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Theorem i1fadd 25212
Description: The sum of two simple functions is a simple function. (Contributed by Mario Carneiro, 18-Jun-2014.)
Hypotheses
Ref Expression
i1fadd.1 (πœ‘ β†’ 𝐹 ∈ dom ∫1)
i1fadd.2 (πœ‘ β†’ 𝐺 ∈ dom ∫1)
Assertion
Ref Expression
i1fadd (πœ‘ β†’ (𝐹 ∘f + 𝐺) ∈ dom ∫1)
Proof of Theorem i1fadd
Dummy variables 𝑦 𝑧 𝑀 𝑣 π‘₯ 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 readdcl 11193 . . . 4 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ (π‘₯ + 𝑦) ∈ ℝ)
21adantl 483 . . 3 ((πœ‘ ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ)) β†’ (π‘₯ + 𝑦) ∈ ℝ)
3 i1fadd.1 . . . 4 (πœ‘ β†’ 𝐹 ∈ dom ∫1)
4 i1ff 25193 . . . 4 (𝐹 ∈ dom ∫1 β†’ 𝐹:β„βŸΆβ„)
53, 4syl 17 . . 3 (πœ‘ β†’ 𝐹:β„βŸΆβ„)
6 i1fadd.2 . . . 4 (πœ‘ β†’ 𝐺 ∈ dom ∫1)
7 i1ff 25193 . . . 4 (𝐺 ∈ dom ∫1 β†’ 𝐺:β„βŸΆβ„)
86, 7syl 17 . . 3 (πœ‘ β†’ 𝐺:β„βŸΆβ„)
9 reex 11201 . . . 4 ℝ ∈ V
109a1i 11 . . 3 (πœ‘ β†’ ℝ ∈ V)
11 inidm 4219 . . 3 (ℝ ∩ ℝ) = ℝ
122, 5, 8, 10, 10, 11off 7688 . 2 (πœ‘ β†’ (𝐹 ∘f + 𝐺):β„βŸΆβ„)
13 i1frn 25194 . . . . . 6 (𝐹 ∈ dom ∫1 β†’ ran 𝐹 ∈ Fin)
143, 13syl 17 . . . . 5 (πœ‘ β†’ ran 𝐹 ∈ Fin)
15 i1frn 25194 . . . . . 6 (𝐺 ∈ dom ∫1 β†’ ran 𝐺 ∈ Fin)
166, 15syl 17 . . . . 5 (πœ‘ β†’ ran 𝐺 ∈ Fin)
17 xpfi 9317 . . . . 5 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) β†’ (ran 𝐹 Γ— ran 𝐺) ∈ Fin)
1814, 16, 17syl2anc 585 . . . 4 (πœ‘ β†’ (ran 𝐹 Γ— ran 𝐺) ∈ Fin)
19 eqid 2733 . . . . . 6 (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)) = (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣))
20 ovex 7442 . . . . . 6 (𝑒 + 𝑣) ∈ V
2119, 20fnmpoi 8056 . . . . 5 (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)) Fn (ran 𝐹 Γ— ran 𝐺)
22 dffn4 6812 . . . . 5 ((𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)) Fn (ran 𝐹 Γ— ran 𝐺) ↔ (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)):(ran 𝐹 Γ— ran 𝐺)–ontoβ†’ran (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)))
2321, 22mpbi 229 . . . 4 (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)):(ran 𝐹 Γ— ran 𝐺)–ontoβ†’ran (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣))
24 fofi 9338 . . . 4 (((ran 𝐹 Γ— ran 𝐺) ∈ Fin ∧ (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)):(ran 𝐹 Γ— ran 𝐺)–ontoβ†’ran (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣))) β†’ ran (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)) ∈ Fin)
2518, 23, 24sylancl 587 . . 3 (πœ‘ β†’ ran (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)) ∈ Fin)
26 eqid 2733 . . . . . . . . 9 (π‘₯ + 𝑦) = (π‘₯ + 𝑦)
27 rspceov 7456 . . . . . . . . 9 ((π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺 ∧ (π‘₯ + 𝑦) = (π‘₯ + 𝑦)) β†’ βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺(π‘₯ + 𝑦) = (𝑒 + 𝑣))
2826, 27mp3an3 1451 . . . . . . . 8 ((π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) β†’ βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺(π‘₯ + 𝑦) = (𝑒 + 𝑣))
29 ovex 7442 . . . . . . . . 9 (π‘₯ + 𝑦) ∈ V
30 eqeq1 2737 . . . . . . . . . 10 (𝑀 = (π‘₯ + 𝑦) β†’ (𝑀 = (𝑒 + 𝑣) ↔ (π‘₯ + 𝑦) = (𝑒 + 𝑣)))
31302rexbidv 3220 . . . . . . . . 9 (𝑀 = (π‘₯ + 𝑦) β†’ (βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺 𝑀 = (𝑒 + 𝑣) ↔ βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺(π‘₯ + 𝑦) = (𝑒 + 𝑣)))
3229, 31elab 3669 . . . . . . . 8 ((π‘₯ + 𝑦) ∈ {𝑀 ∣ βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺 𝑀 = (𝑒 + 𝑣)} ↔ βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺(π‘₯ + 𝑦) = (𝑒 + 𝑣))
3328, 32sylibr 233 . . . . . . 7 ((π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) β†’ (π‘₯ + 𝑦) ∈ {𝑀 ∣ βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺 𝑀 = (𝑒 + 𝑣)})
