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Theorem itg1addlem4OLD 25080
Description: Obsolete version of itg1addlem4 25079. (Contributed by Mario Carneiro, 28-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
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
itg1addlem4OLD (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)))
Distinct variable groups:   𝑖,𝑗,𝑦,𝑧   𝑦,𝐼   𝑦,𝑃,𝑧   𝑖,𝐹,𝑗,𝑦,𝑧   𝑖,𝐺,𝑗,𝑦,𝑧   πœ‘,𝑖,𝑗,𝑦,𝑧
Allowed substitution hints:   𝑃(𝑖,𝑗)   𝐼(𝑧,𝑖,𝑗)

Proof of Theorem itg1addlem4OLD
Dummy variables 𝑀 𝑣 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 i1fadd.1 . . . . 5 (πœ‘ β†’ 𝐹 ∈ dom ∫1)
2 i1fadd.2 . . . . 5 (πœ‘ β†’ 𝐺 ∈ dom ∫1)
31, 2i1fadd 25075 . . . 4 (πœ‘ β†’ (𝐹 ∘f + 𝐺) ∈ dom ∫1)
4 i1frn 25057 . . . . . . . 8 (𝐹 ∈ dom ∫1 β†’ ran 𝐹 ∈ Fin)
51, 4syl 17 . . . . . . 7 (πœ‘ β†’ ran 𝐹 ∈ Fin)
6 i1frn 25057 . . . . . . . 8 (𝐺 ∈ dom ∫1 β†’ ran 𝐺 ∈ Fin)
72, 6syl 17 . . . . . . 7 (πœ‘ β†’ ran 𝐺 ∈ Fin)
8 xpfi 9268 . . . . . . 7 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) β†’ (ran 𝐹 Γ— ran 𝐺) ∈ Fin)
95, 7, 8syl2anc 585 . . . . . 6 (πœ‘ β†’ (ran 𝐹 Γ— ran 𝐺) ∈ Fin)
10 ax-addf 11137 . . . . . . . . . 10 + :(β„‚ Γ— β„‚)βŸΆβ„‚
11 ffn 6673 . . . . . . . . . 10 ( + :(β„‚ Γ— β„‚)βŸΆβ„‚ β†’ + Fn (β„‚ Γ— β„‚))
1210, 11ax-mp 5 . . . . . . . . 9 + Fn (β„‚ Γ— β„‚)
13 i1ff 25056 . . . . . . . . . . . . 13 (𝐹 ∈ dom ∫1 β†’ 𝐹:β„βŸΆβ„)
141, 13syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐹:β„βŸΆβ„)
1514frnd 6681 . . . . . . . . . . 11 (πœ‘ β†’ ran 𝐹 βŠ† ℝ)
16 ax-resscn 11115 . . . . . . . . . . 11 ℝ βŠ† β„‚
1715, 16sstrdi 3961 . . . . . . . . . 10 (πœ‘ β†’ ran 𝐹 βŠ† β„‚)
18 i1ff 25056 . . . . . . . . . . . . 13 (𝐺 ∈ dom ∫1 β†’ 𝐺:β„βŸΆβ„)
192, 18syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐺:β„βŸΆβ„)
2019frnd 6681 . . . . . . . . . . 11 (πœ‘ β†’ ran 𝐺 βŠ† ℝ)
2120, 16sstrdi 3961 . . . . . . . . . 10 (πœ‘ β†’ ran 𝐺 βŠ† β„‚)
22 xpss12 5653 . . . . . . . . . 10 ((ran 𝐹 βŠ† β„‚ ∧ ran 𝐺 βŠ† β„‚) β†’ (ran 𝐹 Γ— ran 𝐺) βŠ† (β„‚ Γ— β„‚))
2317, 21, 22syl2anc 585 . . . . . . . . 9 (πœ‘ β†’ (ran 𝐹 Γ— ran 𝐺) βŠ† (β„‚ Γ— β„‚))
24 fnssres 6629 . . . . . . . . 9 (( + Fn (β„‚ Γ— β„‚) ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† (β„‚ Γ— β„‚)) β†’ ( + β†Ύ (ran 𝐹 Γ— ran 𝐺)) Fn (ran 𝐹 Γ— ran 𝐺))
2512, 23, 24sylancr 588 . . . . . . . 8 (πœ‘ β†’ ( + β†Ύ (ran 𝐹 Γ— ran 𝐺)) Fn (ran 𝐹 Γ— ran 𝐺))
26 itg1add.4 . . . . . . . . 9 𝑃 = ( + β†Ύ (ran 𝐹 Γ— ran 𝐺))
2726fneq1i 6604 . . . . . . . 8 (𝑃 Fn (ran 𝐹 Γ— ran 𝐺) ↔ ( + β†Ύ (ran 𝐹 Γ— ran 𝐺)) Fn (ran 𝐹 Γ— ran 𝐺))
2825, 27sylibr 233 . . . . . . 7 (πœ‘ β†’ 𝑃 Fn (ran 𝐹 Γ— ran 𝐺))
29 dffn4 6767 . . . . . . 7 (𝑃 Fn (ran 𝐹 Γ— ran 𝐺) ↔ 𝑃:(ran 𝐹 Γ— ran 𝐺)–ontoβ†’ran 𝑃)
3028, 29sylib 217 . . . . . 6 (πœ‘ β†’ 𝑃:(ran 𝐹 Γ— ran 𝐺)–ontoβ†’ran 𝑃)
31 fofi 9289 . . . . . 6 (((ran 𝐹 Γ— ran 𝐺) ∈ Fin ∧ 𝑃:(ran 𝐹 Γ— ran 𝐺)–ontoβ†’ran 𝑃) β†’ ran 𝑃 ∈ Fin)
329, 30, 31syl2anc 585 . . . . 5 (πœ‘ β†’ ran 𝑃 ∈ Fin)
33 difss 4096 . . . . 5 (ran 𝑃 βˆ– {0}) βŠ† ran 𝑃
34 ssfi 9124 . . . . 5 ((ran 𝑃 ∈ Fin ∧ (ran 𝑃 βˆ– {0}) βŠ† ran 𝑃) β†’ (ran 𝑃 βˆ– {0}) ∈ Fin)
3532, 33, 34sylancl 587 . . . 4 (πœ‘ β†’ (ran 𝑃 βˆ– {0}) ∈ Fin)
36 ffun 6676 . . . . . . . . . . 11 ( + :(β„‚ Γ— β„‚)βŸΆβ„‚ β†’ Fun + )
3710, 36ax-mp 5 . . . . . . . . . 10 Fun +
3810fdmi 6685 . . . . . . . . . . 11 dom + = (β„‚ Γ— β„‚)
3923, 38sseqtrrdi 4000 . . . . . . . . . 10 (πœ‘ β†’ (ran 𝐹 Γ— ran 𝐺) βŠ† dom + )
40 funfvima2 7186 . . . . . . . . . 10 ((Fun + ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† dom + ) β†’ (⟨π‘₯, π‘¦βŸ© ∈ (ran 𝐹 Γ— ran 𝐺) β†’ ( + β€˜βŸ¨π‘₯, π‘¦βŸ©) ∈ ( + β€œ (ran 𝐹 Γ— ran 𝐺))))
4137, 39, 40sylancr 588 . . . . . . . . 9 (πœ‘ β†’ (⟨π‘₯, π‘¦βŸ© ∈ (ran 𝐹 Γ— ran 𝐺) β†’ ( + β€˜βŸ¨π‘₯, π‘¦βŸ©) ∈ ( + β€œ (ran 𝐹 Γ— ran 𝐺))))
