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Theorem gsum2dlem2 19748
Description: Lemma for gsum2d 19749. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.)
Hypotheses
Ref Expression
gsum2d.b 𝐵 = (Base‘𝐺)
gsum2d.z 0 = (0g𝐺)
gsum2d.g (𝜑𝐺 ∈ CMnd)
gsum2d.a (𝜑𝐴𝑉)
gsum2d.r (𝜑 → Rel 𝐴)
gsum2d.d (𝜑𝐷𝑊)
gsum2d.s (𝜑 → dom 𝐴𝐷)
gsum2d.f (𝜑𝐹:𝐴𝐵)
gsum2d.w (𝜑𝐹 finSupp 0 )
Assertion
Ref Expression
gsum2dlem2 (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))
Distinct variable groups:   𝑗,𝑘,𝐴   𝑗,𝐹,𝑘   𝑗,𝐺,𝑘   𝜑,𝑗,𝑘   𝐵,𝑗,𝑘   𝐷,𝑗,𝑘   0 ,𝑗,𝑘
Allowed substitution hints:   𝑉(𝑗,𝑘)   𝑊(𝑗,𝑘)

Proof of Theorem gsum2dlem2
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsum2d.w . . . 4 (𝜑𝐹 finSupp 0 )
21fsuppimpd 9312 . . 3 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
3 dmfi 9274 . . 3 ((𝐹 supp 0 ) ∈ Fin → dom (𝐹 supp 0 ) ∈ Fin)
42, 3syl 17 . 2 (𝜑 → dom (𝐹 supp 0 ) ∈ Fin)
5 reseq2 5932 . . . . . . . . 9 (𝑥 = ∅ → (𝐴𝑥) = (𝐴 ↾ ∅))
6 res0 5941 . . . . . . . . 9 (𝐴 ↾ ∅) = ∅
75, 6eqtrdi 2792 . . . . . . . 8 (𝑥 = ∅ → (𝐴𝑥) = ∅)
87reseq2d 5937 . . . . . . 7 (𝑥 = ∅ → (𝐹 ↾ (𝐴𝑥)) = (𝐹 ↾ ∅))
9 res0 5941 . . . . . . 7 (𝐹 ↾ ∅) = ∅
108, 9eqtrdi 2792 . . . . . 6 (𝑥 = ∅ → (𝐹 ↾ (𝐴𝑥)) = ∅)
1110oveq2d 7373 . . . . 5 (𝑥 = ∅ → (𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg ∅))
12 mpteq1 5198 . . . . . . 7 (𝑥 = ∅ → (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗 ∈ ∅ ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))
13 mpt0 6643 . . . . . . 7 (𝑗 ∈ ∅ ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = ∅
1412, 13eqtrdi 2792 . . . . . 6 (𝑥 = ∅ → (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = ∅)
1514oveq2d 7373 . . . . 5 (𝑥 = ∅ → (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg ∅))
1611, 15eqeq12d 2752 . . . 4 (𝑥 = ∅ → ((𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) ↔ (𝐺 Σg ∅) = (𝐺 Σg ∅)))
1716imbi2d 340 . . 3 (𝑥 = ∅ → ((𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) ↔ (𝜑 → (𝐺 Σg ∅) = (𝐺 Σg ∅))))
18 reseq2 5932 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
1918reseq2d 5937 . . . . . 6 (𝑥 = 𝑦 → (𝐹 ↾ (𝐴𝑥)) = (𝐹 ↾ (𝐴𝑦)))
2019oveq2d 7373 . . . . 5 (𝑥 = 𝑦 → (𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝐹 ↾ (𝐴𝑦))))
21 mpteq1 5198 . . . . . 6 (𝑥 = 𝑦 → (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))
2221oveq2d 7373 . . . . 5 (𝑥 = 𝑦 → (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))
2320, 22eqeq12d 2752 . . . 4 (𝑥 = 𝑦 → ((𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) ↔ (𝐺 Σg (𝐹 ↾ (𝐴𝑦))) = (𝐺 Σg (𝑗𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))))
2423imbi2d 340 . . 3 (𝑥 = 𝑦 → ((𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) ↔ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴𝑦))) = (𝐺 Σg (𝑗𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))))
25 reseq2 5932 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴𝑥) = (𝐴 ↾ (𝑦 ∪ {𝑧})))
2625reseq2d 5937 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 ↾ (𝐴𝑥)) = (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))))
2726oveq2d 7373 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))))
28 mpteq1 5198 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))
2928oveq2d 7373 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))
3027, 29eqeq12d 2752 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) ↔ (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))))
3130imbi2d 340 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) ↔ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))))
32 reseq2 5932 . . . . . . 7 (𝑥 = dom (𝐹 supp 0 ) → (𝐴𝑥) = (𝐴 ↾ dom (𝐹 supp 0 )))
3332reseq2d 5937 . . . . . 6 (𝑥 = dom (𝐹 supp 0 ) → (𝐹 ↾ (𝐴𝑥)) = (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 ))))