3433adantl 483 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺)) β†’ (π‘₯ + 𝑦) ∈ {𝑀 ∣ βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺 𝑀 = (𝑒 + 𝑣)})
355ffnd 6719 . . . . . . 7 (πœ‘ β†’ 𝐹 Fn ℝ)
36 dffn3 6731 . . . . . . 7 (𝐹 Fn ℝ ↔ 𝐹:β„βŸΆran 𝐹)
3735, 36sylib 217 . . . . . 6 (πœ‘ β†’ 𝐹:β„βŸΆran 𝐹)
388ffnd 6719 . . . . . . 7 (πœ‘ β†’ 𝐺 Fn ℝ)
39 dffn3 6731 . . . . . . 7 (𝐺 Fn ℝ ↔ 𝐺:β„βŸΆran 𝐺)
4038, 39sylib 217 . . . . . 6 (πœ‘ β†’ 𝐺:β„βŸΆran 𝐺)
4134, 37, 40, 10, 10, 11off 7688 . . . . 5 (πœ‘ β†’ (𝐹 ∘f + 𝐺):β„βŸΆ{𝑀 ∣ βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺 𝑀 = (𝑒 + 𝑣)})
4241frnd 6726 . . . 4 (πœ‘ β†’ ran (𝐹 ∘f + 𝐺) βŠ† {𝑀 ∣ βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺 𝑀 = (𝑒 + 𝑣)})
4319rnmpo 7542 . . . 4 ran (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)) = {𝑀 ∣ βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺 𝑀 = (𝑒 + 𝑣)}
4442, 43sseqtrrdi 4034 . . 3 (πœ‘ β†’ ran (𝐹 ∘f + 𝐺) βŠ† ran (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)))
4525, 44ssfid 9267 . 2 (πœ‘ β†’ ran (𝐹 ∘f + 𝐺) ∈ Fin)
4612frnd 6726 . . . . . . 7 (πœ‘ β†’ ran (𝐹 ∘f + 𝐺) βŠ† ℝ)
4746ssdifssd 4143 . . . . . 6 (πœ‘ β†’ (ran (𝐹 ∘f + 𝐺) βˆ– {0}) βŠ† ℝ)
4847sselda 3983 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ 𝑦 ∈ ℝ)
4948recnd 11242 . . . 4 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ 𝑦 ∈ β„‚)
503, 6i1faddlem 25210 . . . 4 ((πœ‘ ∧ 𝑦 ∈ β„‚) β†’ (β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦}) = βˆͺ 𝑧 ∈ ran 𝐺((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})))
5149, 50syldan 592 . . 3 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ (β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦}) = βˆͺ 𝑧 ∈ ran 𝐺((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})))
5216adantr 482 . . . 4 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ ran 𝐺 ∈ Fin)
533ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝐹 ∈ dom ∫1)
54 i1fmbf 25192 . . . . . . . 8 (𝐹 ∈ dom ∫1 β†’ 𝐹 ∈ MblFn)
5553, 54syl 17 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝐹 ∈ MblFn)
565ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝐹:β„βŸΆβ„)
5712ad2antrr 725 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝐹 ∘f + 𝐺):β„βŸΆβ„)
5857frnd 6726 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ran (𝐹 ∘f + 𝐺) βŠ† ℝ)
59 eldifi 4127 . . . . . . . . . 10 (𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0}) β†’ 𝑦 ∈ ran (𝐹 ∘f + 𝐺))
6059ad2antlr 726 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑦 ∈ ran (𝐹 ∘f + 𝐺))
6158, 60sseldd 3984 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑦 ∈ ℝ)
628adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ 𝐺:β„βŸΆβ„)
6362frnd 6726 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ ran 𝐺 βŠ† ℝ)
6463sselda 3983 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ ℝ)
6561, 64resubcld 11642 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦 βˆ’ 𝑧) ∈ ℝ)
66 mbfimasn 25149 . . . . . . 7 ((𝐹 ∈ MblFn ∧ 𝐹:β„βŸΆβ„ ∧ (𝑦 βˆ’ 𝑧) ∈ ℝ) β†’ (◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∈ dom vol)
6755, 56, 65, 66syl3anc 1372 . . . . . 6 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∈ dom vol)
686ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝐺 ∈ dom ∫1)
69 i1fmbf 25192 . . . . . . . 8 (𝐺 ∈ dom ∫1 β†’ 𝐺 ∈ MblFn)
7068, 69syl 17 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝐺 ∈ MblFn)
718ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝐺:β„βŸΆβ„)
72 mbfimasn 25149 . . . . . . 7 ((𝐺 ∈ MblFn ∧ 𝐺:β„βŸΆβ„ ∧ 𝑧 ∈ ℝ) β†’ (◑𝐺 β€œ {𝑧}) ∈ dom vol)
