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Theorem itg1addlem4OLD 25616
Description: Obsolete version of itg1addlem4 25615 as of 6-Oct-2024. (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 25611 . . . 4 (πœ‘ β†’ (𝐹 ∘f + 𝐺) ∈ dom ∫1)
4 i1frn 25593 . . . . . . . 8 (𝐹 ∈ dom ∫1 β†’ ran 𝐹 ∈ Fin)
51, 4syl 17 . . . . . . 7 (πœ‘ β†’ ran 𝐹 ∈ Fin)
6 i1frn 25593 . . . . . . . 8 (𝐺 ∈ dom ∫1 β†’ ran 𝐺 ∈ Fin)
72, 6syl 17 . . . . . . 7 (πœ‘ β†’ ran 𝐺 ∈ Fin)
8 xpfi 9333 . . . . . . 7 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) β†’ (ran 𝐹 Γ— ran 𝐺) ∈ Fin)
95, 7, 8syl2anc 583 . . . . . 6 (πœ‘ β†’ (ran 𝐹 Γ— ran 𝐺) ∈ Fin)
10 ax-addf 11209 . . . . . . . . . 10 + :(β„‚ Γ— β„‚)βŸΆβ„‚
11 ffn 6716 . . . . . . . . . 10 ( + :(β„‚ Γ— β„‚)βŸΆβ„‚ β†’ + Fn (β„‚ Γ— β„‚))
1210, 11ax-mp 5 . . . . . . . . 9 + Fn (β„‚ Γ— β„‚)
13 i1ff 25592 . . . . . . . . . . . . 13 (𝐹 ∈ dom ∫1 β†’ 𝐹:β„βŸΆβ„)
141, 13syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐹:β„βŸΆβ„)
1514frnd 6724 . . . . . . . . . . 11 (πœ‘ β†’ ran 𝐹 βŠ† ℝ)
16 ax-resscn 11187 . . . . . . . . . . 11 ℝ βŠ† β„‚
1715, 16sstrdi 3990 . . . . . . . . . 10 (πœ‘ β†’ ran 𝐹 βŠ† β„‚)
18 i1ff 25592 . . . . . . . . . . . . 13 (𝐺 ∈ dom ∫1 β†’ 𝐺:β„βŸΆβ„)
192, 18syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐺:β„βŸΆβ„)
2019frnd 6724 . . . . . . . . . . 11 (πœ‘ β†’ ran 𝐺 βŠ† ℝ)
2120, 16sstrdi 3990 . . . . . . . . . 10 (πœ‘ β†’ ran 𝐺 βŠ† β„‚)
22 xpss12 5687 . . . . . . . . . 10 ((ran 𝐹 βŠ† β„‚ ∧ ran 𝐺 βŠ† β„‚) β†’ (ran 𝐹 Γ— ran 𝐺) βŠ† (β„‚ Γ— β„‚))
2317, 21, 22syl2anc 583 . . . . . . . . 9 (πœ‘ β†’ (ran 𝐹 Γ— ran 𝐺) βŠ† (β„‚ Γ— β„‚))
24 fnssres 6672 . . . . . . . . 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 586 . . . . . . . 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 6645 . . . . . . . 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 6811 . . . . . . 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 9354 . . . . . 6 (((ran 𝐹 Γ— ran 𝐺) ∈ Fin ∧ 𝑃:(ran 𝐹 Γ— ran 𝐺)–ontoβ†’ran 𝑃) β†’ ran 𝑃 ∈ Fin)
329, 30, 31syl2anc 583 . . . . 5 (πœ‘ β†’ ran 𝑃 ∈ Fin)
33 difss 4127 . . . . 5 (ran 𝑃 βˆ– {0}) βŠ† ran 𝑃
34 ssfi 9189 . . . . 5 ((ran 𝑃 ∈ Fin ∧ (ran 𝑃 βˆ– {0}) βŠ† ran 𝑃) β†’ (ran 𝑃 βˆ– {0}) ∈ Fin)
3532, 33, 34sylancl 585 . . . 4 (πœ‘ β†’ (ran 𝑃 βˆ– {0}) ∈ Fin)
36 ffun 6719 . . . . . . . . . . 11 ( + :(β„‚ Γ— β„‚)βŸΆβ„‚ β†’ Fun + )
3710, 36ax-mp 5 . . . . . . . . . 10 Fun +
3810fdmi 6728 . . . . . . . . . . 11 dom + = (β„‚ Γ— β„‚)
3923, 38sseqtrrdi 4029 . . . . . . . . . 10 (πœ‘ β†’ (ran 𝐹 Γ— ran 𝐺) βŠ† dom + )
40 funfvima2 7237 . . . . . . . . . 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 586 . . . . . . . . 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 5709 . . . . . . . . 9 ((π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) β†’ ⟨π‘₯, π‘¦βŸ© ∈ (ran 𝐹 Γ— ran 𝐺))
4341, 42impel 505 . . . . . . . 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 7417 . . . . . . . 8 (π‘₯ + 𝑦) = ( + β€˜βŸ¨π‘₯, π‘¦βŸ©)
4526rneqi 5933 . . . . . . . . 9 ran 𝑃 = ran ( + β†Ύ (ran 𝐹 Γ— ran 𝐺))
46 df-ima 5685 . . . . . . . . 9 ( + β€œ (ran 𝐹 Γ— ran 𝐺)) = ran ( + β†Ύ (ran 𝐹 Γ— ran 𝐺))
4745, 46eqtr4i 2758 . . . . . . . 8 ran 𝑃 = ( + β€œ (ran 𝐹 Γ— ran 𝐺))
4843, 44, 473eltr4g 2845 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺)) β†’ (π‘₯ + 𝑦) ∈ ran 𝑃)
4914ffnd 6717 . . . . . . . 8 (πœ‘ β†’ 𝐹 Fn ℝ)
50 dffn3 6729 . . . . . . . 8 (𝐹 Fn ℝ ↔ 𝐹:β„βŸΆran 𝐹)
5149, 50sylib 217 . . . . . . 7 (πœ‘ β†’ 𝐹:β„βŸΆran 𝐹)
5219ffnd 6717 . . . . . . . 8 (πœ‘ β†’ 𝐺 Fn ℝ)
