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Theorem itg1addlem4OLD 25208
Description: Obsolete version of itg1addlem4 25207. (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 25203 . . . 4 (πœ‘ β†’ (𝐹 ∘f + 𝐺) ∈ dom ∫1)
4 i1frn 25185 . . . . . . . 8 (𝐹 ∈ dom ∫1 β†’ ran 𝐹 ∈ Fin)
51, 4syl 17 . . . . . . 7 (πœ‘ β†’ ran 𝐹 ∈ Fin)
6 i1frn 25185 . . . . . . . 8 (𝐺 ∈ dom ∫1 β†’ ran 𝐺 ∈ Fin)
72, 6syl 17 . . . . . . 7 (πœ‘ β†’ ran 𝐺 ∈ Fin)
8 xpfi 9313 . . . . . . 7 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) β†’ (ran 𝐹 Γ— ran 𝐺) ∈ Fin)
95, 7, 8syl2anc 584 . . . . . 6 (πœ‘ β†’ (ran 𝐹 Γ— ran 𝐺) ∈ Fin)
10 ax-addf 11185 . . . . . . . . . 10 + :(β„‚ Γ— β„‚)βŸΆβ„‚
11 ffn 6714 . . . . . . . . . 10 ( + :(β„‚ Γ— β„‚)βŸΆβ„‚ β†’ + Fn (β„‚ Γ— β„‚))
1210, 11ax-mp 5 . . . . . . . . 9 + Fn (β„‚ Γ— β„‚)
13 i1ff 25184 . . . . . . . . . . . . 13 (𝐹 ∈ dom ∫1 β†’ 𝐹:β„βŸΆβ„)
141, 13syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐹:β„βŸΆβ„)
1514frnd 6722 . . . . . . . . . . 11 (πœ‘ β†’ ran 𝐹 βŠ† ℝ)
16 ax-resscn 11163 . . . . . . . . . . 11 ℝ βŠ† β„‚
1715, 16sstrdi 3993 . . . . . . . . . 10 (πœ‘ β†’ ran 𝐹 βŠ† β„‚)
18 i1ff 25184 . . . . . . . . . . . . 13 (𝐺 ∈ dom ∫1 β†’ 𝐺:β„βŸΆβ„)
192, 18syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐺:β„βŸΆβ„)
2019frnd 6722 . . . . . . . . . . 11 (πœ‘ β†’ ran 𝐺 βŠ† ℝ)
2120, 16sstrdi 3993 . . . . . . . . . 10 (πœ‘ β†’ ran 𝐺 βŠ† β„‚)
22 xpss12 5690 . . . . . . . . . 10 ((ran 𝐹 βŠ† β„‚ ∧ ran 𝐺 βŠ† β„‚) β†’ (ran 𝐹 Γ— ran 𝐺) βŠ† (β„‚ Γ— β„‚))
2317, 21, 22syl2anc 584 . . . . . . . . 9 (πœ‘ β†’ (ran 𝐹 Γ— ran 𝐺) βŠ† (β„‚ Γ— β„‚))
24 fnssres 6670 . . . . . . . . 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 587 . . . . . . . 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 6643 . . . . . . . 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 6808 . . . . . . 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 9334 . . . . . 6 (((ran 𝐹 Γ— ran 𝐺) ∈ Fin ∧ 𝑃:(ran 𝐹 Γ— ran 𝐺)–ontoβ†’ran 𝑃) β†’ ran 𝑃 ∈ Fin)
329, 30, 31syl2anc 584 . . . . 5 (πœ‘ β†’ ran 𝑃 ∈ Fin)
33 difss 4130 . . . . 5 (ran 𝑃 βˆ– {0}) βŠ† ran 𝑃
34 ssfi 9169 . . . . 5 ((ran 𝑃 ∈ Fin ∧ (ran 𝑃 βˆ– {0}) βŠ† ran 𝑃) β†’ (ran 𝑃 βˆ– {0}) ∈ Fin)
3532, 33, 34sylancl 586 . . . 4 (πœ‘ β†’ (ran 𝑃 βˆ– {0}) ∈ Fin)
36 ffun 6717 . . . . . . . . . . 11 ( + :(β„‚ Γ— β„‚)βŸΆβ„‚ β†’ Fun + )
3710, 36ax-mp 5 . . . . . . . . . 10 Fun +
3810fdmi 6726 . . . . . . . . . . 11 dom + = (β„‚ Γ— β„‚)
3923, 38sseqtrrdi 4032 . . . . . . . . . 10 (πœ‘ β†’ (ran 𝐹 Γ— ran 𝐺) βŠ† dom + )
40 funfvima2 7229 . . . . . . . . . 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 587 . . . . . . . . 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 5712 . . . . . . . . 9 ((π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺) β†’ ⟨π‘₯, π‘¦βŸ© ∈ (ran 𝐹 Γ— ran 𝐺))
4341, 42impel 506 . . . . . . . 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 7408 . . . . . . . 8 (π‘₯ + 𝑦) = ( + β€˜βŸ¨π‘₯, π‘¦βŸ©)
4526rneqi 5934 . . . . . . . . 9 ran 𝑃 = ran ( + β†Ύ (ran 𝐹 Γ— ran 𝐺))
46 df-ima 5688 . . . . . . . . 9 ( + β€œ (ran 𝐹 Γ— ran 𝐺)) = ran ( + β†Ύ (ran 𝐹 Γ— ran 𝐺))
4745, 46eqtr4i 2763 . . . . . . . 8 ran 𝑃 = ( + β€œ (ran 𝐹 Γ— ran 𝐺))
4843, 44, 473eltr4g 2850 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ ran 𝐹 ∧ 𝑦 ∈ ran 𝐺)) β†’ (π‘₯ + 𝑦) ∈ ran 𝑃)
