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Theorem itg1addlem5 23512
Description: Lemma for itg1add . (Contributed by Mario Carneiro, 27-Jun-2014.)
Hypotheses
Ref Expression
i1fadd.1 (𝜑𝐹 ∈ dom ∫1)
i1fadd.2 (𝜑𝐺 ∈ dom ∫1)
itg1add.3 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))
itg1add.4 𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))
Assertion
Ref Expression
itg1addlem5 (𝜑 → (∫1‘(𝐹𝑓 + 𝐺)) = ((∫1𝐹) + (∫1𝐺)))
Distinct variable groups:   𝑖,𝑗,𝐹   𝑖,𝐺,𝑗   𝜑,𝑖,𝑗
Allowed substitution hints:   𝑃(𝑖,𝑗)   𝐼(𝑖,𝑗)

Proof of Theorem itg1addlem5
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 i1fadd.1 . . . 4 (𝜑𝐹 ∈ dom ∫1)
2 i1frn 23489 . . . 4 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
31, 2syl 17 . . 3 (𝜑 → ran 𝐹 ∈ Fin)
4 i1fadd.2 . . . . . 6 (𝜑𝐺 ∈ dom ∫1)
5 i1frn 23489 . . . . . 6 (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
64, 5syl 17 . . . . 5 (𝜑 → ran 𝐺 ∈ Fin)
76adantr 480 . . . 4 ((𝜑𝑦 ∈ ran 𝐹) → ran 𝐺 ∈ Fin)
8 i1ff 23488 . . . . . . . . . 10 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
91, 8syl 17 . . . . . . . . 9 (𝜑𝐹:ℝ⟶ℝ)
10 frn 6091 . . . . . . . . 9 (𝐹:ℝ⟶ℝ → ran 𝐹 ⊆ ℝ)
119, 10syl 17 . . . . . . . 8 (𝜑 → ran 𝐹 ⊆ ℝ)
1211sselda 3636 . . . . . . 7 ((𝜑𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
1312adantr 480 . . . . . 6 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
1413recnd 10106 . . . . 5 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℂ)
15 itg1add.3 . . . . . . . . 9 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))
161, 4, 15itg1addlem2 23509 . . . . . . . 8 (𝜑𝐼:(ℝ × ℝ)⟶ℝ)
1716ad2antrr 762 . . . . . . 7 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
18 i1ff 23488 . . . . . . . . . . 11 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
194, 18syl 17 . . . . . . . . . 10 (𝜑𝐺:ℝ⟶ℝ)
20 frn 6091 . . . . . . . . . 10 (𝐺:ℝ⟶ℝ → ran 𝐺 ⊆ ℝ)
2119, 20syl 17 . . . . . . . . 9 (𝜑 → ran 𝐺 ⊆ ℝ)
2221sselda 3636 . . . . . . . 8 ((𝜑𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
2322adantlr 751 . . . . . . 7 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
2417, 13, 23fovrnd 6848 . . . . . 6 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℝ)
2524recnd 10106 . . . . 5 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℂ)
2614, 25mulcld 10098 . . . 4 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
277, 26fsumcl 14508 . . 3 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
2823recnd 10106 . . . . 5 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℂ)
2928, 25mulcld 10098 . . . 4 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
307, 29fsumcl 14508 . . 3 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
313, 27, 30fsumadd 14514 . 2 (𝜑 → Σ𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))) = (Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
32 itg1add.4 . . . 4 𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))
331, 4, 15, 32itg1addlem4 23511 . . 3 (𝜑 → (∫1‘(𝐹𝑓 + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
3414, 28, 25adddird 10103 . . . . . 6 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = ((𝑦 · (𝑦𝐼𝑧)) + (𝑧 · (𝑦𝐼𝑧))))
3534sumeq2dv 14477 . . . . 5 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺((𝑦 · (𝑦𝐼𝑧)) + (𝑧 · (𝑦𝐼𝑧))))
367, 26, 29fsumadd 14514 . . . . 5 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺((𝑦 · (𝑦𝐼𝑧)) + (𝑧 · (𝑦𝐼𝑧))) = (Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
3735, 36eqtrd 2685 . . . 4 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = (Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
3837sumeq2dv 14477 . . 3 (𝜑 → Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
