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Theorem itg1addlem5 24216
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 24193 . . . 4 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
31, 2syl 17 . . 3 (𝜑 → ran 𝐹 ∈ Fin)
4 i1fadd.2 . . . . . 6 (𝜑𝐺 ∈ dom ∫1)
5 i1frn 24193 . . . . . 6 (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
64, 5syl 17 . . . . 5 (𝜑 → ran 𝐺 ∈ Fin)
76adantr 481 . . . 4 ((𝜑𝑦 ∈ ran 𝐹) → ran 𝐺 ∈ Fin)
8 i1ff 24192 . . . . . . . . . 10 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
91, 8syl 17 . . . . . . . . 9 (𝜑𝐹:ℝ⟶ℝ)
109frnd 6517 . . . . . . . 8 (𝜑 → ran 𝐹 ⊆ ℝ)
1110sselda 3970 . . . . . . 7 ((𝜑𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
1211adantr 481 . . . . . 6 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
1312recnd 10661 . . . . 5 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℂ)
14 itg1add.3 . . . . . . . . 9 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((𝐹 “ {𝑖}) ∩ (𝐺 “ {𝑗})))))
151, 4, 14itg1addlem2 24213 . . . . . . . 8 (𝜑𝐼:(ℝ × ℝ)⟶ℝ)
1615ad2antrr 722 . . . . . . 7 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
17 i1ff 24192 . . . . . . . . . . 11 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
184, 17syl 17 . . . . . . . . . 10 (𝜑𝐺:ℝ⟶ℝ)
1918frnd 6517 . . . . . . . . 9 (𝜑 → ran 𝐺 ⊆ ℝ)
2019sselda 3970 . . . . . . . 8 ((𝜑𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
2120adantlr 711 . . . . . . 7 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
2216, 12, 21fovrnd 7313 . . . . . 6 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℝ)
2322recnd 10661 . . . . 5 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℂ)
2413, 23mulcld 10653 . . . 4 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
257, 24fsumcl 15082 . . 3 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
2621recnd 10661 . . . . 5 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℂ)
2726, 23mulcld 10653 . . . 4 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
287, 27fsumcl 15082 . . 3 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
293, 25, 28fsumadd 15088 . 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))))
30 itg1add.4 . . . 4 𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))
311, 4, 14, 30itg1addlem4 24215 . . 3 (𝜑 → (∫1‘(𝐹f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
3213, 26, 23adddird 10658 . . . . . 6 (((𝜑𝑦 ∈ ran 𝐹) ∧ 𝑧 ∈ ran 𝐺) → ((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = ((𝑦 · (𝑦𝐼𝑧)) + (𝑧 · (𝑦𝐼𝑧))))
3332sumeq2dv 15052 . . . . 5 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺((𝑦 · (𝑦𝐼𝑧)) + (𝑧 · (𝑦𝐼𝑧))))
347, 24, 27fsumadd 15088 . . . . 5 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺((𝑦 · (𝑦𝐼𝑧)) + (𝑧 · (𝑦𝐼𝑧))) = (Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
3533, 34eqtrd 2860 . . . 4 ((𝜑𝑦 ∈ ran 𝐹) → Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = (Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
3635sumeq2dv 15052 . . 3 (𝜑 → Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
3731, 36eqtrd 2860 . 2 (𝜑 → (∫1‘(𝐹f + 𝐺)) = Σ𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
38 itg1val 24199 . . . . 5 (𝐹 ∈ dom ∫1 → (∫1𝐹) = Σ𝑦 ∈ (ran 𝐹 ∖ {0})(𝑦 · (vol‘(𝐹 “ {𝑦}))))
391, 38syl 17 . . . 4 (𝜑 → (∫1𝐹) = Σ𝑦 ∈ (ran 𝐹 ∖ {0})(𝑦 · (vol‘(𝐹 “ {𝑦}))))
4018adantr 481 . . . . . . . . 9 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝐺:ℝ⟶ℝ)
416adantr 481 . . . . . . . . 9 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → ran 𝐺 ∈ Fin)
42 inss2 4209 . . . . . . . . . 10 ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧})
4342a1i 11 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}))
44 i1fima 24194 . . . . . . . . . . . 12 (𝐹 ∈ dom ∫1 → (𝐹 “ {𝑦}) ∈ dom vol)