4914ffnd 6715 . . . . . . . 8 (πœ‘ β†’ 𝐹 Fn ℝ)
50 dffn3 6727 . . . . . . . 8 (𝐹 Fn ℝ ↔ 𝐹:β„βŸΆran 𝐹)
5149, 50sylib 217 . . . . . . 7 (πœ‘ β†’ 𝐹:β„βŸΆran 𝐹)
5219ffnd 6715 . . . . . . . 8 (πœ‘ β†’ 𝐺 Fn ℝ)
53 dffn3 6727 . . . . . . . 8 (𝐺 Fn ℝ ↔ 𝐺:β„βŸΆran 𝐺)
5452, 53sylib 217 . . . . . . 7 (πœ‘ β†’ 𝐺:β„βŸΆran 𝐺)
55 reex 11197 . . . . . . . 8 ℝ ∈ V
5655a1i 11 . . . . . . 7 (πœ‘ β†’ ℝ ∈ V)
57 inidm 4217 . . . . . . 7 (ℝ ∩ ℝ) = ℝ
5848, 51, 54, 56, 56, 57off 7684 . . . . . 6 (πœ‘ β†’ (𝐹 ∘f + 𝐺):β„βŸΆran 𝑃)
5958frnd 6722 . . . . 5 (πœ‘ β†’ ran (𝐹 ∘f + 𝐺) βŠ† ran 𝑃)
6059ssdifd 4139 . . . 4 (πœ‘ β†’ (ran (𝐹 ∘f + 𝐺) βˆ– {0}) βŠ† (ran 𝑃 βˆ– {0}))
6115sselda 3981 . . . . . . . . . 10 ((πœ‘ ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ ℝ)
6220sselda 3981 . . . . . . . . . 10 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ ℝ)
6361, 62anim12dan 619 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ ran 𝐹 ∧ 𝑧 ∈ ran 𝐺)) β†’ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ))
64 readdcl 11189 . . . . . . . . 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 3200 . . . . . . 7 (πœ‘ β†’ βˆ€π‘¦ ∈ ran πΉβˆ€π‘§ ∈ ran 𝐺(𝑦 + 𝑧) ∈ ℝ)
67 funimassov 7580 . . . . . . . 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 587 . . . . . . 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 256 . . . . . 6 (πœ‘ β†’ ( + β€œ (ran 𝐹 Γ— ran 𝐺)) βŠ† ℝ)
7047, 69eqsstrid 4029 . . . . 5 (πœ‘ β†’ ran 𝑃 βŠ† ℝ)
7170ssdifd 4139 . . . 4 (πœ‘ β†’ (ran 𝑃 βˆ– {0}) βŠ† (ℝ βˆ– {0}))
72 itg1val2 25192 . . . 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 1372 . . 3 (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· (volβ€˜(β—‘(𝐹 ∘f + 𝐺) β€œ {𝑀}))))
7419adantr 481 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ 𝐺:β„βŸΆβ„)
757adantr 481 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ ran 𝐺 ∈ Fin)
76 inss2 4228 . . . . . . . . 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 25186 . . . . . . . . . . 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 25186 . . . . . . . . . . 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 25050 . . . . . . . . . 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 584 . . . . . . . . 9 (πœ‘ β†’ ((◑𝐹 β€œ {(𝑀 βˆ’ 𝑧)}) ∩ (◑𝐺 β€œ {𝑧})) ∈ dom vol)
8483ad2antrr 724 . . . . . . . 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 3992 . . . . . . . . . . . . 13 (πœ‘ β†’ (ran 𝑃 βˆ– {0}) βŠ† ℝ)
8685sselda 3981 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ 𝑀 ∈ ℝ)
8786adantr 481 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑀 ∈ ℝ)
8862adantlr 713 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ ℝ)
8987, 88resubcld 11638 . . . . . . . . . 10 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑀 βˆ’ 𝑧) ∈ ℝ)
9087recnd 11238 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑀 ∈ β„‚)
9188recnd 11238 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ β„‚)
9290, 91npcand 11571 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) = 𝑀)
93 eldifsni 4792 . . . . . . . . . . . . 13 (𝑀 ∈ (ran 𝑃 βˆ– {0}) β†’ 𝑀 β‰  0)
9493ad2antlr 725 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑀 β‰  0)
9592, 94eqnetrd 3008 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) β‰  0)
96 oveq12 7414 . . . . . . . . . . . . 13 (((𝑀 βˆ’ 𝑧) = 0 ∧ 𝑧 = 0) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) = (0 + 0))