3933, 38eqtrd 2685 . 2 (𝜑 → (∫1‘(𝐹𝑓 + 𝐺)) = Σ𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
40 itg1val 23495 . . . . 5 (𝐹 ∈ dom ∫1 → (∫1𝐹) = Σ𝑦 ∈ (ran 𝐹 ∖ {0})(𝑦 · (vol‘(𝐹 “ {𝑦}))))
411, 40syl 17 . . . 4 (𝜑 → (∫1𝐹) = Σ𝑦 ∈ (ran 𝐹 ∖ {0})(𝑦 · (vol‘(𝐹 “ {𝑦}))))
4219adantr 480 . . . . . . . . 9 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝐺:ℝ⟶ℝ)
436adantr 480 . . . . . . . . 9 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → ran 𝐺 ∈ Fin)
44 inss2 3867 . . . . . . . . . 10 ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧})
4544a1i 11 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}))
46 i1fima 23490 . . . . . . . . . . . 12 (𝐹 ∈ dom ∫1 → (𝐹 “ {𝑦}) ∈ dom vol)
471, 46syl 17 . . . . . . . . . . 11 (𝜑 → (𝐹 “ {𝑦}) ∈ dom vol)
4847ad2antrr 762 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐹 “ {𝑦}) ∈ dom vol)
49 i1fima 23490 . . . . . . . . . . . 12 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑧}) ∈ dom vol)
504, 49syl 17 . . . . . . . . . . 11 (𝜑 → (𝐺 “ {𝑧}) ∈ dom vol)
5150ad2antrr 762 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐺 “ {𝑧}) ∈ dom vol)
52 inmbl 23356 . . . . . . . . . 10 (((𝐹 “ {𝑦}) ∈ dom vol ∧ (𝐺 “ {𝑧}) ∈ dom vol) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
5348, 51, 52syl2anc 694 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
5411ssdifssd 3781 . . . . . . . . . . . . 13 (𝜑 → (ran 𝐹 ∖ {0}) ⊆ ℝ)
5554sselda 3636 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝑦 ∈ ℝ)
5655adantr 480 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
5721adantr 480 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → ran 𝐺 ⊆ ℝ)
5857sselda 3636 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
59 eldifsni 4353 . . . . . . . . . . . . 13 (𝑦 ∈ (ran 𝐹 ∖ {0}) → 𝑦 ≠ 0)
6059ad2antlr 763 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ≠ 0)
61 simpl 472 . . . . . . . . . . . . 13 ((𝑦 = 0 ∧ 𝑧 = 0) → 𝑦 = 0)
6261necon3ai 2848 . . . . . . . . . . . 12 (𝑦 ≠ 0 → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
6360, 62syl 17 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
641, 4, 15itg1addlem3 23510 . . . . . . . . . . 11 (((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0)) → (𝑦𝐼𝑧) = (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
6556, 58, 63, 64syl21anc 1365 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) = (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
6616ad2antrr 762 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
6766, 56, 58fovrnd 6848 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℝ)
6865, 67eqeltrrd 2731 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
6942, 43, 45, 53, 68itg1addlem1 23504 . . . . . . . 8 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘ 𝑧 ∈ ran 𝐺((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) = Σ𝑧 ∈ ran 𝐺(vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
70 iunin2 4616 . . . . . . . . . 10 𝑧 ∈ ran 𝐺((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ((𝐹 “ {𝑦}) ∩ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}))
711adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝐹 ∈ dom ∫1)
7271, 46syl 17 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑦}) ∈ dom vol)
73 mblss 23345 . . . . . . . . . . . . 13 ((𝐹 “ {𝑦}) ∈ dom vol → (𝐹 “ {𝑦}) ⊆ ℝ)
7472, 73syl 17 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑦}) ⊆ ℝ)
75 iunid 4607 . . . . . . . . . . . . . . 15 𝑧 ∈ ran 𝐺{𝑧} = ran 𝐺
7675imaeq2i 5499 . . . . . . . . . . . . . 14 (𝐺 𝑧 ∈ ran 𝐺{𝑧}) = (𝐺 “ ran 𝐺)
77 imaiun 6543 . . . . . . . . . . . . . 14 (𝐺 𝑧 ∈ ran 𝐺{𝑧}) = 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧})
78 cnvimarndm 5521 . . . . . . . . . . . . . 14 (𝐺 “ ran 𝐺) = dom 𝐺
