Fin)
76adantr 479 . . . 4 ((πœ‘ ∧ 𝑦 ∈ ran 𝐹) β†’ ran 𝐺 ∈ Fin)
8 i1ff 25621 . . . . . . . . . 10 (𝐹 ∈ dom ∫1 β†’ 𝐹:β„βŸΆβ„)
91, 8syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐹:β„βŸΆβ„)
109frnd 6724 . . . . . . . 8 (πœ‘ β†’ ran 𝐹 βŠ† ℝ)
1110sselda 3972 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ ℝ)
1211adantr 479 . . . . . 6 (((πœ‘ ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑦 ∈ ℝ)
1312recnd 11270 . . . . 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 25642 . . . . . . . 8 (πœ‘ β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
1615ad2antrr 724 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
17 i1ff 25621 . . . . . . . . . . 11 (𝐺 ∈ dom ∫1 β†’ 𝐺:β„βŸΆβ„)
184, 17syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐺:β„βŸΆβ„)
1918frnd 6724 . . . . . . . . 9 (πœ‘ β†’ ran 𝐺 βŠ† ℝ)
2019sselda 3972 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ ℝ)
2120adantlr 713 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ ℝ)
2216, 12, 21fovcdmd 7589 . . . . . 6 (((πœ‘ ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦𝐼𝑧) ∈ ℝ)
2322recnd 11270 . . . . 5 (((πœ‘ ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦𝐼𝑧) ∈ β„‚)
2413, 23mulcld 11262 . . . 4 (((πœ‘ ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦 Β· (𝑦𝐼𝑧)) ∈ β„‚)
257, 24fsumcl 15709 . . 3 ((πœ‘ ∧ 𝑦 ∈ ran 𝐹) β†’ Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)) ∈ β„‚)
2621recnd 11270 . . . . 5 (((πœ‘ ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ β„‚)
2726, 23mulcld 11262 . . . 4 (((πœ‘ ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑧 Β· (𝑦𝐼𝑧)) ∈ β„‚)
287, 27fsumcl 15709 . . 3 ((πœ‘ ∧ 𝑦 ∈ ran 𝐹) β†’ Σ𝑧 ∈ ran 𝐺(𝑧 Β· (𝑦𝐼𝑧)) ∈ β„‚)
293, 25, 28fsumadd 15716 . 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 25644 . . 3 (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)))
3213, 26, 23adddird 11267 . . . . . 6 (((πœ‘ ∧ 𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = ((𝑦 Β· (𝑦𝐼𝑧)) + (𝑧 Β· (𝑦𝐼𝑧))))
3332sumeq2dv 15679 . . . . 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 15716 . . . . 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 2765 . . . 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 15679 . . 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 2765 . 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 25628 . . . . 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 479 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ 𝐺:β„βŸΆβ„)
416adantr 479 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ ran 𝐺 ∈ Fin)
42 inss2 4224 . . . . . . . . . 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 25623 . . . . . . . . . . . 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 724 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (◑𝐹 β€œ {𝑦}) ∈ dom vol)
47 i1fima 25623 . . . . . . . . . . . 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 724 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (◑𝐺 β€œ {𝑧}) ∈ dom vol)
50 inmbl 25487 . . . . . . . . . 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 582 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) ∈ dom vol)
5210ssdifssd 4135 . . . . . . . . . . . . 13 (πœ‘ β†’ (ran 𝐹 βˆ– {0}) βŠ† ℝ)
5352sselda 3972 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ 𝑦 ∈ ℝ)
5453adantr 479 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑦 ∈ ℝ)
5519adantr 479 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ ran 𝐺 βŠ† ℝ)
5655sselda 3972 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ ℝ)
57 eldifsni 4789 . . . . . . . . . . . . 13 (𝑦 ∈ (ran 𝐹 βˆ– {0}) β†’ 𝑦 β‰  0)
5857ad2antlr 725 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑦 β‰  0)