97 00id 11385 . . . . . . . . . . . . 13 (0 + 0) = 0
9896, 97eqtrdi 2788 . . . . . . . . . . . 12 (((𝑀 βˆ’ 𝑧) = 0 ∧ 𝑧 = 0) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) = 0)
9998necon3ai 2965 . . . . . . . . . . 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 25206 . . . . . . . . . 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 836 . . . . . . . . 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 25205 . . . . . . . . . . 11 (πœ‘ β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
105104ad2antrr 724 . . . . . . . . . 10 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
106105, 89, 88fovcdmd 7575 . . . . . . . . 9 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) ∈ ℝ)
107103, 106eqeltrrd 2834 . . . . . . . 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 25200 . . . . . . 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 11238 . . . . . . . . 9 ((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ 𝑀 ∈ β„‚)
1101, 2i1faddlem 25201 . . . . . . . . 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 591 . . . . . . . 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 6892 . . . . . . 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 15645 . . . . . . 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 2782 . . . . . 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 7421 . . . . 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 11238 . . . . . 6 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) ∈ β„‚)
11775, 109, 116fsummulc2 15726 . . . . 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 2772 . . . 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 15645 . . 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 11230 . . . . 5 (((πœ‘ ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) ∈ β„‚)
121120anasss 467 . . . 4 ((πœ‘ ∧ (𝑀 ∈ (ran 𝑃 βˆ– {0}) ∧ 𝑧 ∈ ran 𝐺)) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) ∈ β„‚)
12235, 7, 121fsumcom 15717 . . 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 2776 . 2 (πœ‘ β†’ (∫1β€˜(𝐹 ∘f + 𝐺)) = Σ𝑧 ∈ ran 𝐺Σ𝑀 ∈ (ran 𝑃 βˆ– {0})(𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
124 oveq1 7412 . . . . . . 7 (𝑦 = (𝑀 βˆ’ 𝑧) β†’ (𝑦 + 𝑧) = ((𝑀 βˆ’ 𝑧) + 𝑧))
125 oveq1 7412 . . . . . . 7 (𝑦 = (𝑀 βˆ’ 𝑧) β†’ (𝑦𝐼𝑧) = ((𝑀 βˆ’ 𝑧)𝐼𝑧))
126124, 125oveq12d 7423 . . . . . 6 (𝑦 = (𝑀 βˆ’ 𝑧) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = (((𝑀 βˆ’ 𝑧) + 𝑧) Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
12732adantr 481 . . . . . 6 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran 𝑃 ∈ Fin)
12870adantr 481 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran 𝑃 βŠ† ℝ)
129128sselda 3981 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) β†’ 𝑣 ∈ ℝ)
13062adantr 481 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) β†’ 𝑧 ∈ ℝ)
131129, 130resubcld 11638 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) β†’ (𝑣 βˆ’ 𝑧) ∈ ℝ)
132131ex 413 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑣 ∈ ran 𝑃 β†’ (𝑣 βˆ’ 𝑧) ∈ ℝ))
133129recnd 11238 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑣 ∈ ran 𝑃) β†’ 𝑣 ∈ β„‚)
134133adantrr 715 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) β†’ 𝑣 ∈ β„‚)
13570sselda 3981 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑦 ∈ ran 𝑃) β†’ 𝑦 ∈ ℝ)