7976, 77, 783eqtr3i 2681 . . . . . . . . . . . . 13 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}) = dom 𝐺
80 fdm 6089 . . . . . . . . . . . . . 14 (𝐺:ℝ⟶ℝ → dom 𝐺 = ℝ)
8142, 80syl 17 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → dom 𝐺 = ℝ)
8279, 81syl5eq 2697 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}) = ℝ)
8374, 82sseqtr4d 3675 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑦}) ⊆ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}))
84 df-ss 3621 . . . . . . . . . . 11 ((𝐹 “ {𝑦}) ⊆ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}) ↔ ((𝐹 “ {𝑦}) ∩ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧})) = (𝐹 “ {𝑦}))
8583, 84sylib 208 . . . . . . . . . 10 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → ((𝐹 “ {𝑦}) ∩ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧})) = (𝐹 “ {𝑦}))
8670, 85syl5req 2698 . . . . . . . . 9 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑦}) = 𝑧 ∈ ran 𝐺((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})))
8786fveq2d 6233 . . . . . . . 8 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑦})) = (vol‘ 𝑧 ∈ ran 𝐺((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
8865sumeq2dv 14477 . . . . . . . 8 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧) = Σ𝑧 ∈ ran 𝐺(vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
8969, 87, 883eqtr4d 2695 . . . . . . 7 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑦})) = Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧))
9089oveq2d 6706 . . . . . 6 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝑦 · (vol‘(𝐹 “ {𝑦}))) = (𝑦 · Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧)))
9155recnd 10106 . . . . . . 7 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝑦 ∈ ℂ)
9267recnd 10106 . . . . . . 7 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℂ)
9343, 91, 92fsummulc2 14560 . . . . . 6 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝑦 · Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
9490, 93eqtrd 2685 . . . . 5 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝑦 · (vol‘(𝐹 “ {𝑦}))) = Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
9594sumeq2dv 14477 . . . 4 (𝜑 → Σ𝑦 ∈ (ran 𝐹 ∖ {0})(𝑦 · (vol‘(𝐹 “ {𝑦}))) = Σ𝑦 ∈ (ran 𝐹 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
96 difssd 3771 . . . . 5 (𝜑 → (ran 𝐹 ∖ {0}) ⊆ ran 𝐹)
9756recnd 10106 . . . . . . 7 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℂ)
9897, 92mulcld 10098 . . . . . 6 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
9943, 98fsumcl 14508 . . . . 5 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
100 dfin4 3900 . . . . . . . 8 (ran 𝐹 ∩ {0}) = (ran 𝐹 ∖ (ran 𝐹 ∖ {0}))
101 inss2 3867 . . . . . . . 8 (ran 𝐹 ∩ {0}) ⊆ {0}
102100, 101eqsstr3i 3669 . . . . . . 7 (ran 𝐹 ∖ (ran 𝐹 ∖ {0})) ⊆ {0}
103102sseli 3632 . . . . . 6 (𝑦 ∈ (ran 𝐹 ∖ (ran 𝐹 ∖ {0})) → 𝑦 ∈ {0})
104 elsni 4227 . . . . . . . . . . 11 (𝑦 ∈ {0} → 𝑦 = 0)
105104ad2antlr 763 . . . . . . . . . 10 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 = 0)
106105oveq1d 6705 . . . . . . . . 9 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) = (0 · (𝑦𝐼𝑧)))
10716ad2antrr 762 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
108 0re 10078 . . . . . . . . . . . . 13 0 ∈ ℝ
109105, 108syl6eqel 2738 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
11022adantlr 751 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
111107, 109, 110fovrnd 6848 . . . . . . . . . . 11 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℝ)
112111recnd 10106 . . . . . . . . . 10 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℂ)
113112mul02d 10272 . . . . . . . . 9 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (0 · (𝑦𝐼𝑧)) = 0)