42 opelxpi 5675 . . . . . . . . 9 ((π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) β†’ ⟨π‘₯, π‘¦βŸ© ∈ (ran 𝐹 Γ— ran 𝐺))
4341, 42impel 507 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺)) β†’ ( + β€˜βŸ¨π‘₯, π‘¦βŸ©) ∈ ( + β€œ (ran 𝐹 Γ— ran 𝐺)))
44 df-ov 7365 . . . . . . . 8 (π‘₯ + 𝑦) = ( + β€˜βŸ¨π‘₯, π‘¦βŸ©)
4526rneqi 5897 . . . . . . . . 9 ran 𝑃 = ran ( + β†Ύ (ran 𝐹 Γ— ran 𝐺))
46 df-ima 5651 . . . . . . . . 9 ( + β€œ (ran 𝐹 Γ— ran 𝐺)) = ran ( + β†Ύ (ran 𝐹 Γ— ran 𝐺))
4745, 46eqtr4i 2768 . . . . . . . 8 ran 𝑃 = ( + β€œ (ran 𝐹 Γ— ran 𝐺))
4843, 44, 473eltr4g 2855 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺)) β†’ (π‘₯ + 𝑦) ∈ ran 𝑃)
4914ffnd 6674 . . . . . . . 8 (πœ‘ β†’ 𝐹 Fn ℝ)
50 dffn3 6686 . . . . . . . 8 (𝐹 Fn ℝ ↔ 𝐹:β„βŸΆran 𝐹)
5149, 50sylib 217 . . . . . . 7 (πœ‘ β†’ 𝐹:β„βŸΆran 𝐹)
5219ffnd 6674 . . . . . . . 8 (πœ‘ β†’ 𝐺 Fn ℝ)
53 dffn3 6686 . . . . . . . 8 (𝐺 Fn ℝ ↔ 𝐺:β„βŸΆran 𝐺)
5452, 53sylib 217 . . . . . . 7 (πœ‘ β†’ 𝐺:β„βŸΆran 𝐺)
55 reex 11149 . . . . . . . 8 ℝ ∈ V
5655a1i 11 . . . . . . 7 (πœ‘ β†’ ℝ ∈ V)
57 inidm 4183 . . . . . . 7 (ℝ ∩ ℝ) = ℝ
5848, 51, 54, 56, 56, 57off 7640 . . . . . 6 (πœ‘ β†’ (𝐹 ∘f + 𝐺):β„βŸΆran 𝑃)
5958frnd 6681 . . . . 5 (πœ‘ β†’ ran (𝐹 ∘f + 𝐺) βŠ† ran 𝑃)
6059ssdifd 4105 . . . 4 (πœ‘ β†’ (ran (𝐹 ∘f + 𝐺) βˆ– {0}) βŠ† (ran 𝑃 βˆ– {0}))
6115sselda 3949 . . . . . . . . . 10 ((πœ‘ ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ ℝ)
6220sselda 3949 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ ℝ)
6361, 62anim12dan 620 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐺)) β†’ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ))
64 readdcl 11141 . . . . . . . . 9 ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) β†’ (𝑦 + 𝑧) ∈ ℝ)
6563, 64syl 17 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐺)) β†’ (𝑦 + 𝑧) ∈ ℝ)
6665ralrimivva 3198 . . . . . . 7 (πœ‘ β†’ βˆ€π‘¦ ∈ ran πΉβˆ€π‘§ ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ)
67 funimassov 7536 . . . . . . . 8 ((Fun + ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† dom + ) β†’ (( + β€œ (ran 𝐹 Γ— ran 𝐺)) βŠ† ℝ ↔ βˆ€π‘¦ ∈ ran πΉβˆ€π‘§ ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ))
6837, 39, 67sylancr 588 . . . . . . 7 (πœ‘ β†’ (( + β€œ (ran 𝐹 Γ— ran 𝐺)) βŠ† ℝ ↔ βˆ€π‘¦ ∈ ran πΉβˆ€π‘§ ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ))
6966, 68mpbird 257 . . . . . 6 (πœ‘ β†’ ( + β€œ (ran 𝐹 Γ— ran 𝐺)) βŠ† ℝ)
7047, 69eqsstrid 3997 . . . . 5 (πœ‘ β†’ ran 𝑃 βŠ† ℝ)
7170ssdifd 4105 . . . 4 (πœ‘ β†’ (ran 𝑃 βˆ– {0}) βŠ† (ℝ βˆ– {0}))
72 itg1val2 25064 . . . 4 (((𝐹 ∘f + 𝐺) ∈ dom ∫1 ∧ ((ran 𝑃 βˆ– {0}) ∈ Fin ∧ (ran (𝐹 ∘f + 𝐺) βˆ– {0}) βŠ† (ran 𝑃 βˆ– {0}) ∧ (ran 𝑃 βˆ– {0}) βŠ† (ℝ βˆ– {0}))) β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀}))))
733, 35, 60, 71, 72syl13anc 1373 . . 3 (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀}))))
7419adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ 𝐺:β„βŸΆβ„)
757adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ ran 𝐺 ∈ Fin)
76 inss2 4194 . . . . . . . . 9 ((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† (◑𝐺 β€œ {𝑧})
7776a1i 11 . . . . . . . 8 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† (◑𝐺 β€œ {𝑧}))
78 i1fima 25058 . . . . . . . . . . 11 (𝐹 ∈ dom ∫1 β†’ (◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∈ dom vol)
791, 78syl 17 . . . . . . . . . 10 (πœ‘ β†’ (◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∈ dom vol)
80 i1fima 25058 . . . . . . . . . . 11 (𝐺 ∈ dom ∫1 β†’ (◑𝐺 β€œ {𝑧}) ∈ dom vol)
812, 80syl 17 . . . . . . . . . 10 (πœ‘ β†’ (◑𝐺 β€œ {𝑧}) ∈ dom vol)