136135ad2ant2rl 747 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) β†’ 𝑦 ∈ ℝ)
137136recnd 11238 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) β†’ 𝑦 ∈ β„‚)
13862recnd 11238 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ 𝑧 ∈ β„‚)
139138adantr 481 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) β†’ 𝑧 ∈ β„‚)
140134, 137, 139subcan2ad 11612 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃)) β†’ ((𝑣 βˆ’ 𝑧) = (𝑦 βˆ’ 𝑧) ↔ 𝑣 = 𝑦))
141140ex 413 . . . . . . . 8 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ((𝑣 ∈ ran 𝑃 ∧ 𝑦 ∈ ran 𝑃) β†’ ((𝑣 βˆ’ 𝑧) = (𝑦 βˆ’ 𝑧) ↔ 𝑣 = 𝑦)))
142132, 141dom2lem 8984 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–1-1→ℝ)
143 f1f1orn 6841 . . . . . . 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 7412 . . . . . . . 8 (𝑣 = 𝑀 β†’ (𝑣 βˆ’ 𝑧) = (𝑀 βˆ’ 𝑧))
146 eqid 2732 . . . . . . . 8 (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) = (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))
147 ovex 7438 . . . . . . . 8 (𝑀 βˆ’ 𝑧) ∈ V
148145, 146, 147fvmpt 6995 . . . . . . 7 (𝑀 ∈ ran 𝑃 β†’ ((𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))β€˜π‘€) = (𝑀 βˆ’ 𝑧))
149148adantl 482 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ ((𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))β€˜π‘€) = (𝑀 βˆ’ 𝑧))
150 f1f 6784 . . . . . . . . . . 11 ((𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran 𝑃–1-1→ℝ β†’ (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)):ran π‘ƒβŸΆβ„)
151 frn 6721 . . . . . . . . . . 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 3981 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ 𝑦 ∈ ℝ)
15462adantr 481 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ 𝑧 ∈ ℝ)
155153, 154readdcld 11239 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ (𝑦 + 𝑧) ∈ ℝ)
156104ad2antrr 724 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
157156, 153, 154fovcdmd 7575 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ (𝑦𝐼𝑧) ∈ ℝ)
158155, 157remulcld 11240 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧))) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) ∈ ℝ)
159158recnd 11238 . . . . . 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 15665 . . . . 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 3981 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ 𝑀 ∈ ℝ)
162161recnd 11238 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ 𝑀 ∈ β„‚)
163138adantr 481 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ 𝑧 ∈ β„‚)
164162, 163npcand 11571 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ ran 𝑃) β†’ ((𝑀 βˆ’ 𝑧) + 𝑧) = 𝑀)
165164oveq1d 7420 . . . . . 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 15645 . . . . 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 2772 . . . 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 724 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ (ran 𝐹 Γ— ran 𝐺) βŠ† dom + )
169 simpr 485 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ ran 𝐹)
170 simplr 767 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ ran 𝐺)
171169, 170opelxpd 5713 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ βŸ¨π‘¦, π‘§βŸ© ∈ (ran 𝐹 Γ— ran 𝐺))
