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Theorem i1fadd 25062
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 11135 . . . 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 25043 . . . 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 25043 . . . 4 (𝐺 ∈ dom ∫1 β†’ 𝐺:β„βŸΆβ„)
86, 7syl 17 . . 3 (πœ‘ β†’ 𝐺:β„βŸΆβ„)
9 reex 11143 . . . 4 ℝ ∈ V
109a1i 11 . . 3 (πœ‘ β†’ ℝ ∈ V)
11 inidm 4179 . . 3 (ℝ ∩ ℝ) = ℝ
122, 5, 8, 10, 10, 11off 7636 . 2 (πœ‘ β†’ (𝐹 ∘f + 𝐺):β„βŸΆβ„)
13 i1frn 25044 . . . . . 6 (𝐹 ∈ dom ∫1 β†’ ran 𝐹 ∈ Fin)
143, 13syl 17 . . . . 5 (πœ‘ β†’ ran 𝐹 ∈ Fin)
15 i1frn 25044 . . . . . 6 (𝐺 ∈ dom ∫1 β†’ ran 𝐺 ∈ Fin)
166, 15syl 17 . . . . 5 (πœ‘ β†’ ran 𝐺 ∈ Fin)
17 xpfi 9262 . . . . 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 2737 . . . . . 6 (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)) = (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣))
20 ovex 7391 . . . . . 6 (𝑒 + 𝑣) ∈ V
2119, 20fnmpoi 8003 . . . . 5 (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)) Fn (ran 𝐹 Γ— ran 𝐺)
22 dffn4 6763 . . . . 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 9283 . . . 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 2737 . . . . . . . . 9 (π‘₯ + 𝑦) = (π‘₯ + 𝑦)
27 rspceov 7405 . . . . . . . . 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 7391 . . . . . . . . 9 (π‘₯ + 𝑦) ∈ V
30 eqeq1 2741 . . . . . . . . . 10 (𝑀 = (π‘₯ + 𝑦) β†’ (𝑀 = (𝑒 + 𝑣) ↔ (π‘₯ + 𝑦) = (𝑒 + 𝑣)))
31302rexbidv 3214 . . . . . . . . 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 3631 . . . . . . . 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 6670 . . . . . . 7 (πœ‘ β†’ 𝐹 Fn ℝ)
36 dffn3 6682 . . . . . . 7 (𝐹 Fn ℝ ↔ 𝐹:β„βŸΆran 𝐹)
3735, 36sylib 217 . . . . . 6 (πœ‘ β†’ 𝐹:β„βŸΆran 𝐹)
388ffnd 6670 . . . . . . 7 (πœ‘ β†’ 𝐺 Fn ℝ)
39 dffn3 6682 . . . . . . 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 7636 . . . . 5 (πœ‘ β†’ (𝐹 ∘f + 𝐺):β„βŸΆ{𝑀 ∣ βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺 𝑀 = (𝑒 + 𝑣)})
4241frnd 6677 . . . 4 (πœ‘ β†’ ran (𝐹 ∘f + 𝐺) βŠ† {𝑀 ∣ βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺 𝑀 = (𝑒 + 𝑣)})
4319rnmpo 7490 . . . 4 ran (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)) = {𝑀 ∣ βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺 𝑀 = (𝑒 + 𝑣)}
4442, 43sseqtrrdi 3996 . . 3 (πœ‘ β†’ ran (𝐹 ∘f + 𝐺) βŠ† ran (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)))
4525, 44ssfid 9212 . 2 (πœ‘ β†’ ran (𝐹 ∘f + 𝐺) ∈ Fin)
4612frnd 6677 . . . . . . 7 (πœ‘ β†’ ran (𝐹 ∘f + 𝐺) βŠ† ℝ)
4746ssdifssd 4103 . . . . . 6 (πœ‘ β†’ (ran (𝐹 ∘f + 𝐺) βˆ– {0}) βŠ† ℝ)
4847sselda 3945 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ 𝑦 ∈ ℝ)
4948recnd 11184 . . . 4 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ 𝑦 ∈ β„‚)
503, 6i1faddlem 25060 . . . 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 25042 . . . . . . . 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 6677 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ran (𝐹 ∘f + 𝐺) βŠ† ℝ)
59 eldifi 4087 . . . . . . . . . 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 3946 . . . . . . . 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 6677 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ ran 𝐺 βŠ† ℝ)
6463sselda 3945 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ ℝ)
6561, 64resubcld 11584 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦 βˆ’ 𝑧) ∈ ℝ)
66 mbfimasn 24999 . . . . . . 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 25042 . . . . . . . 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 24999 . . . . . . 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 24909 . . . . . 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 3144 . . . 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 24911 . . . 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 2838 . 2 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ (β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦}) ∈ dom vol)