82 inmbl 24922 . . . . . . . . . 10 (((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∈ dom vol ∧ (◑𝐺 β€œ {𝑧}) ∈ dom vol) β†’ ((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) ∈ dom vol)
8379, 81, 82syl2anc 585 . . . . . . . . 9 (πœ‘ β†’ ((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) ∈ dom vol)
8483ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) ∈ dom vol)
8533, 70sstrid 3960 . . . . . . . . . . . . 13 (πœ‘ β†’ (ran 𝑃 βˆ– {0}) βŠ† ℝ)
8685sselda 3949 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ 𝑀 ∈ ℝ)
8786adantr 482 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑀 ∈ ℝ)
8862adantlr 714 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ ℝ)
8987, 88resubcld 11590 . . . . . . . . . 10 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑀 βˆ’ 𝑧) ∈ ℝ)
9087recnd 11190 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑀 ∈ β„‚)
9188recnd 11190 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ β„‚)
9290, 91npcand 11523 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) = 𝑀)
93 eldifsni 4755 . . . . . . . . . . . . 13 (𝑀 ∈ (ran 𝑃 βˆ– {0}) β†’ 𝑀 β‰  0)
9493ad2antlr 726 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑀 β‰  0)
9592, 94eqnetrd 3012 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) β‰  0)
96 oveq12 7371 . . . . . . . . . . . . 13 (((𝑀 βˆ’ 𝑧) = 0 ∧ 𝑧 = 0) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) = (0 + 0))
97 00id 11337 . . . . . . . . . . . . 13 (0 + 0) = 0
9896, 97eqtrdi 2793 . . . . . . . . . . . 12 (((𝑀 βˆ’ 𝑧) = 0 ∧ 𝑧 = 0) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) = 0)
9998necon3ai 2969 . . . . . . . . . . 11 (((𝑀 βˆ’ 𝑧) + 𝑧) β‰  0 β†’ Β¬ ((𝑀 βˆ’ 𝑧) = 0 ∧ 𝑧 = 0))
10095, 99syl 17 . . . . . . . . . 10 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ Β¬ ((𝑀 βˆ’ 𝑧) = 0 ∧ 𝑧 = 0))
101 itg1add.3 . . . . . . . . . . 11 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (volβ€˜((◑𝐹 β€œ {𝑖}) ∩ (◑𝐺 β€œ {𝑗})))))
1021, 2, 101itg1addlem3 25078 . . . . . . . . . 10 ((((𝑀 βˆ’ 𝑧) ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ Β¬ ((𝑀 βˆ’ 𝑧) = 0 ∧ 𝑧 = 0)) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) = (volβ€˜((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))))
10389, 88, 100, 102syl21anc 837 . . . . . . . . 9 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) = (volβ€˜((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))))
1041, 2, 101itg1addlem2 25077 . . . . . . . . . . 11 (πœ‘ β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
105104ad2antrr 725 . . . . . . . . . 10 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
106105, 89, 88fovcdmd 7531 . . . . . . . . 9 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) ∈ ℝ)
107103, 106eqeltrrd 2839 . . . . . . . 8 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (volβ€˜((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))) ∈ ℝ)
10874, 75, 77, 84, 107itg1addlem1 25072 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ (volβ€˜βˆͺ 𝑧 ∈ ran 𝐺((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))) = Σ𝑧 ∈ ran 𝐺(volβ€˜((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))))
10986recnd 11190 . . . . . . . . 9 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ 𝑀 ∈ β„‚)
1101, 2i1faddlem 25073 . . . . . . . . 9 ((πœ‘ ∧ 𝑀 ∈ β„‚) β†’ (β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀}) = βˆͺ 𝑧 ∈ ran 𝐺((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})))
111109, 110syldan 592 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ (β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀}) = βˆͺ 𝑧 ∈ ran 𝐺((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})))
112111fveq2d 6851 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀})) = (volβ€˜βˆͺ 𝑧 ∈ ran 𝐺((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))))