172 funfvima2 7229 . . . . . . . . . . . 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 688 . . . . . . . . . . 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 7408 . . . . . . . . . 10 (𝑦 + 𝑧) = ( + β€˜βŸ¨π‘¦, π‘§βŸ©)
176174, 175, 473eltr4g 2850 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦 + 𝑧) ∈ ran 𝑃)
17761adantlr 713 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ ℝ)
178177recnd 11238 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 ∈ β„‚)
179138adantr 481 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ β„‚)
180178, 179pncand 11568 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ ((𝑦 + 𝑧) βˆ’ 𝑧) = 𝑦)
181180eqcomd 2738 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑦 = ((𝑦 + 𝑧) βˆ’ 𝑧))
182 oveq1 7412 . . . . . . . . . 10 (𝑣 = (𝑦 + 𝑧) β†’ (𝑣 βˆ’ 𝑧) = ((𝑦 + 𝑧) βˆ’ 𝑧))
183182rspceeqv 3632 . . . . . . . . 9 (((𝑦 + 𝑧) ∈ ran 𝑃 ∧ 𝑦 = ((𝑦 + 𝑧) βˆ’ 𝑧)) β†’ βˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧))
184176, 181, 183syl2anc 584 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ βˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧))
185184ralrimiva 3146 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ βˆ€π‘¦ ∈ ran πΉβˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧))
186 ssabral 4058 . . . . . . 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 5952 . . . . . 6 ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) = {𝑦 ∣ βˆƒπ‘£ ∈ ran 𝑃 𝑦 = (𝑣 βˆ’ 𝑧)}
189187, 188sseqtrrdi 4032 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran 𝐹 βŠ† ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)))
19062adantr 481 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝑧 ∈ ℝ)
191177, 190readdcld 11239 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦 + 𝑧) ∈ ℝ)
192104ad2antrr 724 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
193192, 177, 190fovcdmd 7575 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ (𝑦𝐼𝑧) ∈ ℝ)
194191, 193remulcld 11240 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) ∈ ℝ)
195194recnd 11238 . . . . 5 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) ∈ β„‚)
196152ssdifd 4139 . . . . . . 7 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ (ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) βˆ– ran 𝐹) βŠ† (ℝ βˆ– ran 𝐹))
197196sselda 3981 . . . . . 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 4125 . . . . . . . . . . . . 13 (𝑦 ∈ (ℝ βˆ– ran 𝐹) β†’ 𝑦 ∈ ℝ)
199198ad2antrl 726 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ 𝑦 ∈ ℝ)
20062adantr 481 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ 𝑧 ∈ ℝ)
201 simprr 771 . . . . . . . . . . . 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 25206 . . . . . . . . . . . 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 836 . . . . . . . . . . 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 4227 . . . . . . . . . . . . . . 15 ((◑𝐹 β€œ {𝑦}) ∩ (◑𝐺 β€œ {𝑧})) βŠ† (◑𝐹 β€œ {𝑦})
205 eldifn 4126 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (ℝ βˆ– ran 𝐹) β†’ Β¬ 𝑦 ∈ ran 𝐹)
206205ad2antrl 726 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ Β¬ 𝑦 ∈ ran 𝐹)
207 vex 3478 . . . . . . . . . . . . . . . . . . . . 21 𝑣 ∈ V
208207eliniseg 6090 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ V β†’ (𝑣 ∈ (◑𝐹 β€œ {𝑦}) ↔ 𝑣𝐹𝑦))
209208elv 3480 . . . . . . . . . . . . . . . . . . 19 (𝑣 ∈ (◑𝐹 β€œ {𝑦}) ↔ 𝑣𝐹𝑦)