59 simpl 481 . . . . . . . . . . . . 13 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ 𝑦 = 0)
6059necon3ai 2955 . . . . . . . . . . . 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 25643 . . . . . . . . . . 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 836 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦𝐼𝑧) = (volβ€˜((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))))
6415ad2antrr 724 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
6564, 54, 56fovcdmd 7589 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦𝐼𝑧) ∈ ℝ)
6663, 65eqeltrrd 2826 . . . . . . . . 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 25637 . . . . . . . 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 5069 . . . . . . . . . 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 479 . . . . . . . . . . . . . 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 25476 . . . . . . . . . . . . 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 5058 . . . . . . . . . . . . . . 15 βˆͺ 𝑧 ∈ ran 𝐺{𝑧} = ran 𝐺
7473imaeq2i 6056 . . . . . . . . . . . . . 14 (◑𝐺 β€œ βˆͺ 𝑧 ∈ ran 𝐺{𝑧}) = (◑𝐺 β€œ ran 𝐺)
75 imaiun 7250 . . . . . . . . . . . . . 14 (◑𝐺 β€œ βˆͺ 𝑧 ∈ ran 𝐺{𝑧}) = βˆͺ 𝑧 ∈ ran 𝐺(◑𝐺 β€œ {𝑧})
76 cnvimarndm 6081 . . . . . . . . . . . . . 14 (◑𝐺 β€œ ran 𝐺) = dom 𝐺
7774, 75, 763eqtr3i 2761 . . . . . . . . . . . . 13 βˆͺ 𝑧 ∈ ran 𝐺(◑𝐺 β€œ {𝑧}) = dom 𝐺
7840fdmd 6727 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ dom 𝐺 = ℝ)
7977, 78eqtrid 2777 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ βˆͺ 𝑧 ∈ ran 𝐺(◑𝐺 β€œ {𝑧}) = ℝ)
8072, 79sseqtrrd 4014 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ (◑𝐹 β€œ {𝑦}) βŠ† βˆͺ 𝑧 ∈ ran 𝐺(◑𝐺 β€œ {𝑧}))
81 dfss2 3958 . . . . . . . . . . 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 2778 . . . . . . . . 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 6895 . . . . . . . 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 15679 . . . . . . . 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 2775 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ (volβ€˜(◑𝐹 β€œ {𝑦})) = Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧))
8786oveq2d 7431 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ (𝑦 Β· (volβ€˜(◑𝐹 β€œ {𝑦}))) = (𝑦 Β· Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧)))
8853recnd 11270 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ 𝑦 ∈ β„‚)
8965recnd 11270 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦𝐼𝑧) ∈ β„‚)
9041, 88, 89fsummulc2 15760 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ (𝑦 Β· Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)))
9187, 90eqtrd 2765 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ (𝑦 Β· (volβ€˜(◑𝐹 β€œ {𝑦}))) = Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)))
9291sumeq2dv 15679 . . . 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 4125 . . . . 5 (πœ‘ β†’ (ran 𝐹 βˆ– {0}) βŠ† ran 𝐹)
9454recnd 11270 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑦 ∈ β„‚)
9594, 89mulcld 11262 . . . . . 6 (((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦 Β· (𝑦𝐼𝑧)) ∈ β„‚)
9641, 95fsumcl 15709 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– {0})) β†’ Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)) ∈ β„‚)
97 dfin4 4262 . . . . . . . 8 (ran 𝐹 ∩ {0}) = (ran 𝐹 βˆ– (ran 𝐹 βˆ– {0}))
98 inss2 4224 . . . . . . . 8 (ran 𝐹 ∩ {0}) βŠ† {0}
9997, 98eqsstrri 4008 . . . . . . 7 (ran 𝐹 βˆ– (ran 𝐹 βˆ– {0})) βŠ† {0}
10099sseli 3968 . . . . . 6 (𝑦 ∈ (ran 𝐹 βˆ– (ran 𝐹 βˆ– {0})) β†’ 𝑦 ∈ {0})
101 elsni 4641 . . . . . . . . . . 11 (𝑦 ∈ {0} β†’ 𝑦 = 0)
102101ad2antlr 725 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑦 = 0)
103102oveq1d 7430 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦 Β· (𝑦𝐼𝑧)) = (0 Β· (𝑦𝐼𝑧)))
10415ad2antrr 724 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
105 0re 11244 . . . . . . . . . . . . 13 0 ∈ ℝ
106102, 105eqeltrdi 2833 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑦 ∈ ℝ)
10720adantlr 713 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ ℝ)
108104, 106, 107fovcdmd 7589 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦𝐼𝑧) ∈ ℝ)
109108recnd 11270 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦𝐼𝑧) ∈ β„‚)
110109mul02d 11440 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) β†’ (0 Β· (𝑦𝐼𝑧)) = 0)
111103, 110eqtrd 2765 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦 Β· (𝑦𝐼𝑧)) = 0)
112111sumeq2dv 15679 . . . . . . 7 ((πœ‘ ∧ 𝑦 ∈ {0}) β†’ Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺0)
1136adantr 479 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ {0}) β†’ ran 𝐺 ∈ Fin)
114113olcd 872 . . . . . . . 8 ((πœ‘ ∧ 𝑦 ∈ {0}) β†’ (ran 𝐺 βŠ† (β„€β‰₯β€˜0) ∨ ran 𝐺 ∈ Fin))
115 sumz 15698 . . . . . . . 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 2765 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ {0}) β†’ Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)) = 0)
118100, 117sylan2 591 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (ran 𝐹 βˆ– (ran 𝐹 βˆ– {0}))) β†’ Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)) = 0)
11993, 96, 118, 3fsumss 15701 . . . 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 2769 . . 3 (πœ‘ β†’ (∫1β€˜πΉ) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 Β· (𝑦𝐼𝑧)))
121 itg1val 25628 . . . . 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 479 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ 𝐹:β„βŸΆβ„)
1243adantr 479 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ ran 𝐹 ∈ Fin)
125 inss1 4223 . . . . . . . . . 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 724 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ (◑𝐹 β€œ {𝑦}) ∈ dom vol)
12848ad2antrr 724 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ (◑𝐺 β€œ {𝑧}) ∈ dom vol)
129127, 128, 50syl2anc 582 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) ∈ dom vol)
13010adantr 479 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ ran 𝐹 βŠ† ℝ)
131130sselda 3972 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ ℝ)
13219ssdifssd 4135 . . . . . . . . . . . . 13 (πœ‘ β†’ (ran 𝐺 βˆ– {0}) βŠ† ℝ)
133132sselda 3972 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ 𝑧 ∈ ℝ)
134133adantr 479 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ ℝ)
135 eldifsni 4789 . . . . . . . . . . . . 13 (𝑧 ∈ (ran 𝐺 βˆ– {0}) β†’ 𝑧 β‰  0)
136135ad2antlr 725 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 β‰  0)
137 simpr 483 . . . . . . . . . . . . 13 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ 𝑧 = 0)
138137necon3ai 2955 . . . . . . . . . . . 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 836 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦𝐼𝑧) = (volβ€˜((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))))
14115ad2antrr 724 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
142141, 131, 134fovcdmd 7589 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦𝐼𝑧) ∈ ℝ)
143140, 142eqeltrrd 2826 . . . . . . . . 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 25637 . . . . . . . 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 4195 . . . . . . . . . . . . 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 5012 . . . . . . . . . . 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 5069 . . . . . . . . . . 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 2753 . . . . . . . . . 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 6080 . . . . . . . . . . . . 13 (◑𝐺 β€œ {𝑧}) βŠ† dom 𝐺