7370, 71, 64, 72syl3anc 1372 . . . . . 6 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (◑𝐺 β€œ {𝑧}) ∈ dom vol)
74 inmbl 25059 . . . . . 6 (((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∈ dom vol ∧ (◑𝐺 β€œ {𝑧}) ∈ dom vol) β†’ ((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) ∈ dom vol)
7567, 73, 74syl2anc 585 . . . . 5 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) ∈ dom vol)
7675ralrimiva 3147 . . . 4 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ βˆ€π‘§ ∈ ran 𝐺((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) ∈ dom vol)
77 finiunmbl 25061 . . . 4 ((ran 𝐺 ∈ Fin ∧ βˆ€π‘§ ∈ ran 𝐺((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) ∈ dom vol) β†’ βˆͺ 𝑧 ∈ ran 𝐺((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) ∈ dom vol)
7852, 76, 77syl2anc 585 . . 3 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ βˆͺ 𝑧 ∈ ran 𝐺((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) ∈ dom vol)
7951, 78eqeltrd 2834 . 2 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ (β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦}) ∈ dom vol)
80 mblvol 25047 . . . 4 ((β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦}) ∈ dom vol β†’ (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦})) = (vol*β€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦})))
8179, 80syl 17 . . 3 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦})) = (vol*β€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦})))
82 mblss 25048 . . . . 5 ((β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦}) ∈ dom vol β†’ (β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦}) βŠ† ℝ)
8379, 82syl 17 . . . 4 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ (β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦}) βŠ† ℝ)
84 inss1 4229 . . . . . . . 8 ((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† (◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)})
8567adantrr 716 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ (◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∈ dom vol)
86 mblss 25048 . . . . . . . . 9 ((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∈ dom vol β†’ (◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) βŠ† ℝ)
8785, 86syl 17 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ (◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) βŠ† ℝ)
88 mblvol 25047 . . . . . . . . . 10 ((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∈ dom vol β†’ (volβ€˜(◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)})) = (vol*β€˜(◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)})))
8985, 88syl 17 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ (volβ€˜(◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)})) = (vol*β€˜(◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)})))
90 simprr 772 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ 𝑧 = 0)
9190oveq2d 7425 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ (𝑦 βˆ’ 𝑧) = (𝑦 βˆ’ 0))
9249adantr 482 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ 𝑦 ∈ β„‚)
9392subid1d 11560 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ (𝑦 βˆ’ 0) = 𝑦)
9491, 93eqtrd 2773 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ (𝑦 βˆ’ 𝑧) = 𝑦)
9594sneqd 4641 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ {(𝑦 βˆ’ 𝑧)} = {𝑦})
9695imaeq2d 6060 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ (◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) = (◑𝐹 β€œ {𝑦}))
9796fveq2d 6896 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ (volβ€˜(◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)})) = (volβ€˜(◑𝐹 β€œ {𝑦})))