210 vex 3478 . . . . . . . . . . . . . . . . . . . 20 𝑦 ∈ V
211207, 210brelrn 5939 . . . . . . . . . . . . . . . . . . 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 3987 . . . . . . . . . . . . . . 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 3992 . . . . . . . . . . . . . 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 4397 . . . . . . . . . . . . . 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 6892 . . . . . . . . . . . 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 25047 . . . . . . . . . . . . . 14 βˆ… ∈ dom vol
221 mblvol 25038 . . . . . . . . . . . . . 14 (βˆ… ∈ dom vol β†’ (volβ€˜βˆ…) = (vol*β€˜βˆ…))
222220, 221ax-mp 5 . . . . . . . . . . . . 13 (volβ€˜βˆ…) = (vol*β€˜βˆ…)
223 ovol0 25001 . . . . . . . . . . . . 13 (vol*β€˜βˆ…) = 0
224222, 223eqtri 2760 . . . . . . . . . . . 12 (volβ€˜βˆ…) = 0
225219, 224eqtrdi 2788 . . . . . . . . . . 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 2772 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (𝑦𝐼𝑧) = 0)
227226oveq2d 7421 . . . . . . . . 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 11239 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (𝑦 + 𝑧) ∈ ℝ)
229228recnd 11238 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ (𝑦 + 𝑧) ∈ β„‚)
230229mul01d 11409 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ ((𝑦 + 𝑧) Β· 0) = 0)
231227, 230eqtrd 2772 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ (𝑦 ∈ (ℝ βˆ– ran 𝐹) ∧ Β¬ (𝑦 = 0 ∧ 𝑧 = 0))) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = 0)
232231expr 457 . . . . . . 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 7414 . . . . . . . . . 10 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ (𝑦 + 𝑧) = (0 + 0))
234233, 97eqtrdi 2788 . . . . . . . . 9 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ (𝑦 + 𝑧) = 0)
235 oveq12 7414 . . . . . . . . . 10 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ (𝑦𝐼𝑧) = (0𝐼0))
236 0re 11212 . . . . . . . . . . 11 0 ∈ ℝ
237 iftrue 4533 . . . . . . . . . . . 12 ((𝑖 = 0 ∧ 𝑗 = 0) β†’ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (volβ€˜((◑𝐹 β€œ {𝑖}) ∩ (◑𝐺 β€œ {𝑗})))) = 0)
238 c0ex 11204 . . . . . . . . . . . 12 0 ∈ V
239237, 101, 238ovmpoa 7559 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ 0 ∈ ℝ) β†’ (0𝐼0) = 0)
240236, 236, 239mp2an 690 . . . . . . . . . 10 (0𝐼0) = 0
241235, 240eqtrdi 2788 . . . . . . . . 9 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ (𝑦𝐼𝑧) = 0)
242234, 241oveq12d 7423 . . . . . . . 8 ((𝑦 = 0 ∧ 𝑧 = 0) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) = (0 Β· 0))
243 0cn 11202 . . . . . . . . 9 0 ∈ β„‚
244243mul01i 11400 . . . . . . . 8 (0 Β· 0) = 0
245242, 244eqtrdi 2788 . . . . . . 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 591 . . . . 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 6837 . . . . . . 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 9334 . . . . . 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 584 . . . . 5 ((πœ‘ ∧ 𝑧 ∈ ran 𝐺) β†’ ran (𝑣 ∈ ran 𝑃 ↦ (𝑣 βˆ’ 𝑧)) ∈ Fin)
252189, 195, 247, 251fsumss 15667 . . . 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 650 . . . . 5 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ (ran 𝑃 βˆ– {0})) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) ∈ β„‚)