451, 44syl 17 . . . . . . . . . . 11 (𝜑 → (𝐹 “ {𝑦}) ∈ dom vol)
4645ad2antrr 722 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐹 “ {𝑦}) ∈ dom vol)
47 i1fima 24194 . . . . . . . . . . . 12 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑧}) ∈ dom vol)
484, 47syl 17 . . . . . . . . . . 11 (𝜑 → (𝐺 “ {𝑧}) ∈ dom vol)
4948ad2antrr 722 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐺 “ {𝑧}) ∈ dom vol)
50 inmbl 24058 . . . . . . . . . 10 (((𝐹 “ {𝑦}) ∈ dom vol ∧ (𝐺 “ {𝑧}) ∈ dom vol) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
5146, 49, 50syl2anc 584 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
5210ssdifssd 4122 . . . . . . . . . . . . 13 (𝜑 → (ran 𝐹 ∖ {0}) ⊆ ℝ)
5352sselda 3970 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝑦 ∈ ℝ)
5453adantr 481 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
5519adantr 481 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → ran 𝐺 ⊆ ℝ)
5655sselda 3970 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
57 eldifsni 4720 . . . . . . . . . . . . 13 (𝑦 ∈ (ran 𝐹 ∖ {0}) → 𝑦 ≠ 0)
5857ad2antlr 723 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ≠ 0)
59 simpl 483 . . . . . . . . . . . . 13 ((𝑦 = 0 ∧ 𝑧 = 0) → 𝑦 = 0)
6059necon3ai 3045 . . . . . . . . . . . 12 (𝑦 ≠ 0 → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
6158, 60syl 17 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
621, 4, 14itg1addlem3 24214 . . . . . . . . . . 11 (((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) ∧ ¬ (𝑦 = 0 ∧ 𝑧 = 0)) → (𝑦𝐼𝑧) = (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
6354, 56, 61, 62syl21anc 835 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) = (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
6415ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
6564, 54, 56fovrnd 7313 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℝ)
6663, 65eqeltrrd 2918 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
6740, 41, 43, 51, 66itg1addlem1 24208 . . . . . . . 8 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘ 𝑧 ∈ ran 𝐺((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) = Σ𝑧 ∈ ran 𝐺(vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
68 iunin2 4989 . . . . . . . . . 10 𝑧 ∈ ran 𝐺((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ((𝐹 “ {𝑦}) ∩ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}))
691adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝐹 ∈ dom ∫1)
7069, 44syl 17 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑦}) ∈ dom vol)
71 mblss 24047 . . . . . . . . . . . . 13 ((𝐹 “ {𝑦}) ∈ dom vol → (𝐹 “ {𝑦}) ⊆ ℝ)
7270, 71syl 17 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑦}) ⊆ ℝ)
73 iunid 4980 . . . . . . . . . . . . . . 15 𝑧 ∈ ran 𝐺{𝑧} = ran 𝐺
7473imaeq2i 5924 . . . . . . . . . . . . . 14 (𝐺 𝑧 ∈ ran 𝐺{𝑧}) = (𝐺 “ ran 𝐺)
75 imaiun 7001 . . . . . . . . . . . . . 14 (𝐺 𝑧 ∈ ran 𝐺{𝑧}) = 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧})
76 cnvimarndm 5947 . . . . . . . . . . . . . 14 (𝐺 “ ran 𝐺) = dom 𝐺
7774, 75, 763eqtr3i 2856 . . . . . . . . . . . . 13 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}) = dom 𝐺
7840fdmd 6519 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → dom 𝐺 = ℝ)
7977, 78syl5eq 2872 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}) = ℝ)
8072, 79sseqtrrd 4011 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑦}) ⊆ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}))
81 df-ss 3955 . . . . . . . . . . 11 ((𝐹 “ {𝑦}) ⊆ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧}) ↔ ((𝐹 “ {𝑦}) ∩ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧})) = (𝐹 “ {𝑦}))
8280, 81sylib 219 . . . . . . . . . 10 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → ((𝐹 “ {𝑦}) ∩ 𝑧 ∈ ran 𝐺(𝐺 “ {𝑧})) = (𝐹 “ {𝑦}))