53 dffn3 6729 . . . . . . . 8 (𝐺 Fn ℝ ↔ 𝐺:β„βŸΆran 𝐺)
5452, 53sylib 217 . . . . . . 7 (πœ‘ β†’ 𝐺:β„βŸΆran 𝐺)
55 reex 11221 . . . . . . . 8 ℝ ∈ V
5655a1i 11 . . . . . . 7 (πœ‘ β†’ ℝ ∈ V)
57 inidm 4214 . . . . . . 7 (ℝ ∩ ℝ) = ℝ
5848, 51, 54, 56, 56, 57off 7697 . . . . . 6 (πœ‘ β†’ (𝐹 ∘f + 𝐺):β„βŸΆran 𝑃)
5958frnd 6724 . . . . 5 (πœ‘ β†’ ran (𝐹 ∘f + 𝐺) βŠ† ran 𝑃)
6059ssdifd 4136 . . . 4 (πœ‘ β†’ (ran (𝐹 ∘f + 𝐺) βˆ– {0}) βŠ† (ran 𝑃 βˆ– {0}))
6115sselda 3978 . . . . . . . . . 10 ((πœ‘ ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ ℝ)
6220sselda 3978 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ ℝ)
6361, 62anim12dan 618 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐺)) β†’ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ))
64 readdcl 11213 . . . . . . . . 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 3195 . . . . . . 7 (πœ‘ β†’ βˆ€π‘¦ ∈ ran πΉβˆ€π‘§ ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ)
67 funimassov 7592 . . . . . . . 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 586 . . . . . . 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 4026 . . . . 5 (πœ‘ β†’ ran 𝑃 βŠ† ℝ)
7170ssdifd 4136 . . . 4 (πœ‘ β†’ (ran 𝑃 βˆ– {0}) βŠ† (ℝ βˆ– {0}))
72 itg1val2 25600 . . . 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 1370 . . 3 (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀}))))
7419adantr 480 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ 𝐺:β„βŸΆβ„)
757adantr 480 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ ran 𝐺 ∈ Fin)
76 inss2 4225 . . . . . . . . 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 25594 . . . . . . . . . . 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 25594 . . . . . . . . . . 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 25458 . . . . . . . . . 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 583 . . . . . . . . 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 3989 . . . . . . . . . . . . 13 (πœ‘ β†’ (ran 𝑃 βˆ– {0}) βŠ† ℝ)
8685sselda 3978 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ 𝑀 ∈ ℝ)
8786adantr 480 . . . . . . . . . . 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 11664 . . . . . . . . . 10 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑀 βˆ’ 𝑧) ∈ ℝ)
9087recnd 11264 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑀 ∈ β„‚)
9188recnd 11264 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ β„‚)
9290, 91npcand 11597 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) = 𝑀)
93 eldifsni 4789 . . . . . . . . . . . . 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 3003 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) β‰  0)
96 oveq12 7423 . . . . . . . . . . . . 13 (((𝑀 βˆ’ 𝑧) = 0 ∧ 𝑧 = 0) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) = (0 + 0))
97 00id 11411 . . . . . . . . . . . . 13 (0 + 0) = 0
9896, 97eqtrdi 2783 . . . . . . . . . . . 12 (((𝑀 βˆ’ 𝑧) = 0 ∧ 𝑧 = 0) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) = 0)
9998necon3ai 2960 . . . . . . . . . . 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 25614 . . . . . . . . . 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 25613 . . . . . . . . . . 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 7587 . . . . . . . . 9 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) ∈ ℝ)
107103, 106eqeltrrd 2829 . . . . . . . 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 25608 . . . . . . 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 11264 . . . . . . . . 9 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ 𝑀 ∈ β„‚)