255 dfin4 4266 . . . . . . . 8 (ran 𝑃 ∩ {0}) = (ran 𝑃 βˆ– (ran 𝑃 βˆ– {0}))
256 inss2 4228 . . . . . . . 8 (ran 𝑃 ∩ {0}) βŠ† {0}
257255, 256eqsstrri 4016 . . . . . . 7 (ran 𝑃 βˆ– (ran 𝑃 βˆ– {0})) βŠ† {0}
258257sseli 3977 . . . . . 6 (𝑀 ∈ (ran 𝑃 βˆ– (ran 𝑃 βˆ– {0})) β†’ 𝑀 ∈ {0})
259 elsni 4644 . . . . . . . . 9 (𝑀 ∈ {0} β†’ 𝑀 = 0)
260259adantl 482 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ 𝑀 = 0)
261260oveq1d 7420 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = (0 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)))
262104ad2antrr 724 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ 𝐼:(ℝ Γ— ℝ)βŸΆβ„)
263260, 236eqeltrdi 2841 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ 𝑀 ∈ ℝ)
26462adantr 481 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ 𝑧 ∈ ℝ)
265263, 264resubcld 11638 . . . . . . . . . 10 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ (𝑀 βˆ’ 𝑧) ∈ ℝ)
266262, 265, 264fovcdmd 7575 . . . . . . . . 9 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) ∈ ℝ)
267266recnd 11238 . . . . . . . 8 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ ((𝑀 βˆ’ 𝑧)𝐼𝑧) ∈ β„‚)
268267mul02d 11408 . . . . . . 7 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ (0 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = 0)
269261, 268eqtrd 2772 . . . . . 6 (((πœ‘ ∧ 𝑧 ∈ ran 𝐺) ∧ 𝑀 ∈ {0}) β†’ (𝑀 Β· ((𝑀 βˆ’ 𝑧)𝐼𝑧)) = 0)
270258, 269sylan2 593 . . . . 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 15667 . . . 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 2782 . . 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 15645 . 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 467 . . 3 ((πœ‘ ∧ (𝑧 ∈ ran 𝐺 ∧ 𝑦 ∈ ran 𝐹)) β†’ ((𝑦 + 𝑧) Β· (𝑦𝐼𝑧)) ∈ β„‚)
2757, 5, 274fsumcom 15717 . 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 2778 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 396   = wceq 1541   ∈ wcel 2106  {cab 2709   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   βˆ– cdif 3944   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  ifcif 4527  {csn 4627  βŸ¨cop 4633  βˆͺ ciun 4996   class class class wbr 5147   ↦ cmpt 5230   Γ— cxp 5673  β—‘ccnv 5674  dom cdm 5675  ran crn 5676   β†Ύ cres 5677   β€œ cima 5678  Fun wfun 6534   Fn wfn 6535  βŸΆwf 6536  β€“1-1β†’wf1 6537  β€“ontoβ†’wfo 6538  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407   ∘f cof 7664  Fincfn 8935  β„‚cc 11104  β„cr 11105  0cc0 11106   + caddc 11109   Β· cmul 11111   βˆ’ cmin 11440  Ξ£csu 15628  vol*covol 24970  volcvol 24971  βˆ«1citg1 25123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184  ax-addf 11185
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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-disj 5113  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 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-inf 9434  df-oi 9501  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-q 12929  df-rp 12971  df-xadd 13089  df-ioo 13324  df-ico 13326  df-icc 13327  df-fz 13481  df-fzo 13624  df-fl 13753  df-seq 13963  df-exp 14024  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-sum 15629  df-xmet 20929  df-met 20930  df-ovol 24972  df-vol 24973  df-mbf 25127  df-itg1 25128
This theorem is referenced by: (None)
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