113103sumeq2dv 15595 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ Σ𝑧 ∈ ran 𝐺((𝑀 βˆ’ 𝑧)𝐼𝑧) = Σ𝑧 ∈ ran 𝐺(volβ€˜((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧}))))
114108, 112, 1133eqtr4d 2787 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀})) = Σ𝑧 ∈ ran 𝐺((𝑀 βˆ’ 𝑧)𝐼𝑧))
115114oveq2d 7378 . . . . 5 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ (𝑀 Β· (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀}))) = (𝑀 Β· Σ𝑧 ∈ ran 𝐺((𝑀 βˆ’ 𝑧)𝐼𝑧)))
116106recnd 11190 . . . . . 6 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) ∈ β„‚)
11775, 109, 116fsummulc2 15676 . . . . 5 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ (𝑀 Β· Σ𝑧 ∈ ran 𝐺((𝑀 βˆ’ 𝑧)𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
118115, 117eqtrd 2777 . . . 4 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ (𝑀 Β· (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀}))) = Σ𝑧 ∈ ran 𝐺(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
119118sumeq2dv 15595 . . 3 (πœ‘ β†’ Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀}))) = Σ𝑀 ∈ (ran 𝑃 βˆ– {0})Σ𝑧 ∈ ran 𝐺(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
12090, 116mulcld 11182 . . . . 5 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) ∈ β„‚)
121120anasss 468 . . . 4 ((πœ‘ ∧ (𝑀 ∈ (ran 𝑃 βˆ– {0}) ∧ 𝑧 ∈ ran 𝐺)) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) ∈ β„‚)
12235, 7, 121fsumcom 15667 . . 3 (πœ‘ β†’ Σ𝑀 ∈ (ran 𝑃 βˆ– {0})Σ𝑧 ∈ ran 𝐺(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
12373, 119, 1223eqtrd 2781 . 2 (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = Σ𝑧 ∈ ran 𝐺Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
124 oveq1 7369 . . . . . . 7 (𝑦 = (𝑀 βˆ’ 𝑧) β†’ (𝑦 + 𝑧) = ((𝑀 βˆ’ 𝑧) + 𝑧))
125 oveq1 7369 . . . . . . 7 (𝑦 = (𝑀 βˆ’ 𝑧) β†’ (𝑦𝐼𝑧) = ((𝑀 βˆ’ 𝑧)𝐼𝑧))
126124, 125oveq12d 7380 . . . . . 6 (𝑦 = (𝑀 βˆ’ 𝑧) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = (((𝑀 βˆ’ 𝑧) + 𝑧) Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
12732adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran 𝑃 ∈ Fin)
12870adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran 𝑃 βŠ† ℝ)
129128sselda 3949 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) β†’ 𝑣 ∈ ℝ)
13062adantr 482 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) β†’ 𝑧 ∈ ℝ)
131129, 130resubcld 11590 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) β†’ (𝑣 βˆ’ 𝑧) ∈ ℝ)
132131ex 414 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑣 ∈ ran 𝑃 β†’ (𝑣 βˆ’ 𝑧) ∈ ℝ))
133129recnd 11190 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) β†’ 𝑣 ∈ β„‚)
134133adantrr 716 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) β†’ 𝑣 ∈ β„‚)
13570sselda 3949 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑦 ∈ ran 𝑃) β†’ 𝑦 ∈ ℝ)
136135ad2ant2rl 748 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) β†’ 𝑦 ∈ ℝ)
137136recnd 11190 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) β†’ 𝑦 ∈ β„‚)
13862recnd 11190 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ β„‚)
139138adantr 482 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) β†’ 𝑧 ∈ β„‚)
140134, 137, 139subcan2ad 11564 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) β†’ ((𝑣 βˆ’ 𝑧) = (𝑦 βˆ’ 𝑧) ↔ 𝑣 = 𝑦))
141140ex 414 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃) β†’ ((𝑣 βˆ’ 𝑧) = (𝑦 βˆ’ 𝑧) ↔ 𝑣 = 𝑦)))
142132, 141dom2lem 8939 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–1-1→ℝ)
143 f1f1orn 6800 . . . . . . 7 ((𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–1-1→ℝ β†’ (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–1-1-ontoβ†’ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)))