8368, 82syl5req 2873 . . . . . . . . 9 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑦}) = 𝑧 ∈ ran 𝐺((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})))
8483fveq2d 6670 . . . . . . . 8 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑦})) = (vol‘ 𝑧 ∈ ran 𝐺((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
8563sumeq2dv 15052 . . . . . . . 8 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧) = Σ𝑧 ∈ ran 𝐺(vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
8667, 84, 853eqtr4d 2870 . . . . . . 7 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑦})) = Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧))
8786oveq2d 7167 . . . . . 6 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝑦 · (vol‘(𝐹 “ {𝑦}))) = (𝑦 · Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧)))
8853recnd 10661 . . . . . . 7 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → 𝑦 ∈ ℂ)
8965recnd 10661 . . . . . . 7 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℂ)
9041, 88, 89fsummulc2 15131 . . . . . 6 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝑦 · Σ𝑧 ∈ ran 𝐺(𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
9187, 90eqtrd 2860 . . . . 5 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → (𝑦 · (vol‘(𝐹 “ {𝑦}))) = Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
9291sumeq2dv 15052 . . . 4 (𝜑 → Σ𝑦 ∈ (ran 𝐹 ∖ {0})(𝑦 · (vol‘(𝐹 “ {𝑦}))) = Σ𝑦 ∈ (ran 𝐹 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
93 difssd 4112 . . . . 5 (𝜑 → (ran 𝐹 ∖ {0}) ⊆ ran 𝐹)
9454recnd 10661 . . . . . . 7 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℂ)
9594, 89mulcld 10653 . . . . . 6 (((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
9641, 95fsumcl 15082 . . . . 5 ((𝜑𝑦 ∈ (ran 𝐹 ∖ {0})) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) ∈ ℂ)
97 dfin4 4247 . . . . . . . 8 (ran 𝐹 ∩ {0}) = (ran 𝐹 ∖ (ran 𝐹 ∖ {0}))
98 inss2 4209 . . . . . . . 8 (ran 𝐹 ∩ {0}) ⊆ {0}
9997, 98eqsstrri 4005 . . . . . . 7 (ran 𝐹 ∖ (ran 𝐹 ∖ {0})) ⊆ {0}
10099sseli 3966 . . . . . 6 (𝑦 ∈ (ran 𝐹 ∖ (ran 𝐹 ∖ {0})) → 𝑦 ∈ {0})
101 elsni 4580 . . . . . . . . . . 11 (𝑦 ∈ {0} → 𝑦 = 0)
102101ad2antlr 723 . . . . . . . . . 10 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 = 0)
103102oveq1d 7166 . . . . . . . . 9 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) = (0 · (𝑦𝐼𝑧)))
10415ad2antrr 722 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝐼:(ℝ × ℝ)⟶ℝ)
105 0re 10635 . . . . . . . . . . . . 13 0 ∈ ℝ
106102, 105syl6eqel 2925 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
10720adantlr 711 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
108104, 106, 107fovrnd 7313 . . . . . . . . . . 11 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℝ)
109108recnd 10661 . . . . . . . . . 10 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝐼𝑧) ∈ ℂ)
110109mul02d 10830 . . . . . . . . 9 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (0 · (𝑦𝐼𝑧)) = 0)
111103, 110eqtrd 2860 . . . . . . . 8 (((𝜑𝑦 ∈ {0}) ∧ 𝑧 ∈ ran 𝐺) → (𝑦 · (𝑦𝐼𝑧)) = 0)
112111sumeq2dv 15052 . . . . . . 7 ((𝜑𝑦 ∈ {0}) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺0)
1136adantr 481 . . . . . . . . 9 ((𝜑𝑦 ∈ {0}) → ran 𝐺 ∈ Fin)
114113olcd 872 . . . . . . . 8 ((𝜑𝑦 ∈ {0}) → (ran 𝐺 ⊆ (ℤ‘0) ∨ ran 𝐺 ∈ Fin))
115 sumz 15071 . . . . . . . 8 ((ran 𝐺 ⊆ (ℤ‘0) ∨ ran 𝐺 ∈ Fin) → Σ𝑧 ∈ ran 𝐺0 = 0)
116114, 115syl 17 . . . . . . 7 ((𝜑𝑦 ∈ {0}) → Σ𝑧 ∈ ran 𝐺0 = 0)
117112, 116eqtrd 2860 . . . . . 6 ((𝜑𝑦 ∈ {0}) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = 0)
118100, 117sylan2 592 . . . . 5 ((𝜑𝑦 ∈ (ran 𝐹 ∖ (ran 𝐹 ∖ {0}))) → Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = 0)