114106, 113eqtrd 2685 . . . . . . . 8 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) = 0)
115114sumeq2dv 14477 . . . . . . 7 ((𝜑𝑦 ∈ {0}) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺0)
1166adantr 480 . . . . . . . . 9 ((𝜑𝑦 ∈ {0}) → ran 𝐺 ∈ Fin)
117116olcd 407 . . . . . . . 8 ((𝜑𝑦 ∈ {0}) → (ran 𝐺 ⊆ (ℤ‘0) ∨ ran 𝐺 ∈ Fin))
118 sumz 14497 . . . . . . . 8 ((ran 𝐺 ⊆ (ℤ‘0) ∨ ran 𝐺 ∈ Fin) → Σ𝑧 ∈ ran 𝐺0 = 0)
119117, 118syl 17 . . . . . . 7 ((𝜑𝑦 ∈ {0}) → Σ𝑧 ∈ ran 𝐺0 = 0)
120115, 119eqtrd 2685 . . . . . 6 ((𝜑𝑦 ∈ {0}) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = 0)
121103, 120sylan2 490 . . . . 5 ((𝜑𝑦 ∈ (ran 𝐹 ∖ (ran 𝐹 ∖ {0}))) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = 0)
12296, 99, 121, 3fsumss 14500 . . . 4 (𝜑 → Σ𝑦 ∈ (ran 𝐹 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
12341, 95, 1223eqtrd 2689 . . 3 (𝜑 → (∫1𝐹) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
124 itg1val 23495 . . . . 5 (𝐺 ∈ dom ∫1 → (∫1𝐺) = Σ𝑧 ∈ (ran 𝐺 ∖ {0})(𝑧 · (vol‘(𝐺 “ {𝑧}))))
1254, 124syl 17 . . . 4 (𝜑 → (∫1𝐺) = Σ𝑧 ∈ (ran 𝐺 ∖ {0})(𝑧 · (vol‘(𝐺 “ {𝑧}))))
1269adantr 480 . . . . . . . . 9 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝐹:ℝ⟶ℝ)
1273adantr 480 . . . . . . . . 9 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → ran 𝐹 ∈ Fin)
128 inss1 3866 . . . . . . . . . 10 ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {𝑦})
129128a1i 11 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {𝑦}))
13047ad2antrr 762 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝐹 “ {𝑦}) ∈ dom vol)
13150ad2antrr 762 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝐺 “ {𝑧}) ∈ dom vol)
132130, 131, 52syl2anc 694 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
13311adantr 480 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → ran 𝐹 ⊆ ℝ)
134133sselda 3636 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
13521ssdifssd 3781 . . . . . . . . . . . . 13 (𝜑 → (ran 𝐺 ∖ {0}) ⊆ ℝ)
136135sselda 3636 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝑧 ∈ ℝ)
137136adantr 480 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
138 eldifsni 4353 . . . . . . . . . . . . 13 (𝑧 ∈ (ran 𝐺 ∖ {0}) → 𝑧 ≠ 0)
139138ad2antlr 763 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ≠ 0)
140 simpr 476 . . . . . . . . . . . . 13 ((𝑦 = 0 ∧ 𝑧 = 0) → 𝑧 = 0)
141140necon3ai 2848 . . . . . . . . . . . 12 (𝑧 ≠ 0 → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
142139, 141syl 17 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
143134, 137, 142, 64syl21anc 1365 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) = (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
14416ad2antrr 762 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
145144, 134, 137fovrnd 6848 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
146143, 145eqeltrrd 2731 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
147126, 127, 129, 132, 146itg1addlem1 23504 . . . . . . . 8 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘ 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) = Σ𝑦 ∈ ran 𝐹(vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
148 incom 3838 . . . . . . . . . . . . 13 ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ((𝐺 “ {𝑧}) ∩ (𝐹 “ {𝑦}))
149148a1i 11 . . . . . . . . . . . 12 (𝑦 ∈ ran 𝐹 → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ((𝐺 “ {𝑧}) ∩ (𝐹 “ {𝑦})))
150149iuneq2i 4571 . . . . . . . . . . 11 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = 𝑦 ∈ ran 𝐹((𝐺 “ {𝑧}) ∩ (𝐹 “ {𝑦}))
151 iunin2 4616 . . . . . . . . . . 11 𝑦 ∈ ran 𝐹((𝐺 “ {𝑧}) ∩ (𝐹 “ {𝑦})) = ((𝐺 “ {𝑧}) ∩ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}))