15118fdmd 6727 . . . . . . . . . . . . . 14 (πœ‘ β†’ dom 𝐺 = ℝ)
152151adantr 479 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ dom 𝐺 = ℝ)
153150, 152sseqtrid 4025 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (◑𝐺 β€œ {𝑧}) βŠ† ℝ)
154 iunid 5058 . . . . . . . . . . . . . . 15 βˆͺ 𝑦 ∈ ran 𝐹{𝑦} = ran 𝐹
155154imaeq2i 6056 . . . . . . . . . . . . . 14 (◑𝐹 β€œ βˆͺ 𝑦 ∈ ran 𝐹{𝑦}) = (◑𝐹 β€œ ran 𝐹)
156 imaiun 7250 . . . . . . . . . . . . . 14 (◑𝐹 β€œ βˆͺ 𝑦 ∈ ran 𝐹{𝑦}) = βˆͺ 𝑦 ∈ ran 𝐹(◑𝐹 β€œ {𝑦})
157 cnvimarndm 6081 . . . . . . . . . . . . . 14 (◑𝐹 β€œ ran 𝐹) = dom 𝐹
158155, 156, 1573eqtr3i 2761 . . . . . . . . . . . . 13 βˆͺ 𝑦 ∈ ran 𝐹(◑𝐹 β€œ {𝑦}) = dom 𝐹
1599fdmd 6727 . . . . . . . . . . . . . 14 (πœ‘ β†’ dom 𝐹 = ℝ)
160159adantr 479 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ dom 𝐹 = ℝ)
161158, 160eqtrid 2777 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ βˆͺ 𝑦 ∈ ran 𝐹(◑𝐹 β€œ {𝑦}) = ℝ)
162153, 161sseqtrrd 4014 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (◑𝐺 β€œ {𝑧}) βŠ† βˆͺ 𝑦 ∈ ran 𝐹(◑𝐹 β€œ {𝑦}))
163 dfss2 3958 . . . . . . . . . . 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 2778 . . . . . . . . 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 6895 . . . . . . . 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 15679 . . . . . . . 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 2775 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (volβ€˜(◑𝐺 β€œ {𝑧})) = Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧))
169168oveq2d 7431 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (𝑧 Β· (volβ€˜(◑𝐺 β€œ {𝑧}))) = (𝑧 Β· Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧)))
170133recnd 11270 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ 𝑧 ∈ β„‚)
171142recnd 11270 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦𝐼𝑧) ∈ β„‚)
172124, 170, 171fsummulc2 15760 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (𝑧 Β· Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)))
173169, 172eqtrd 2765 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (𝑧 Β· (volβ€˜(◑𝐺 β€œ {𝑧}))) = Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)))
174173sumeq2dv 15679 . . . 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 4125 . . . . . 6 (πœ‘ β†’ (ran 𝐺 βˆ– {0}) βŠ† ran 𝐺)
176170adantr 479 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ β„‚)
177176, 171mulcld 11262 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑧 Β· (𝑦𝐼𝑧)) ∈ β„‚)
178124, 177fsumcl 15709 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)) ∈ β„‚)
179 dfin4 4262 . . . . . . . . 9 (ran 𝐺 ∩ {0}) = (ran 𝐺 βˆ– (ran 𝐺 βˆ– {0}))
180 inss2 4224 . . . . . . . . 9 (ran 𝐺 ∩ {0}) βŠ† {0}
181179, 180eqsstrri 4008 . . . . . . . 8 (ran 𝐺 βˆ– (ran 𝐺 βˆ– {0})) βŠ† {0}
182181sseli 3968 . . . . . . 7 (𝑧 ∈ (ran 𝐺 βˆ– (ran 𝐺 βˆ– {0})) β†’ 𝑧 ∈ {0})
183 elsni 4641 . . . . . . . . . . . 12 (𝑧 ∈ {0} β†’ 𝑧 = 0)
184183ad2antlr 725 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 = 0)
185184oveq1d 7430 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑧 Β· (𝑦𝐼𝑧)) = (0 Β· (𝑦𝐼𝑧)))
18615ad2antrr 724 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
18711adantlr 713 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ ℝ)
188184, 105eqeltrdi 2833 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ ℝ)
189186, 187, 188fovcdmd 7589 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦𝐼𝑧) ∈ ℝ)
190189recnd 11270 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦𝐼𝑧) ∈ β„‚)
191190mul02d 11440 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) β†’ (0 Β· (𝑦𝐼𝑧)) = 0)
192185, 191eqtrd 2765 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑧 Β· (𝑦𝐼𝑧)) = 0)