144142, 143syl 17 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–1-1-ontoβ†’ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)))
145 oveq1 7369 . . . . . . . 8 (𝑣 = 𝑀 β†’ (𝑣 βˆ’ 𝑧) = (𝑀 βˆ’ 𝑧))
146 eqid 2737 . . . . . . . 8 (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) = (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))
147 ovex 7395 . . . . . . . 8 (𝑀 βˆ’ 𝑧) ∈ V
148145, 146, 147fvmpt 6953 . . . . . . 7 (𝑀 ∈ ran 𝑃 β†’ ((𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))β€˜π‘€) = (𝑀 βˆ’ 𝑧))
149148adantl 483 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ ((𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))β€˜π‘€) = (𝑀 βˆ’ 𝑧))
150 f1f 6743 . . . . . . . . . . 11 ((𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–1-1→ℝ β†’ (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran π‘ƒβŸΆβ„)
151 frn 6680 . . . . . . . . . . 11 ((𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran π‘ƒβŸΆβ„ β†’ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) βŠ† ℝ)
152142, 150, 1513syl 18 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) βŠ† ℝ)
153152sselda 3949 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ 𝑦 ∈ ℝ)
15462adantr 482 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ 𝑧 ∈ ℝ)
155153, 154readdcld 11191 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ (𝑦 + 𝑧) ∈ ℝ)
156104ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
157156, 153, 154fovcdmd 7531 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ (𝑦𝐼𝑧) ∈ ℝ)
158155, 157remulcld 11192 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) ∈ ℝ)
159158recnd 11190 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) ∈ β„‚)
160126, 127, 144, 149, 159fsumf1o 15615 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ Σ𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = Σ𝑀 ∈ ran 𝑃(((𝑀 βˆ’ 𝑧) + 𝑧) Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
161128sselda 3949 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ 𝑀 ∈ ℝ)
162161recnd 11190 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ 𝑀 ∈ β„‚)
163138adantr 482 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ 𝑧 ∈ β„‚)
164162, 163npcand 11523 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) = 𝑀)
165164oveq1d 7377 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ (((𝑀 βˆ’ 𝑧) + 𝑧) Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
166165sumeq2dv 15595 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ Σ𝑀 ∈ ran 𝑃(((𝑀 βˆ’ 𝑧) + 𝑧) Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = Σ𝑀 ∈ ran 𝑃(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
167160, 166eqtrd 2777 . . . 4 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ Σ𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = Σ𝑀 ∈ ran 𝑃(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
16839ad2antrr 725 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ (ran 𝐹 Γ— ran 𝐺) βŠ† dom + )
169 simpr 486 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ ran 𝐹)
170 simplr 768 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ ran 𝐺)
171169, 170opelxpd 5676 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ βŸ¨π‘¦, π‘§βŸ© ∈ (ran 𝐹 Γ— ran 𝐺))
172 funfvima2 7186 . . . . . . . . . . . 12 ((Fun + ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† dom + ) β†’ (βŸ¨π‘¦, π‘§βŸ© ∈ (ran 𝐹 Γ— ran 𝐺) β†’ ( + β€˜βŸ¨π‘¦, π‘§βŸ©) ∈ ( + β€œ (ran 𝐹 Γ— ran 𝐺))))
17337, 172mpan 689 . . . . . . . . . . 11 ((ran 𝐹 Γ— ran 𝐺) βŠ† dom + β†’ (βŸ¨π‘¦, π‘§βŸ© ∈ (ran 𝐹 Γ— ran 𝐺) β†’ ( + β€˜βŸ¨π‘¦, π‘§βŸ©) ∈ ( + β€œ (ran 𝐹 Γ— ran 𝐺))))
174168, 171, 173sylc 65 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ ( + β€˜βŸ¨π‘¦, π‘§βŸ©) ∈ ( + β€œ (ran 𝐹 Γ— ran 𝐺)))