98 i1fima2sn 25197 . . . . . . . . . . . 12 ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ (volβ€˜(◑𝐹 β€œ {𝑦})) ∈ ℝ)
993, 98sylan 581 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ (volβ€˜(◑𝐹 β€œ {𝑦})) ∈ ℝ)
10099adantr 482 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ (volβ€˜(◑𝐹 β€œ {𝑦})) ∈ ℝ)
10197, 100eqeltrd 2834 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ (volβ€˜(◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)})) ∈ ℝ)
10289, 101eqeltrrd 2835 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ (vol*β€˜(◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)})) ∈ ℝ)
103 ovolsscl 25003 . . . . . . . 8 ((((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† (◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∧ (◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) βŠ† ℝ ∧ (vol*β€˜(◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)})) ∈ ℝ) β†’ (vol*β€˜((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))) ∈ ℝ)
10484, 87, 102, 103mp3an2i 1467 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ (vol*β€˜((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))) ∈ ℝ)
105104expr 458 . . . . . 6 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑧 = 0 β†’ (vol*β€˜((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))) ∈ ℝ))
106 eldifsn 4791 . . . . . . . 8 (𝑧 ∈ (ran 𝐺 βˆ– {0}) ↔ (𝑧 ∈ ran 𝐺 ∧ 𝑧 β‰  0))
107 inss2 4230 . . . . . . . . 9 ((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† (◑𝐺 β€œ {𝑧})
108 eldifi 4127 . . . . . . . . . 10 (𝑧 ∈ (ran 𝐺 βˆ– {0}) β†’ 𝑧 ∈ ran 𝐺)
109 mblss 25048 . . . . . . . . . . 11 ((◑𝐺 β€œ {𝑧}) ∈ dom vol β†’ (◑𝐺 β€œ {𝑧}) βŠ† ℝ)
11073, 109syl 17 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (◑𝐺 β€œ {𝑧}) βŠ† ℝ)
111108, 110sylan2 594 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (◑𝐺 β€œ {𝑧}) βŠ† ℝ)
112 i1fima 25195 . . . . . . . . . . . . 13 (𝐺 ∈ dom ∫1 β†’ (◑𝐺 β€œ {𝑧}) ∈ dom vol)
1136, 112syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ (◑𝐺 β€œ {𝑧}) ∈ dom vol)
114113ad2antrr 725 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (◑𝐺 β€œ {𝑧}) ∈ dom vol)
115 mblvol 25047 . . . . . . . . . . 11 ((◑𝐺 β€œ {𝑧}) ∈ dom vol β†’ (volβ€˜(◑𝐺 β€œ {𝑧})) = (vol*β€˜(◑𝐺 β€œ {𝑧})))
116114, 115syl 17 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (volβ€˜(◑𝐺 β€œ {𝑧})) = (vol*β€˜(◑𝐺 β€œ {𝑧})))
1176adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ 𝐺 ∈ dom ∫1)
118 i1fima2sn 25197 . . . . . . . . . . 11 ((𝐺 ∈ dom ∫1 ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (volβ€˜(◑𝐺 β€œ {𝑧})) ∈ ℝ)
119117, 118sylan 581 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (volβ€˜(◑𝐺 β€œ {𝑧})) ∈ ℝ)
120116, 119eqeltrrd 2835 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (vol*β€˜(◑𝐺 β€œ {𝑧})) ∈ ℝ)
121 ovolsscl 25003 . . . . . . . . 9 ((((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† (◑𝐺 β€œ {𝑧}) ∧ (◑𝐺 β€œ {𝑧}) βŠ† ℝ ∧ (vol*β€˜(◑𝐺 β€œ {𝑧})) ∈ ℝ) β†’ (vol*β€˜((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))) ∈ ℝ)
122107, 111, 120, 121mp3an2i 1467 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (vol*β€˜((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))) ∈ ℝ)
123106, 122sylan2br 596 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 β‰  0)) β†’ (vol*β€˜((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))) ∈ ℝ)
124123expr 458 . . . . . 6 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑧 β‰  0 β†’ (vol*β€˜((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))) ∈ ℝ))
125105, 124pm2.61dne 3029 . . . . 5 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (vol*β€˜((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))) ∈ ℝ)