193192sumeq2dv 15679 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ {0}) β†’ Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹0)
1943adantr 479 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ {0}) β†’ ran 𝐹 ∈ Fin)
195194olcd 872 . . . . . . . . 9 ((πœ‘ ∧ 𝑧 ∈ {0}) β†’ (ran 𝐹 βŠ† (β„€β‰₯β€˜0) ∨ ran 𝐹 ∈ Fin))
196 sumz 15698 . . . . . . . . 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 2765 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ {0}) β†’ Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)) = 0)
199182, 198sylan2 591 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ (ran 𝐺 βˆ– (ran 𝐺 βˆ– {0}))) β†’ Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)) = 0)
200175, 178, 199, 6fsumss 15701 . . . . 5 (πœ‘ β†’ Σ𝑧 ∈ (ran 𝐺 βˆ– {0})Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)))
20120adantr 479 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ ℝ)
202201recnd 11270 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ β„‚)
20315ad2antrr 724 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
20410adantr 479 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran 𝐹 βŠ† ℝ)
205204sselda 3972 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ ℝ)
206203, 205, 201fovcdmd 7589 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦𝐼𝑧) ∈ ℝ)
207206recnd 11270 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦𝐼𝑧) ∈ β„‚)
208202, 207mulcld 11262 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑧 Β· (𝑦𝐼𝑧)) ∈ β„‚)
209208anasss 465 . . . . . 6 ((πœ‘ ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐹)) β†’ (𝑧 Β· (𝑦𝐼𝑧)) ∈ β„‚)
2106, 3, 209fsumcom 15751 . . . . 5 (πœ‘ β†’ Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹(𝑧 Β· (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 Β· (𝑦𝐼𝑧)))
211200, 210eqtrd 2765 . . . 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 2769 . . 3 (πœ‘ β†’ (∫1β€˜πΊ) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 Β· (𝑦𝐼𝑧)))
213120, 212oveq12d 7433 . 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 2775 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 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098   β‰  wne 2930   βˆ– cdif 3937   ∩ cin 3939   βŠ† wss 3940  ifcif 4524  {csn 4624  βˆͺ ciun 4991   Γ— cxp 5670  β—‘ccnv 5671  dom cdm 5672  ran crn 5673   β†Ύ cres 5674   β€œ cima 5675  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7415   ∈ cmpo 7417   ∘f cof 7679  Fincfn 8960  β„‚cc 11134  β„cr 11135  0cc0 11136   + caddc 11139   Β· cmul 11141  β„€β‰₯cuz 12850  Ξ£csu 15662  volcvol 25408  βˆ«1citg1 25560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-inf2 9662  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213  ax-pre-sup 11214  ax-addf 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-disj 5109  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-of 7681  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-2o 8484  df-er 8721  df-map 8843  df-pm 8844  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-sup 9463  df-inf 9464  df-oi 9531  df-dju 9922  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-div 11900  df-nn 12241  df-2 12303  df-3 12304  df-n0 12501  df-z 12587  df-uz 12851  df-q 12961  df-rp 13005  df-xadd 13123  df-ioo 13358  df-ico 13360  df-icc 13361  df-fz 13515  df-fzo 13658  df-fl 13787  df-seq 13997  df-exp 14057  df-hash 14320  df-cj 15076  df-re 15077  df-im 15078  df-sqrt 15212  df-abs 15213  df-clim 15462  df-sum 15663  df-xmet 21274  df-met 21275  df-ovol 25409  df-vol 25410  df-mbf 25564  df-itg1 25565
This theorem is referenced by:  itg1add  25647
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