175 df-ov 7365 . . . . . . . . . 10 (𝑦 + 𝑧) = ( + β€˜βŸ¨π‘¦, π‘§βŸ©)
176174, 175, 473eltr4g 2855 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦 + 𝑧) ∈ ran 𝑃)
17761adantlr 714 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ ℝ)
178177recnd 11190 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ β„‚)
179138adantr 482 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ β„‚)
180178, 179pncand 11520 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ ((𝑦 + 𝑧) βˆ’ 𝑧) = 𝑦)
181180eqcomd 2743 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 = ((𝑦 + 𝑧) βˆ’ 𝑧))
182 oveq1 7369 . . . . . . . . . 10 (𝑣 = (𝑦 + 𝑧) β†’ (𝑣 βˆ’ 𝑧) = ((𝑦 + 𝑧) βˆ’ 𝑧))
183182rspceeqv 3600 . . . . . . . . 9 (((𝑦 + 𝑧) ∈ ran 𝑃 ∧ 𝑦 = ((𝑦 + 𝑧) βˆ’ 𝑧)) β†’ βˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧))
184176, 181, 183syl2anc 585 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ βˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧))
185184ralrimiva 3144 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ βˆ€π‘¦ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧))
186 ssabral 4024 . . . . . . 7 (ran 𝐹 βŠ† {𝑦 ∣ βˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧)} ↔ βˆ€π‘¦ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧))
187185, 186sylibr 233 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran 𝐹 βŠ† {𝑦 ∣ βˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧)})
188146rnmpt 5915 . . . . . 6 ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) = {𝑦 ∣ βˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧)}
189187, 188sseqtrrdi 4000 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran 𝐹 βŠ† ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)))
19062adantr 482 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ ℝ)
191177, 190readdcld 11191 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦 + 𝑧) ∈ ℝ)
192104ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
193192, 177, 190fovcdmd 7531 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦𝐼𝑧) ∈ ℝ)
194191, 193remulcld 11192 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) ∈ ℝ)
195194recnd 11190 . . . . 5 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) ∈ β„‚)
196152ssdifd 4105 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) βˆ– ran 𝐹) βŠ† (ℝ βˆ– ran 𝐹))
197196sselda 3949 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) βˆ– ran 𝐹)) β†’ 𝑦 ∈ (ℝ βˆ– ran 𝐹))
198 eldifi 4091 . . . . . . . . . . . . 13 (𝑦 ∈ (ℝ βˆ– ran 𝐹) β†’ 𝑦 ∈ ℝ)
199198ad2antrl 727 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ 𝑦 ∈ ℝ)
20062adantr 482 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ 𝑧 ∈ ℝ)
201 simprr 772 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))
2021, 2, 101itg1addlem3 25078 . . . . . . . . . . . 12 (((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0)) β†’ (𝑦𝐼𝑧) = (volβ€˜((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))))
203199, 200, 201, 202syl21anc 837 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (𝑦𝐼𝑧) = (volβ€˜((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))))
204 inss1 4193 . . . . . . . . . . . . . . 15 ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† (◑𝐹 β€œ {𝑦})
205 eldifn 4092 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (ℝ βˆ– ran 𝐹) β†’ Β¬ 𝑦 ∈ ran 𝐹)
206205ad2antrl 727 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ Β¬ 𝑦 ∈ ran 𝐹)
207 vex 3452 . . . . . . . . . . . . . . . . . . . . 21 𝑣 ∈ V
208207eliniseg 6051 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ V β†’ (𝑣 ∈ (◑𝐹 β€œ {𝑦}) ↔ 𝑣𝐹𝑦))
209208elv 3454 . . . . . . . . . . . . . . . . . . 19 (𝑣 ∈ (◑𝐹 β€œ {𝑦}) ↔ 𝑣𝐹𝑦)