152150, 151eqtri 2673 . . . . . . . . . 10 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ((𝐺 “ {𝑧}) ∩ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}))
153 cnvimass 5520 . . . . . . . . . . . . 13 (𝐺 “ {𝑧}) ⊆ dom 𝐺
15419, 80syl 17 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐺 = ℝ)
155154adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → dom 𝐺 = ℝ)
156153, 155syl5sseq 3686 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ⊆ ℝ)
157 iunid 4607 . . . . . . . . . . . . . . 15 𝑦 ∈ ran 𝐹{𝑦} = ran 𝐹
158157imaeq2i 5499 . . . . . . . . . . . . . 14 (𝐹 𝑦 ∈ ran 𝐹{𝑦}) = (𝐹 “ ran 𝐹)
159 imaiun 6543 . . . . . . . . . . . . . 14 (𝐹 𝑦 ∈ ran 𝐹{𝑦}) = 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦})
160 cnvimarndm 5521 . . . . . . . . . . . . . 14 (𝐹 “ ran 𝐹) = dom 𝐹
161158, 159, 1603eqtr3i 2681 . . . . . . . . . . . . 13 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}) = dom 𝐹
162 fdm 6089 . . . . . . . . . . . . . . 15 (𝐹:ℝ⟶ℝ → dom 𝐹 = ℝ)
1639, 162syl 17 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐹 = ℝ)
164163adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → dom 𝐹 = ℝ)
165161, 164syl5eq 2697 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}) = ℝ)
166156, 165sseqtr4d 3675 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ⊆ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}))
167 df-ss 3621 . . . . . . . . . . 11 ((𝐺 “ {𝑧}) ⊆ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}) ↔ ((𝐺 “ {𝑧}) ∩ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦})) = (𝐺 “ {𝑧}))
168166, 167sylib 208 . . . . . . . . . 10 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → ((𝐺 “ {𝑧}) ∩ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦})) = (𝐺 “ {𝑧}))
169152, 168syl5req 2698 . . . . . . . . 9 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) = 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})))
170169fveq2d 6233 . . . . . . . 8 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) = (vol‘ 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
171143sumeq2dv 14477 . . . . . . . 8 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧) = Σ𝑦 ∈ ran 𝐹(vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
172147, 170, 1713eqtr4d 2695 . . . . . . 7 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) = Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧))
173172oveq2d 6706 . . . . . 6 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝑧 · (vol‘(𝐺 “ {𝑧}))) = (𝑧 · Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧)))
174136recnd 10106 . . . . . . 7 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝑧 ∈ ℂ)
175145recnd 10106 . . . . . . 7 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℂ)
176127, 174, 175fsummulc2 14560 . . . . . 6 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝑧 · Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
177173, 176eqtrd 2685 . . . . 5 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝑧 · (vol‘(𝐺 “ {𝑧}))) = Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
178177sumeq2dv 14477 . . . 4 (𝜑 → Σ𝑧 ∈ (ran 𝐺 ∖ {0})(𝑧 · (vol‘(𝐺 “ {𝑧}))) = Σ𝑧 ∈ (ran 𝐺 ∖ {0})Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
179 difssd 3771 . . . . . 6 (𝜑 → (ran 𝐺 ∖ {0}) ⊆ ran 𝐺)
180174adantr 480 . . . . . . . 8 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
181180, 175mulcld 10098 . . . . . . 7 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
182127, 181fsumcl 14508 . . . . . 6 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
183 dfin4 3900 . . . . . . . . 9 (ran 𝐺 ∩ {0}) = (ran 𝐺 ∖ (ran 𝐺 ∖ {0}))
184 inss2 3867 . . . . . . . . 9 (ran 𝐺 ∩ {0}) ⊆ {0}
185183, 184eqsstr3i 3669 . . . . . . . 8 (ran 𝐺 ∖ (ran 𝐺 ∖ {0})) ⊆ {0}