1101, 2i1faddlem 25609 . . . . . . . . 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 590 . . . . . . . 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 6895 . . . . . . 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 15673 . . . . . . 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 2777 . . . . . 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 7430 . . . . 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 11264 . . . . . 6 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) ∈ β„‚)
11775, 109, 116fsummulc2 15754 . . . . 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 2767 . . . 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 15673 . . 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 11256 . . . . 5 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) ∈ β„‚)
121120anasss 466 . . . 4 ((πœ‘ ∧ (𝑀 ∈ (ran 𝑃 βˆ– {0}) ∧ 𝑧 ∈ ran 𝐺)) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) ∈ β„‚)
12235, 7, 121fsumcom 15745 . . 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 2771 . 2 (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = Σ𝑧 ∈ ran 𝐺Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
124 oveq1 7421 . . . . . . 7 (𝑦 = (𝑀 βˆ’ 𝑧) β†’ (𝑦 + 𝑧) = ((𝑀 βˆ’ 𝑧) + 𝑧))
125 oveq1 7421 . . . . . . 7 (𝑦 = (𝑀 βˆ’ 𝑧) β†’ (𝑦𝐼𝑧) = ((𝑀 βˆ’ 𝑧)𝐼𝑧))
126124, 125oveq12d 7432 . . . . . 6 (𝑦 = (𝑀 βˆ’ 𝑧) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = (((𝑀 βˆ’ 𝑧) + 𝑧) Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
12732adantr 480 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran 𝑃 ∈ Fin)
12870adantr 480 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran 𝑃 βŠ† ℝ)
129128sselda 3978 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) β†’ 𝑣 ∈ ℝ)
13062adantr 480 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) β†’ 𝑧 ∈ ℝ)
131129, 130resubcld 11664 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) β†’ (𝑣 βˆ’ 𝑧) ∈ ℝ)
132131ex 412 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑣 ∈ ran 𝑃 β†’ (𝑣 βˆ’ 𝑧) ∈ ℝ))
133129recnd 11264 . . . . . . . . . . 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 3978 . . . . . . . . . . . 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 11264 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) β†’ 𝑦 ∈ β„‚)
13862recnd 11264 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ β„‚)
139138adantr 480 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) β†’ 𝑧 ∈ β„‚)
140134, 137, 139subcan2ad 11638 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) β†’ ((𝑣 βˆ’ 𝑧) = (𝑦 βˆ’ 𝑧) ↔ 𝑣 = 𝑦))
141140ex 412 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃) β†’ ((𝑣 βˆ’ 𝑧) = (𝑦 βˆ’ 𝑧) ↔ 𝑣 = 𝑦)))
142132, 141dom2lem 9004 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–1-1→ℝ)
143 f1f1orn 6844 . . . . . . 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 7421 . . . . . . . 8 (𝑣 = 𝑀 β†’ (𝑣 βˆ’ 𝑧) = (𝑀 βˆ’ 𝑧))
146 eqid 2727 . . . . . . . 8 (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) = (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))
147 ovex 7447 . . . . . . . 8 (𝑀 βˆ’ 𝑧) ∈ V
148145, 146, 147fvmpt 6999 . . . . . . 7 (𝑀 ∈ ran 𝑃 β†’ ((𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))β€˜π‘€) = (𝑀 βˆ’ 𝑧))
149148adantl 481 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ ((𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))β€˜π‘€) = (𝑀 βˆ’ 𝑧))
