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Theorem i1fadd 25444
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 11195 . . . 4 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ (π‘₯ + 𝑦) ∈ ℝ)
21adantl 480 . . 3 ((πœ‘ ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ)) β†’ (π‘₯ + 𝑦) ∈ ℝ)
3 i1fadd.1 . . . 4 (πœ‘ β†’ 𝐹 ∈ dom ∫1)
4 i1ff 25425 . . . 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 25425 . . . 4 (𝐺 ∈ dom ∫1 β†’ 𝐺:β„βŸΆβ„)
86, 7syl 17 . . 3 (πœ‘ β†’ 𝐺:β„βŸΆβ„)
9 reex 11203 . . . 4 ℝ ∈ V
109a1i 11 . . 3 (πœ‘ β†’ ℝ ∈ V)
11 inidm 4217 . . 3 (ℝ ∩ ℝ) = ℝ
122, 5, 8, 10, 10, 11off 7690 . 2 (πœ‘ β†’ (𝐹 ∘f + 𝐺):β„βŸΆβ„)
13 i1frn 25426 . . . . . 6 (𝐹 ∈ dom ∫1 β†’ ran 𝐹 ∈ Fin)
143, 13syl 17 . . . . 5 (πœ‘ β†’ ran 𝐹 ∈ Fin)
15 i1frn 25426 . . . . . 6 (𝐺 ∈ dom ∫1 β†’ ran 𝐺 ∈ Fin)
166, 15syl 17 . . . . 5 (πœ‘ β†’ ran 𝐺 ∈ Fin)
17 xpfi 9319 . . . . 5 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) β†’ (ran 𝐹 Γ— ran 𝐺) ∈ Fin)
1814, 16, 17syl2anc 582 . . . 4 (πœ‘ β†’ (ran 𝐹 Γ— ran 𝐺) ∈ Fin)
19 eqid 2730 . . . . . 6 (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)) = (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣))
20 ovex 7444 . . . . . 6 (𝑒 + 𝑣) ∈ V
2119, 20fnmpoi 8058 . . . . 5 (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)) Fn (ran 𝐹 Γ— ran 𝐺)
22 dffn4 6810 . . . . 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 9340 . . . 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 584 . . 3 (πœ‘ β†’ ran (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)) ∈ Fin)
26 eqid 2730 . . . . . . . . 9 (π‘₯ + 𝑦) = (π‘₯ + 𝑦)
27 rspceov 7458 . . . . . . . . 9 ((π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺 ∧ (π‘₯ + 𝑦) = (π‘₯ + 𝑦)) β†’ βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺(π‘₯ + 𝑦) = (𝑒 + 𝑣))
2826, 27mp3an3 1448 . . . . . . . 8 ((π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) β†’ βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺(π‘₯ + 𝑦) = (𝑒 + 𝑣))
29 ovex 7444 . . . . . . . . 9 (π‘₯ + 𝑦) ∈ V
30 eqeq1 2734 . . . . . . . . . 10 (𝑀 = (π‘₯ + 𝑦) β†’ (𝑀 = (𝑒 + 𝑣) ↔ (π‘₯ + 𝑦) = (𝑒 + 𝑣)))
31302rexbidv 3217 . . . . . . . . 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 3667 . . . . . . . 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 480 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺)) β†’ (π‘₯ + 𝑦) ∈ {𝑀 ∣ βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺 𝑀 = (𝑒 + 𝑣)})
355ffnd 6717 . . . . . . 7 (πœ‘ β†’ 𝐹 Fn ℝ)
36 dffn3 6729 . . . . . . 7 (𝐹 Fn ℝ ↔ 𝐹:β„βŸΆran 𝐹)
3735, 36sylib 217 . . . . . 6 (πœ‘ β†’ 𝐹:β„βŸΆran 𝐹)
388ffnd 6717 . . . . . . 7 (πœ‘ β†’ 𝐺 Fn ℝ)
39 dffn3 6729 . . . . . . 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 7690 . . . . 5 (πœ‘ β†’ (𝐹 ∘f + 𝐺):β„βŸΆ{𝑀 ∣ βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺 𝑀 = (𝑒 + 𝑣)})
4241frnd 6724 . . . 4 (πœ‘ β†’ ran (𝐹 ∘f + 𝐺) βŠ† {𝑀 ∣ βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺 𝑀 = (𝑒 + 𝑣)})
4319rnmpo 7544 . . . 4 ran (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)) = {𝑀 ∣ βˆƒπ‘’ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝐺 𝑀 = (𝑒 + 𝑣)}
4442, 43sseqtrrdi 4032 . . 3 (πœ‘ β†’ ran (𝐹 ∘f + 𝐺) βŠ† ran (𝑒 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑒 + 𝑣)))
4525, 44ssfid 9269 . 2 (πœ‘ β†’ ran (𝐹 ∘f + 𝐺) ∈ Fin)
4612frnd 6724 . . . . . . 7 (πœ‘ β†’ ran (𝐹 ∘f + 𝐺) βŠ† ℝ)
4746ssdifssd 4141 . . . . . 6 (πœ‘ β†’ (ran (𝐹 ∘f + 𝐺) βˆ– {0}) βŠ† ℝ)
4847sselda 3981 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ 𝑦 ∈ ℝ)
4948recnd 11246 . . . 4 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ 𝑦 ∈ β„‚)
503, 6i1faddlem 25442 . . . 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 589 . . 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 479 . . . 4 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ ran 𝐺 ∈ Fin)
533ad2antrr 722 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝐹 ∈ dom ∫1)