80 mblvol 24897 . . . 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 24898 . . . . 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 4189 . . . . . . . 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 24898 . . . . . . . . 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 24897 . . . . . . . . . 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 7374 . . . . . . . . . . . . . 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 11502 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ (𝑦 βˆ’ 0) = 𝑦)
9491, 93eqtrd 2777 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ (𝑦 βˆ’ 𝑧) = 𝑦)
9594sneqd 4599 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ {(𝑦 βˆ’ 𝑧)} = {𝑦})
9695imaeq2d 6014 . . . . . . . . . . 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 6847 . . . . . . . . . 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 25047 . . . . . . . . . . . 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 2838 . . . . . . . . 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 2839 . . . . . . . 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 24853 . . . . . . . 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 4748 . . . . . . . 8 (𝑧 ∈ (ran 𝐺 βˆ– {0}) ↔ (𝑧 ∈ ran 𝐺 ∧ 𝑧 β‰  0))
107 inss2 4190 . . . . . . . . 9 ((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† (◑𝐺 β€œ {𝑧})
108 eldifi 4087 . . . . . . . . . 10 (𝑧 ∈ (ran 𝐺 βˆ– {0}) β†’ 𝑧 ∈ ran 𝐺)
109 mblss 24898 . . . . . . . . . . 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 25045 . . . . . . . . . . . . 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 24897 . . . . . . . . . . 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 25047 . . . . . . . . . . 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 2839 . . . . . . . . 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 24853 . . . . . . . . 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 3032 . . . . 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 15620 . . . 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 6847 . . . . 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 3956 . . . . . . . 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 3144 . . . . . 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 24868 . . . . . 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 5128 . . . 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 24850 . . . 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 2838 . 2 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦})) ∈ ℝ)
13712, 45, 79, 136i1fd 25048 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 2714   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074  Vcvv 3446   βˆ– cdif 3908   ∩ cin 3910   βŠ† wss 3911  {csn 4587  βˆͺ ciun 4955   class class class wbr 5106   Γ— cxp 5632  β—‘ccnv 5633  dom cdm 5634  ran crn 5635   β€œ cima 5637   Fn wfn 6492  βŸΆwf 6493  β€“ontoβ†’wfo 6495  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360   ∘f cof 7616  Fincfn 8884  β„‚cc 11050  β„cr 11051  0cc0 11052   + caddc 11055   ≀ cle 11191   βˆ’ cmin 11386  Ξ£csu 15571  vol*covol 24829  volcvol 24830  MblFncmbf 24981  βˆ«1citg1 24982
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9578  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129  ax-pre-sup 11130
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 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-er 8649  df-map 8768  df-pm 8769  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-sup 9379  df-inf 9380  df-oi 9447  df-dju 9838  df-card 9876  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-div 11814  df-nn 12155  df-2 12217  df-3 12218  df-n0 12415  df-z 12501  df-uz 12765  df-q 12875  df-rp 12917  df-xadd 13035  df-ioo 13269  df-ico 13271  df-icc 13272  df-fz 13426  df-fzo 13569  df-fl 13698  df-seq 13908  df-exp 13969  df-hash 14232  df-cj 14985  df-re 14986  df-im 14987  df-sqrt 15121  df-abs 15122  df-clim 15371  df-sum 15572  df-xmet 20792  df-met 20793  df-ovol 24831  df-vol 24832  df-mbf 24986  df-itg1 24987
This theorem is referenced by:  itg1addlem4  25066  itg1addlem4OLD  25067  i1fsub  25076  itg2splitlem  25116  itg2split  25117  itg2addlem  25126  itg2addnc  36135  ftc1anclem3  36156  ftc1anclem5  36158  ftc1anclem8  36161
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