210 vex 3452 . . . . . . . . . . . . . . . . . . . 20 𝑦 ∈ V
211207, 210brelrn 5902 . . . . . . . . . . . . . . . . . . 19 (𝑣𝐹𝑦 β†’ 𝑦 ∈ ran 𝐹)
212209, 211sylbi 216 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ (◑𝐹 β€œ {𝑦}) β†’ 𝑦 ∈ ran 𝐹)
213206, 212nsyl 140 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ Β¬ 𝑣 ∈ (◑𝐹 β€œ {𝑦}))
214213pm2.21d 121 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (𝑣 ∈ (◑𝐹 β€œ {𝑦}) β†’ 𝑣 ∈ βˆ…))
215214ssrdv 3955 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (◑𝐹 β€œ {𝑦}) βŠ† βˆ…)
216204, 215sstrid 3960 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† βˆ…)
217 ss0 4363 . . . . . . . . . . . . . 14 (((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† βˆ… β†’ ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) = βˆ…)
218216, 217syl 17 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) = βˆ…)
219218fveq2d 6851 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (volβ€˜((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))) = (volβ€˜βˆ…))
220 0mbl 24919 . . . . . . . . . . . . . 14 βˆ… ∈ dom vol
221 mblvol 24910 . . . . . . . . . . . . . 14 (βˆ… ∈ dom vol β†’ (volβ€˜βˆ…) = (vol*β€˜βˆ…))
222220, 221ax-mp 5 . . . . . . . . . . . . 13 (volβ€˜βˆ…) = (vol*β€˜βˆ…)
223 ovol0 24873 . . . . . . . . . . . . 13 (vol*β€˜βˆ…) = 0
224222, 223eqtri 2765 . . . . . . . . . . . 12 (volβ€˜βˆ…) = 0
225219, 224eqtrdi 2793 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (volβ€˜((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧}))) = 0)
226203, 225eqtrd 2777 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (𝑦𝐼𝑧) = 0)
227226oveq2d 7378 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = ((𝑦 + 𝑧) Β· 0))
228199, 200readdcld 11191 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (𝑦 + 𝑧) ∈ ℝ)
229228recnd 11190 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (𝑦 + 𝑧) ∈ β„‚)
230229mul01d 11361 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ ((𝑦 + 𝑧) Β· 0) = 0)
231227, 230eqtrd 2777 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = 0)
232231expr 458 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ℝ βˆ– ran 𝐹)) β†’ (Β¬ (𝑦 = 0 ∧ 𝑧 = 0) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = 0))
233 oveq12 7371 . . . . . . . . . 10 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ (𝑦 + 𝑧) = (0 + 0))
234233, 97eqtrdi 2793 . . . . . . . . 9 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ (𝑦 + 𝑧) = 0)
235 oveq12 7371 . . . . . . . . . 10 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ (𝑦𝐼𝑧) = (0𝐼0))
236 0re 11164 . . . . . . . . . . 11 0 ∈ ℝ
237 iftrue 4497 . . . . . . . . . . . 12 ((𝑖 = 0 ∧ 𝑗 = 0) β†’ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (volβ€˜((◑𝐹 β€œ {𝑖}) ∩ (◑𝐺 β€œ {𝑗})))) = 0)
238 c0ex 11156 . . . . . . . . . . . 12 0 ∈ V
239237, 101, 238ovmpoa 7515 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ 0 ∈ ℝ) β†’ (0𝐼0) = 0)
240236, 236, 239mp2an 691 . . . . . . . . . 10 (0𝐼0) = 0
241235, 240eqtrdi 2793 . . . . . . . . 9 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ (𝑦𝐼𝑧) = 0)
242234, 241oveq12d 7380 . . . . . . . 8 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = (0 Β· 0))
243 0cn 11154 . . . . . . . . 9 0 ∈ β„‚
244243mul01i 11352 . . . . . . . 8 (0 Β· 0) = 0
245242, 244eqtrdi 2793 . . . . . . 7 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = 0)
246232, 245pm2.61d2 181 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ℝ βˆ– ran 𝐹)) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = 0)
247197, 246syldan 592 . . . . 5 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) βˆ– ran 𝐹)) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = 0)