54 i1fmbf 25424 . . . . . . . 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 722 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝐹:β„βŸΆβ„)
5712ad2antrr 722 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝐹 ∘f + 𝐺):β„βŸΆβ„)
5857frnd 6724 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ran (𝐹 ∘f + 𝐺) βŠ† ℝ)
59 eldifi 4125 . . . . . . . . . 10 (𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0}) β†’ 𝑦 ∈ ran (𝐹 ∘f + 𝐺))
6059ad2antlr 723 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑦 ∈ ran (𝐹 ∘f + 𝐺))
6158, 60sseldd 3982 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑦 ∈ ℝ)
628adantr 479 . . . . . . . . . 10 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ 𝐺:β„βŸΆβ„)
6362frnd 6724 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ ran 𝐺 βŠ† ℝ)
6463sselda 3981 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ ℝ)
6561, 64resubcld 11646 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑦 βˆ’ 𝑧) ∈ ℝ)
66 mbfimasn 25381 . . . . . . 7 ((𝐹 ∈ MblFn ∧ 𝐹:β„βŸΆβ„ ∧ (𝑦 βˆ’ 𝑧) ∈ ℝ) β†’ (◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∈ dom vol)
6755, 56, 65, 66syl3anc 1369 . . . . . 6 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∈ dom vol)
686ad2antrr 722 . . . . . . . 8 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝐺 ∈ dom ∫1)
69 i1fmbf 25424 . . . . . . . 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 722 . . . . . . 7 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝐺:β„βŸΆβ„)
72 mbfimasn 25381 . . . . . . 7 ((𝐺 ∈ MblFn ∧ 𝐺:β„βŸΆβ„ ∧ 𝑧 ∈ ℝ) β†’ (◑𝐺 β€œ {𝑧}) ∈ dom vol)
7370, 71, 64, 72syl3anc 1369 . . . . . 6 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (◑𝐺 β€œ {𝑧}) ∈ dom vol)
74 inmbl 25291 . . . . . 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 582 . . . . 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 25293 . . . 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 582 . . 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 2831 . 2 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ (β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦}) ∈ dom vol)
80 mblvol 25279 . . . 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 25280 . . . . 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 4227 . . . . . . . 8 ((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† (◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)})
8567adantrr 713 . . . . . . . . 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 25280 . . . . . . . . 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 25279 . . . . . . . . . 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 769 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ 𝑧 = 0)
9190oveq2d 7427 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ (𝑦 βˆ’ 𝑧) = (𝑦 βˆ’ 0))
9249adantr 479 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ 𝑦 ∈ β„‚)
9392subid1d 11564 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ (𝑦 βˆ’ 0) = 𝑦)
9491, 93eqtrd 2770 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ (𝑦 βˆ’ 𝑧) = 𝑦)
9594sneqd 4639 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ {(𝑦 βˆ’ 𝑧)} = {𝑦})
9695imaeq2d 6058 . . . . . . . . . . 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 6894 . . . . . . . . . 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 25429 . . . . . . . . . . . 12 ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ (volβ€˜(◑𝐹 β€œ {𝑦})) ∈ ℝ)
993, 98sylan 578 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ (volβ€˜(◑𝐹 β€œ {𝑦})) ∈ ℝ)
10099adantr 479 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑧 = 0)) β†’ (volβ€˜(◑𝐹 β€œ {𝑦})) ∈ ℝ)
10197, 100eqeltrd 2831 . . . . . . . . 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 2832 . . . . . . . 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 25235 . . . . . . . 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 1464 . . . . . . 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 455 . . . . . 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 4789 . . . . . . . 8 (𝑧 ∈ (ran 𝐺 βˆ– {0}) ↔ (𝑧 ∈ ran 𝐺 ∧ 𝑧 β‰  0))