11993, 96, 118, 3fsumss 15074 . . . 4 (𝜑 → Σ𝑦 ∈ (ran 𝐹 ∖ {0})Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
12039, 92, 1193eqtrd 2864 . . 3 (𝜑 → (∫1𝐹) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)))
121 itg1val 24199 . . . . 5 (𝐺 ∈ dom ∫1 → (∫1𝐺) = Σ𝑧 ∈ (ran 𝐺 ∖ {0})(𝑧 · (vol‘(𝐺 “ {𝑧}))))
1224, 121syl 17 . . . 4 (𝜑 → (∫1𝐺) = Σ𝑧 ∈ (ran 𝐺 ∖ {0})(𝑧 · (vol‘(𝐺 “ {𝑧}))))
1239adantr 481 . . . . . . . . 9 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝐹:ℝ⟶ℝ)
1243adantr 481 . . . . . . . . 9 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → ran 𝐹 ∈ Fin)
125 inss1 4208 . . . . . . . . . 10 ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {𝑦})
126125a1i 11 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {𝑦}))
12745ad2antrr 722 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝐹 “ {𝑦}) ∈ dom vol)
12848ad2antrr 722 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝐺 “ {𝑧}) ∈ dom vol)
129127, 128, 50syl2anc 584 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
13010adantr 481 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → ran 𝐹 ⊆ ℝ)
131130sselda 3970 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
13219ssdifssd 4122 . . . . . . . . . . . . 13 (𝜑 → (ran 𝐺 ∖ {0}) ⊆ ℝ)
133132sselda 3970 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝑧 ∈ ℝ)
134133adantr 481 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
135 eldifsni 4720 . . . . . . . . . . . . 13 (𝑧 ∈ (ran 𝐺 ∖ {0}) → 𝑧 ≠ 0)
136135ad2antlr 723 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ≠ 0)
137 simpr 485 . . . . . . . . . . . . 13 ((𝑦 = 0 ∧ 𝑧 = 0) → 𝑧 = 0)
138137necon3ai 3045 . . . . . . . . . . . 12 (𝑧 ≠ 0 → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
139136, 138syl 17 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → ¬ (𝑦 = 0 ∧ 𝑧 = 0))
140131, 134, 139, 62syl21anc 835 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) = (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
14115ad2antrr 722 . . . . . . . . . . 11 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
142141, 131, 134fovrnd 7313 . . . . . . . . . 10 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
143140, 142eqeltrrd 2918 . . . . . . . . 9 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
144123, 124, 126, 129, 143itg1addlem1 24208 . . . . . . . 8 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘ 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))) = Σ𝑦 ∈ ran 𝐹(vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
145 incom 4181 . . . . . . . . . . . . 13 ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ((𝐺 “ {𝑧}) ∩ (𝐹 “ {𝑦}))
146145a1i 11 . . . . . . . . . . . 12 (𝑦 ∈ ran 𝐹 → ((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ((𝐺 “ {𝑧}) ∩ (𝐹 “ {𝑦})))
147146iuneq2i 4936 . . . . . . . . . . 11 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = 𝑦 ∈ ran 𝐹((𝐺 “ {𝑧}) ∩ (𝐹 “ {𝑦}))
148 iunin2 4989 . . . . . . . . . . 11 𝑦 ∈ ran 𝐹((𝐺 “ {𝑧}) ∩ (𝐹 “ {𝑦})) = ((𝐺 “ {𝑧}) ∩ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}))
149147, 148eqtri 2848 . . . . . . . . . 10 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})) = ((𝐺 “ {𝑧}) ∩ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}))
150 cnvimass 5946 . . . . . . . . . . . . 13 (𝐺 “ {𝑧}) ⊆ dom 𝐺
15118fdmd 6519 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐺 = ℝ)
152151adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → dom 𝐺 = ℝ)
153150, 152sseqtrid 4022 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ⊆ ℝ)
154 iunid 4980 . . . . . . . . . . . . . . 15 𝑦 ∈ ran 𝐹{𝑦} = ran 𝐹
155154imaeq2i 5924 . . . . . . . . . . . . . 14 (𝐹 𝑦 ∈ ran 𝐹{𝑦}) = (𝐹 “ ran 𝐹)
156 imaiun 7001 . . . . . . . . . . . . . 14 (𝐹 𝑦 ∈ ran 𝐹{𝑦}) = 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦})