186185sseli 3632 . . . . . . 7 (𝑧 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {0})) → 𝑧 ∈ {0})
187 elsni 4227 . . . . . . . . . . . 12 (𝑧 ∈ {0} → 𝑧 = 0)
188187ad2antlr 763 . . . . . . . . . . 11 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 = 0)
189188oveq1d 6705 . . . . . . . . . 10 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) = (0 · (𝑦𝐼𝑧)))
19016ad2antrr 762 . . . . . . . . . . . . 13 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
19112adantlr 751 . . . . . . . . . . . . 13 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
192188, 108syl6eqel 2738 . . . . . . . . . . . . 13 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
193190, 191, 192fovrnd 6848 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
194193recnd 10106 . . . . . . . . . . 11 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℂ)
195194mul02d 10272 . . . . . . . . . 10 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (0 · (𝑦𝐼𝑧)) = 0)
196189, 195eqtrd 2685 . . . . . . . . 9 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) = 0)
197196sumeq2dv 14477 . . . . . . . 8 ((𝜑𝑧 ∈ {0}) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹0)
1983adantr 480 . . . . . . . . . 10 ((𝜑𝑧 ∈ {0}) → ran 𝐹 ∈ Fin)
199198olcd 407 . . . . . . . . 9 ((𝜑𝑧 ∈ {0}) → (ran 𝐹 ⊆ (ℤ‘0) ∨ ran 𝐹 ∈ Fin))
200 sumz 14497 . . . . . . . . 9 ((ran 𝐹 ⊆ (ℤ‘0) ∨ ran 𝐹 ∈ Fin) → Σ𝑦 ∈ ran 𝐹0 = 0)
201199, 200syl 17 . . . . . . . 8 ((𝜑𝑧 ∈ {0}) → Σ𝑦 ∈ ran 𝐹0 = 0)
202197, 201eqtrd 2685 . . . . . . 7 ((𝜑𝑧 ∈ {0}) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = 0)
203186, 202sylan2 490 . . . . . 6 ((𝜑𝑧 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {0}))) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = 0)
204179, 182, 203, 6fsumss 14500 . . . . 5 (𝜑 → Σ𝑧 ∈ (ran 𝐺 ∖ {0})Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
20522adantr 480 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
206205recnd 10106 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
20716ad2antrr 762 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
20811adantr 480 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ran 𝐺) → ran 𝐹 ⊆ ℝ)
209208sselda 3636 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
210207, 209, 205fovrnd 6848 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
211210recnd 10106 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℂ)
212206, 211mulcld 10098 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
213212anasss 680 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ran 𝐺𝑦 ∈ ran 𝐹)) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
2146, 3, 213fsumcom 14551 . . . . 5 (𝜑 → Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)))
215204, 214eqtrd 2685 . . . 4 (𝜑 → Σ𝑧 ∈ (ran 𝐺 ∖ {0})Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)))
216125, 178, 2153eqtrd 2689 . . 3 (𝜑 → (∫1𝐺) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)))
217123, 216oveq12d 6708 . 2 (𝜑 → ((∫1𝐹) + (∫1𝐺)) = (Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
21831, 39, 2173eqtr4d 2695 1 (𝜑 → (∫1‘(𝐹𝑓 + 𝐺)) = ((∫1𝐹) + (∫1𝐺)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  cdif 3604  cin 3606  wss 3607  ifcif 4119  {csn 4210   ciun 4552   × cxp 5141  ccnv 5142  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  𝑓 cof 6937  Fincfn 7997  cc 9972  cr 9973  0cc0 9974   + caddc 9977   · cmul 9979  cuz 11725  Σcsu 14460  volcvol 23278  1citg1 23429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xadd 11985  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-xmet 19787  df-met 19788  df-ovol 23279  df-vol 23280  df-mbf 23433  df-itg1 23434
This theorem is referenced by:  itg1add  23513
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