150 f1f 6787 . . . . . . . . . . 11 ((𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–1-1→ℝ β†’ (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran π‘ƒβŸΆβ„)
151 frn 6723 . . . . . . . . . . 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 3978 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ 𝑦 ∈ ℝ)
15462adantr 480 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ 𝑧 ∈ ℝ)
155153, 154readdcld 11265 . . . . . . . 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 7587 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ (𝑦𝐼𝑧) ∈ ℝ)
158155, 157remulcld 11266 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) ∈ ℝ)
159158recnd 11264 . . . . . 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 15693 . . . . 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 3978 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ 𝑀 ∈ ℝ)
162161recnd 11264 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ 𝑀 ∈ β„‚)
163138adantr 480 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ 𝑧 ∈ β„‚)
164162, 163npcand 11597 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) = 𝑀)
165164oveq1d 7429 . . . . . 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 15673 . . . . 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 2767 . . . 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 484 . . . . . . . . . . . 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 5711 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ βŸ¨π‘¦, π‘§βŸ© ∈ (ran 𝐹 Γ— ran 𝐺))
172 funfvima2 7237 . . . . . . . . . . . 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 7417 . . . . . . . . . 10 (𝑦 + 𝑧) = ( + β€˜βŸ¨π‘¦, π‘§βŸ©)
176174, 175, 473eltr4g 2845 . . . . . . . . 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 11264 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ β„‚)
179138adantr 480 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ β„‚)
180178, 179pncand 11594 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ ((𝑦 + 𝑧) βˆ’ 𝑧) = 𝑦)
181180eqcomd 2733 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 = ((𝑦 + 𝑧) βˆ’ 𝑧))
182 oveq1 7421 . . . . . . . . . 10 (𝑣 = (𝑦 + 𝑧) β†’ (𝑣 βˆ’ 𝑧) = ((𝑦 + 𝑧) βˆ’ 𝑧))
183182rspceeqv 3629 . . . . . . . . 9 (((𝑦 + 𝑧) ∈ ran 𝑃 ∧ 𝑦 = ((𝑦 + 𝑧) βˆ’ 𝑧)) β†’ βˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧))
184176, 181, 183syl2anc 583 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ βˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧))
185184ralrimiva 3141 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ βˆ€π‘¦ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧))
186 ssabral 4055 . . . . . . 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 5951 . . . . . 6 ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) = {𝑦 ∣ βˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧)}
189187, 188sseqtrrdi 4029 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran 𝐹 βŠ† ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)))
19062adantr 480 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ ℝ)
191177, 190readdcld 11265 . . . . . . 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 7587 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦𝐼𝑧) ∈ ℝ)
194191, 193remulcld 11266 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) ∈ ℝ)
195194recnd 11264 . . . . 5 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) ∈ β„‚)
196152ssdifd 4136 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) βˆ– ran 𝐹) βŠ† (ℝ βˆ– ran 𝐹))