157 cnvimarndm 5947 . . . . . . . . . . . . . 14 (𝐹 “ ran 𝐹) = dom 𝐹
158155, 156, 1573eqtr3i 2856 . . . . . . . . . . . . 13 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}) = dom 𝐹
1599fdmd 6519 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐹 = ℝ)
160159adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → dom 𝐹 = ℝ)
161158, 160syl5eq 2872 . . . . . . . . . . . 12 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}) = ℝ)
162153, 161sseqtrrd 4011 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ⊆ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}))
163 df-ss 3955 . . . . . . . . . . 11 ((𝐺 “ {𝑧}) ⊆ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦}) ↔ ((𝐺 “ {𝑧}) ∩ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦})) = (𝐺 “ {𝑧}))
164162, 163sylib 219 . . . . . . . . . 10 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → ((𝐺 “ {𝑧}) ∩ 𝑦 ∈ ran 𝐹(𝐹 “ {𝑦})) = (𝐺 “ {𝑧}))
165149, 164syl5req 2873 . . . . . . . . 9 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) = 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧})))
166165fveq2d 6670 . . . . . . . 8 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) = (vol‘ 𝑦 ∈ ran 𝐹((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
167140sumeq2dv 15052 . . . . . . . 8 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧) = Σ𝑦 ∈ ran 𝐹(vol‘((𝐹 “ {𝑦}) ∩ (𝐺 “ {𝑧}))))
168144, 166, 1673eqtr4d 2870 . . . . . . 7 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) = Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧))
169168oveq2d 7167 . . . . . 6 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝑧 · (vol‘(𝐺 “ {𝑧}))) = (𝑧 · Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧)))
170133recnd 10661 . . . . . . 7 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → 𝑧 ∈ ℂ)
171142recnd 10661 . . . . . . 7 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℂ)
172124, 170, 171fsummulc2 15131 . . . . . 6 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝑧 · Σ𝑦 ∈ ran 𝐹(𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
173169, 172eqtrd 2860 . . . . 5 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝑧 · (vol‘(𝐺 “ {𝑧}))) = Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
174173sumeq2dv 15052 . . . 4 (𝜑 → Σ𝑧 ∈ (ran 𝐺 ∖ {0})(𝑧 · (vol‘(𝐺 “ {𝑧}))) = Σ𝑧 ∈ (ran 𝐺 ∖ {0})Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
175 difssd 4112 . . . . . 6 (𝜑 → (ran 𝐺 ∖ {0}) ⊆ ran 𝐺)
176170adantr 481 . . . . . . . 8 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
177176, 171mulcld 10653 . . . . . . 7 (((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
178124, 177fsumcl 15082 . . . . . 6 ((𝜑𝑧 ∈ (ran 𝐺 ∖ {0})) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
179 dfin4 4247 . . . . . . . . 9 (ran 𝐺 ∩ {0}) = (ran 𝐺 ∖ (ran 𝐺 ∖ {0}))
180 inss2 4209 . . . . . . . . 9 (ran 𝐺 ∩ {0}) ⊆ {0}
181179, 180eqsstrri 4005 . . . . . . . 8 (ran 𝐺 ∖ (ran 𝐺 ∖ {0})) ⊆ {0}
182181sseli 3966 . . . . . . 7 (𝑧 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {0})) → 𝑧 ∈ {0})
183 elsni 4580 . . . . . . . . . . . 12 (𝑧 ∈ {0} → 𝑧 = 0)
184183ad2antlr 723 . . . . . . . . . . 11 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 = 0)
185184oveq1d 7166 . . . . . . . . . 10 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) = (0 · (𝑦𝐼𝑧)))
18615ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
18711adantlr 711 . . . . . . . . . . . . 13 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
188184, 105syl6eqel 2925 . . . . . . . . . . . . 13 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
189186, 187, 188fovrnd 7313 . . . . . . . . . . . 12 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