248 f1ofo 6796 . . . . . . 7 ((𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–1-1-ontoβ†’ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) β†’ (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–ontoβ†’ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)))
249144, 248syl 17 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–ontoβ†’ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)))
250 fofi 9289 . . . . . 6 ((ran 𝑃 ∈ Fin ∧ (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–ontoβ†’ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) ∈ Fin)
251127, 249, 250syl2anc 585 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) ∈ Fin)
252189, 195, 247, 251fsumss 15617 . . . 4 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)))
25333a1i 11 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ (ran 𝑃 βˆ– {0}) βŠ† ran 𝑃)
254120an32s 651 . . . . 5 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) ∈ β„‚)
255 dfin4 4232 . . . . . . . 8 (ran 𝑃 ∩ {0}) = (ran 𝑃 βˆ– (ran 𝑃 βˆ– {0}))
256 inss2 4194 . . . . . . . 8 (ran 𝑃 ∩ {0}) βŠ† {0}
257255, 256eqsstrri 3984 . . . . . . 7 (ran 𝑃 βˆ– (ran 𝑃 βˆ– {0})) βŠ† {0}
258257sseli 3945 . . . . . 6 (𝑀 ∈ (ran 𝑃 βˆ– (ran 𝑃 βˆ– {0})) β†’ 𝑀 ∈ {0})
259 elsni 4608 . . . . . . . . 9 (𝑀 ∈ {0} β†’ 𝑀 = 0)
260259adantl 483 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ 𝑀 = 0)
261260oveq1d 7377 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = (0 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
262104ad2antrr 725 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
263260, 236eqeltrdi 2846 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ 𝑀 ∈ ℝ)
26462adantr 482 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ 𝑧 ∈ ℝ)
265263, 264resubcld 11590 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ (𝑀 βˆ’ 𝑧) ∈ ℝ)
266262, 265, 264fovcdmd 7531 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) ∈ ℝ)
267266recnd 11190 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) ∈ β„‚)
268267mul02d 11360 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ (0 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = 0)
269261, 268eqtrd 2777 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = 0)
270258, 269sylan2 594 . . . . 5 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ (ran 𝑃 βˆ– (ran 𝑃 βˆ– {0}))) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = 0)
271253, 254, 270, 127fsumss 15617 . . . 4 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = Σ𝑀 ∈ ran 𝑃(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
272167, 252, 2713eqtr4d 2787 . . 3 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
273272sumeq2dv 15595 . 2 (πœ‘ β†’ Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
274195anasss 468 . . 3 ((πœ‘ ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐹)) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) ∈ β„‚)
2757, 5, 274fsumcom 15667 . 2 (πœ‘ β†’ Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)))
276123, 273, 2753eqtr2d 2783 1 (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2714   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074  Vcvv 3448   βˆ– cdif 3912   ∩ cin 3914   βŠ† wss 3915  βˆ…c0 4287  ifcif 4491  {csn 4591  βŸ¨cop 4597  βˆͺ ciun 4959   class class class wbr 5110   ↦ cmpt 5193   Γ— cxp 5636  β—‘ccnv 5637  dom cdm 5638  ran crn 5639   β†Ύ cres 5640   β€œ cima 5641  Fun wfun 6495   Fn wfn 6496  βŸΆwf 6497  β€“1-1β†’wf1 6498  β€“ontoβ†’wfo 6499  β€“1-1-ontoβ†’wf1o 6500  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   βˆ’ cmin 11392  Ξ£csu 15577  vol*covol 24842  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: (None)
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