197196sselda 3978 . . . . . 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 4122 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 25614 . . . . . . . . . . . 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 4224 . . . . . . . . . . . . . . 15 ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† (◑𝐹 β€œ {𝑦})
205 eldifn 4123 . . . . . . . . . . . . . . . . . . 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 3473 . . . . . . . . . . . . . . . . . . . . 21 𝑣 ∈ V
208207eliniseg 6092 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ V β†’ (𝑣 ∈ (◑𝐹 β€œ {𝑦}) ↔ 𝑣𝐹𝑦))
209208elv 3475 . . . . . . . . . . . . . . . . . . 19 (𝑣 ∈ (◑𝐹 β€œ {𝑦}) ↔ 𝑣𝐹𝑦)
210 vex 3473 . . . . . . . . . . . . . . . . . . . 20 𝑦 ∈ V
211207, 210brelrn 5938 . . . . . . . . . . . . . . . . . . 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 3984 . . . . . . . . . . . . . . 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 3989 . . . . . . . . . . . . . 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 4394 . . . . . . . . . . . . . 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 6895 . . . . . . . . . . . 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 25455 . . . . . . . . . . . . . 14 βˆ… ∈ dom vol
221 mblvol 25446 . . . . . . . . . . . . . 14 (βˆ… ∈ dom vol β†’ (volβ€˜βˆ…) = (vol*β€˜βˆ…))
222220, 221ax-mp 5 . . . . . . . . . . . . 13 (volβ€˜βˆ…) = (vol*β€˜βˆ…)
223 ovol0 25409 . . . . . . . . . . . . 13 (vol*β€˜βˆ…) = 0
224222, 223eqtri 2755 . . . . . . . . . . . 12 (volβ€˜βˆ…) = 0
225219, 224eqtrdi 2783 . . . . . . . . . . 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 2767 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (𝑦𝐼𝑧) = 0)
227226oveq2d 7430 . . . . . . . . 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 11265 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (𝑦 + 𝑧) ∈ ℝ)
229228recnd 11264 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (𝑦 + 𝑧) ∈ β„‚)
230229mul01d 11435 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ ((𝑦 + 𝑧) Β· 0) = 0)
231227, 230eqtrd 2767 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = 0)
232231expr 456 . . . . . . 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 7423 . . . . . . . . . 10 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ (𝑦 + 𝑧) = (0 + 0))
234233, 97eqtrdi 2783 . . . . . . . . 9 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ (𝑦 + 𝑧) = 0)
235 oveq12 7423 . . . . . . . . . 10 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ (𝑦𝐼𝑧) = (0𝐼0))
236 0re 11238 . . . . . . . . . . 11 0 ∈ ℝ
237 iftrue 4530 . . . . . . . . . . . 12 ((𝑖 = 0 ∧ 𝑗 = 0) β†’ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (volβ€˜((◑𝐹 β€œ {𝑖}) ∩ (◑𝐺 β€œ {𝑗})))) = 0)
238 c0ex 11230 . . . . . . . . . . . 12 0 ∈ V
239237, 101, 238ovmpoa 7570 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ 0 ∈ ℝ) β†’ (0𝐼0) = 0)
240236, 236, 239mp2an 691 . . . . . . . . . 10 (0𝐼0) = 0
241235, 240eqtrdi 2783 . . . . . . . . 9 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ (𝑦𝐼𝑧) = 0)
242234, 241oveq12d 7432 . . . . . . . 8 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = (0 Β· 0))
243 0cn 11228 . . . . . . . . 9 0 ∈ β„‚
244243mul01i 11426 . . . . . . . 8 (0 Β· 0) = 0
245242, 244eqtrdi 2783 . . . . . . 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 590 . . . . 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 6840 . . . . . . 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 9354 . . . . . 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 583 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) ∈ Fin)