190189recnd 10661 . . . . . . . . . . 11 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℂ)
191190mul02d 10830 . . . . . . . . . 10 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (0 · (𝑦𝐼𝑧)) = 0)
192185, 191eqtrd 2860 . . . . . . . . 9 (((𝜑𝑧 ∈ {0}) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) = 0)
193192sumeq2dv 15052 . . . . . . . 8 ((𝜑𝑧 ∈ {0}) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹0)
1943adantr 481 . . . . . . . . . 10 ((𝜑𝑧 ∈ {0}) → ran 𝐹 ∈ Fin)
195194olcd 872 . . . . . . . . 9 ((𝜑𝑧 ∈ {0}) → (ran 𝐹 ⊆ (ℤ‘0) ∨ ran 𝐹 ∈ Fin))
196 sumz 15071 . . . . . . . . 9 ((ran 𝐹 ⊆ (ℤ‘0) ∨ ran 𝐹 ∈ Fin) → Σ𝑦 ∈ ran 𝐹0 = 0)
197195, 196syl 17 . . . . . . . 8 ((𝜑𝑧 ∈ {0}) → Σ𝑦 ∈ ran 𝐹0 = 0)
198193, 197eqtrd 2860 . . . . . . 7 ((𝜑𝑧 ∈ {0}) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = 0)
199182, 198sylan2 592 . . . . . 6 ((𝜑𝑧 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {0}))) → Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = 0)
200175, 178, 199, 6fsumss 15074 . . . . 5 (𝜑 → Σ𝑧 ∈ (ran 𝐺 ∖ {0})Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)))
20120adantr 481 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℝ)
202201recnd 10661 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑧 ∈ ℂ)
20315ad2antrr 722 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝐼:(ℝ × ℝ)⟶ℝ)
20410adantr 481 . . . . . . . . . . 11 ((𝜑𝑧 ∈ ran 𝐺) → ran 𝐹 ⊆ ℝ)
205204sselda 3970 . . . . . . . . . 10 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → 𝑦 ∈ ℝ)
206203, 205, 201fovrnd 7313 . . . . . . . . 9 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℝ)
207206recnd 10661 . . . . . . . 8 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑦𝐼𝑧) ∈ ℂ)
208202, 207mulcld 10653 . . . . . . 7 (((𝜑𝑧 ∈ ran 𝐺) ∧ 𝑦 ∈ ran 𝐹) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
209208anasss 467 . . . . . 6 ((𝜑 ∧ (𝑧 ∈ ran 𝐺𝑦 ∈ ran 𝐹)) → (𝑧 · (𝑦𝐼𝑧)) ∈ ℂ)
2106, 3, 209fsumcom 15122 . . . . 5 (𝜑 → Σ𝑧 ∈ ran 𝐺Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)))
211200, 210eqtrd 2860 . . . 4 (𝜑 → Σ𝑧 ∈ (ran 𝐺 ∖ {0})Σ𝑦 ∈ ran 𝐹(𝑧 · (𝑦𝐼𝑧)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)))
212122, 174, 2113eqtrd 2864 . . 3 (𝜑 → (∫1𝐺) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧)))
213120, 212oveq12d 7169 . 2 (𝜑 → ((∫1𝐹) + (∫1𝐺)) = (Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑦 · (𝑦𝐼𝑧)) + Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺(𝑧 · (𝑦𝐼𝑧))))
21429, 37, 2133eqtr4d 2870 1 (𝜑 → (∫1‘(𝐹f + 𝐺)) = ((∫1𝐹) + (∫1𝐺)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 843   = wceq 1530  wcel 2106  wne 3020  cdif 3936  cin 3938  wss 3939  ifcif 4469  {csn 4563   ciun 4916   × cxp 5551  ccnv 5552  dom cdm 5553  ran crn 5554  cres 5555  cima 5556  wf 6347  cfv 6351  (class class class)co 7151  cmpo 7153  f cof 7400  Fincfn 8501  cc 10527  cr 10528  0cc0 10529   + caddc 10532   · cmul 10534  cuz 12235  Σcsu 15035  volcvol 23979  1citg1 24131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-13 2385  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-addf 10608
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-disj 5028  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-map 8401  df-pm 8402  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-inf 8899  df-oi 8966  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12383  df-xadd 12501  df-ioo 12735  df-ico 12737  df-icc 12738  df-fz 12886  df-fzo 13027  df-fl 13155  df-seq 13363  df-exp 13423  df-hash 13684  df-cj 14451  df-re 14452  df-im 14453  df-sqrt 14587  df-abs 14588  df-clim 14838  df-sum 15036  df-xmet 20454  df-met 20455  df-ovol 23980  df-vol 23981  df-mbf 24135  df-itg1 24136
This theorem is referenced by:  itg1add  24217
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