12652, 125fsumrecl 15680 . . . 4 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ Σ𝑧 ∈ ran 𝐺(vol*β€˜((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))) ∈ ℝ)
12751fveq2d 6896 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ (vol*β€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦})) = (vol*β€˜βˆͺ 𝑧 ∈ ran 𝐺((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))))
128107, 110sstrid 3994 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† ℝ)
129128, 125jca 513 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† ℝ ∧ (vol*β€˜((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))) ∈ ℝ))
130129ralrimiva 3147 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ βˆ€π‘§ ∈ ran 𝐺(((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† ℝ ∧ (vol*β€˜((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))) ∈ ℝ))
131 ovolfiniun 25018 . . . . . 6 ((ran 𝐺 ∈ Fin ∧ βˆ€π‘§ ∈ ran 𝐺(((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† ℝ ∧ (vol*β€˜((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))) ∈ ℝ)) β†’ (vol*β€˜βˆͺ 𝑧 ∈ ran 𝐺((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))) ≀ Σ𝑧 ∈ ran 𝐺(vol*β€˜((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))))
13252, 130, 131syl2anc 585 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ (vol*β€˜βˆͺ 𝑧 ∈ ran 𝐺((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))) ≀ Σ𝑧 ∈ ran 𝐺(vol*β€˜((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))))
133127, 132eqbrtrd 5171 . . . 4 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ (vol*β€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦})) ≀ Σ𝑧 ∈ ran 𝐺(vol*β€˜((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))))
134 ovollecl 25000 . . . 4 (((β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦}) βŠ† ℝ ∧ Σ𝑧 ∈ ran 𝐺(vol*β€˜((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))) ∈ ℝ ∧ (vol*β€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦})) ≀ Σ𝑧 ∈ ran 𝐺(vol*β€˜((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})))) β†’ (vol*β€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦})) ∈ ℝ)
13583, 126, 133, 134syl3anc 1372 . . 3 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ (vol*β€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦})) ∈ ℝ)
13681, 135eqeltrd 2834 . 2 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦})) ∈ ℝ)
13712, 45, 79, 136i1fd 25198 1 (πœ‘ β†’ (𝐹 ∘f + 𝐺) ∈ dom ∫1)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949  {csn 4629  βˆͺ ciun 4998   class class class wbr 5149   Γ— cxp 5675  β—‘ccnv 5676  dom cdm 5677  ran crn 5678   β€œ cima 5680   Fn wfn 6539  βŸΆwf 6540  β€“ontoβ†’wfo 6542  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411   ∘f cof 7668  Fincfn 8939  β„‚cc 11108  β„cr 11109  0cc0 11110   + caddc 11113   ≀ cle 11249   βˆ’ cmin 11444  Ξ£csu 15632  vol*covol 24979  volcvol 24980  MblFncmbf 25131  βˆ«1citg1 25132
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-inf 9438  df-oi 9505  df-dju 9896  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-xadd 13093  df-ioo 13328  df-ico 13330  df-icc 13331  df-fz 13485  df-fzo 13628  df-fl 13757  df-seq 13967  df-exp 14028  df-hash 14291  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-sum 15633  df-xmet 20937  df-met 20938  df-ovol 24981  df-vol 24982  df-mbf 25136  df-itg1 25137
This theorem is referenced by:  itg1addlem4  25216  itg1addlem4OLD  25217  i1fsub  25226  itg2splitlem  25266  itg2split  25267  itg2addlem  25276  itg2addnc  36542  ftc1anclem3  36563  ftc1anclem5  36565  ftc1anclem8  36568
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