252189, 195, 247, 251fsumss 15695 . . . 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 4263 . . . . . . . 8 (ran 𝑃 ∩ {0}) = (ran 𝑃 βˆ– (ran 𝑃 βˆ– {0}))
256 inss2 4225 . . . . . . . 8 (ran 𝑃 ∩ {0}) βŠ† {0}
257255, 256eqsstrri 4013 . . . . . . 7 (ran 𝑃 βˆ– (ran 𝑃 βˆ– {0})) βŠ† {0}
258257sseli 3974 . . . . . 6 (𝑀 ∈ (ran 𝑃 βˆ– (ran 𝑃 βˆ– {0})) β†’ 𝑀 ∈ {0})
259 elsni 4641 . . . . . . . . 9 (𝑀 ∈ {0} β†’ 𝑀 = 0)
260259adantl 481 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ 𝑀 = 0)
261260oveq1d 7429 . . . . . . 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 2836 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ 𝑀 ∈ ℝ)
26462adantr 480 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ 𝑧 ∈ ℝ)
265263, 264resubcld 11664 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ (𝑀 βˆ’ 𝑧) ∈ ℝ)
266262, 265, 264fovcdmd 7587 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) ∈ ℝ)
267266recnd 11264 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) ∈ β„‚)
268267mul02d 11434 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ (0 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = 0)
269261, 268eqtrd 2767 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = 0)
270258, 269sylan2 592 . . . . 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 15695 . . . 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 2777 . . 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 15673 . 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 466 . . 3 ((πœ‘ ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐹)) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) ∈ β„‚)
2757, 5, 274fsumcom 15745 . 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 2773 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 395   = wceq 1534   ∈ wcel 2099  {cab 2704   β‰  wne 2935  βˆ€wral 3056  βˆƒwrex 3065  Vcvv 3469   βˆ– cdif 3941   ∩ cin 3943   βŠ† wss 3944  βˆ…c0 4318  ifcif 4524  {csn 4624  βŸ¨cop 4630  βˆͺ ciun 4991   class class class wbr 5142   ↦ cmpt 5225   Γ— cxp 5670  β—‘ccnv 5671  dom cdm 5672  ran crn 5673   β†Ύ cres 5674   β€œ cima 5675  Fun wfun 6536   Fn wfn 6537  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€“ontoβ†’wfo 6540  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416   ∘f cof 7677  Fincfn 8955  β„‚cc 11128  β„cr 11129  0cc0 11130   + caddc 11133   Β· cmul 11135   βˆ’ cmin 11466  Ξ£csu 15656  vol*covol 25378  volcvol 25379  βˆ«1citg1 25531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208  ax-addf 11209
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  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 5108  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  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 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 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8718  df-map 8838  df-pm 8839  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-inf 9458  df-oi 9525  df-dju 9916  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-z 12581  df-uz 12845  df-q 12955  df-rp 12999  df-xadd 13117  df-ioo 13352  df-ico 13354  df-icc 13355  df-fz 13509  df-fzo 13652  df-fl 13781  df-seq 13991  df-exp 14051  df-hash 14314  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-clim 15456  df-sum 15657  df-xmet 21259  df-met 21260  df-ovol 25380  df-vol 25381  df-mbf 25535  df-itg1 25536
This theorem is referenced by: (None)
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