107 inss2 4228 . . . . . . . . 9 ((◑𝐹 β€œ {(𝑦 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† (◑𝐺 β€œ {𝑧})
108 eldifi 4125 . . . . . . . . . 10 (𝑧 ∈ (ran 𝐺 βˆ– {0}) β†’ 𝑧 ∈ ran 𝐺)
109 mblss 25280 . . . . . . . . . . 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 591 . . . . . . . . 9 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (◑𝐺 β€œ {𝑧}) βŠ† ℝ)
112 i1fima 25427 . . . . . . . . . . . . 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 722 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (◑𝐺 β€œ {𝑧}) ∈ dom vol)
115 mblvol 25279 . . . . . . . . . . 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 479 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ 𝐺 ∈ dom ∫1)
118 i1fima2sn 25429 . . . . . . . . . . 11 ((𝐺 ∈ dom ∫1 ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (volβ€˜(◑𝐺 β€œ {𝑧})) ∈ ℝ)
119117, 118sylan 578 . . . . . . . . . 10 (((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) ∧ 𝑧 ∈ (ran 𝐺 βˆ– {0})) β†’ (volβ€˜(◑𝐺 β€œ {𝑧})) ∈ ℝ)
120116, 119eqeltrrd 2832 . . . . . . . . 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 25235 . . . . . . . . 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 1464 . . . . . . . 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 593 . . . . . . 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 455 . . . . . 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 3026 . . . . 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 15684 . . . 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 6894 . . . . 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 3992 . . . . . . . 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 510 . . . . . . 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 25250 . . . . . 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 582 . . . . 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 5169 . . . 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 25232 . . . 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 1369 . . 3 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ (vol*β€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦})) ∈ ℝ)
13681, 135eqeltrd 2831 . 2 ((πœ‘ ∧ 𝑦 ∈ (ran (𝐹 ∘f + 𝐺) βˆ– {0})) β†’ (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑦})) ∈ ℝ)
13712, 45, 79, 136i1fd 25430 1 (πœ‘ β†’ (𝐹 ∘f + 𝐺) ∈ dom ∫1)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {cab 2707   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068  Vcvv 3472   βˆ– cdif 3944   ∩ cin 3946   βŠ† wss 3947  {csn 4627  βˆͺ ciun 4996   class class class wbr 5147   Γ— cxp 5673  β—‘ccnv 5674  dom cdm 5675  ran crn 5676   β€œ cima 5678   Fn wfn 6537  βŸΆwf 6538  β€“ontoβ†’wfo 6540  β€˜cfv 6542  (class class class)co 7411   ∈ cmpo 7413   ∘f cof 7670  Fincfn 8941  β„‚cc 11110  β„cr 11111  0cc0 11112   + caddc 11115   ≀ cle 11253   βˆ’ cmin 11448  Ξ£csu 15636  vol*covol 25211  volcvol 25212  MblFncmbf 25363  βˆ«1citg1 25364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  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 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-oi 9507  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-q 12937  df-rp 12979  df-xadd 13097  df-ioo 13332  df-ico 13334  df-icc 13335  df-fz 13489  df-fzo 13632  df-fl 13761  df-seq 13971  df-exp 14032  df-hash 14295  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436  df-sum 15637  df-xmet 21137  df-met 21138  df-ovol 25213  df-vol 25214  df-mbf 25368  df-itg1 25369
This theorem is referenced by:  itg1addlem4  25448  itg1addlem4OLD  25449  i1fsub  25458  itg2splitlem  25498  itg2split  25499  itg2addlem  25508  itg2addnc  36845  ftc1anclem3  36866  ftc1anclem5  36868  ftc1anclem8  36871
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