3433oveq2d 7373 . . . . 5 (𝑥 = dom (𝐹 supp 0 ) → (𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))))
35 mpteq1 5198 . . . . . 6 (𝑥 = dom (𝐹 supp 0 ) → (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))
3635oveq2d 7373 . . . . 5 (𝑥 = dom (𝐹 supp 0 ) → (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))
3734, 36eqeq12d 2752 . . . 4 (𝑥 = dom (𝐹 supp 0 ) → ((𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) ↔ (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))))
3837imbi2d 340 . . 3 (𝑥 = dom (𝐹 supp 0 ) → ((𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) ↔ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))))
39 eqidd 2737 . . 3 (𝜑 → (𝐺 Σg ∅) = (𝐺 Σg ∅))
40 oveq1 7364 . . . . . 6 ((𝐺 Σg (𝐹 ↾ (𝐴𝑦))) = (𝐺 Σg (𝑗𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) → ((𝐺 Σg (𝐹 ↾ (𝐴𝑦)))(+g𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))) = ((𝐺 Σg (𝑗𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))(+g𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))))
41 gsum2d.b . . . . . . . . 9 𝐵 = (Base‘𝐺)
42 gsum2d.z . . . . . . . . 9 0 = (0g𝐺)
43 eqid 2736 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
44 gsum2d.g . . . . . . . . . 10 (𝜑𝐺 ∈ CMnd)
4544adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝐺 ∈ CMnd)
46 gsum2d.a . . . . . . . . . . 11 (𝜑𝐴𝑉)
4746resexd 5984 . . . . . . . . . 10 (𝜑 → (𝐴 ↾ (𝑦 ∪ {𝑧})) ∈ V)
4847adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐴 ↾ (𝑦 ∪ {𝑧})) ∈ V)
49 gsum2d.f . . . . . . . . . . 11 (𝜑𝐹:𝐴𝐵)
50 resss 5962 . . . . . . . . . . 11 (𝐴 ↾ (𝑦 ∪ {𝑧})) ⊆ 𝐴
51 fssres 6708 . . . . . . . . . . 11 ((𝐹:𝐴𝐵 ∧ (𝐴 ↾ (𝑦 ∪ {𝑧})) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))):(𝐴 ↾ (𝑦 ∪ {𝑧}))⟶𝐵)
5249, 50, 51sylancl 586 . . . . . . . . . 10 (𝜑 → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))):(𝐴 ↾ (𝑦 ∪ {𝑧}))⟶𝐵)
5352adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))):(𝐴 ↾ (𝑦 ∪ {𝑧}))⟶𝐵)
5449ffund 6672 . . . . . . . . . . . 12 (𝜑 → Fun 𝐹)
5554funresd 6544 . . . . . . . . . . 11 (𝜑 → Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))))
5655adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))))
572adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹 supp 0 ) ∈ Fin)
5849, 46fexd 7177 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
5942fvexi 6856 . . . . . . . . . . . . 13 0 ∈ V
60 ressuppss 8114 . . . . . . . . . . . . 13 ((𝐹 ∈ V ∧ 0 ∈ V) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ⊆ (𝐹 supp 0 ))
6158, 59, 60sylancl 586 . . . . . . . . . . . 12 (𝜑 → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ⊆ (𝐹 supp 0 ))
6261adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ⊆ (𝐹 supp 0 ))
6357, 62ssfid 9211 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ∈ Fin)
6458resexd 5984 . . . . . . . . . . . 12 (𝜑 → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∈ V)
65 isfsupp 9309 . . . . . . . . . . . 12 (((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∈ V ∧ 0 ∈ V) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) finSupp 0 ↔ (Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∧ ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ∈ Fin)))
6664, 59, 65sylancl 586 . . . . . . . . . . 11 (𝜑 → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) finSupp 0 ↔ (Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∧ ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ∈ Fin)))
6766adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) finSupp 0 ↔ (Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∧ ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ∈ Fin)))
6856, 63, 67mpbir2and 711 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) finSupp 0 )
69 simprr 771 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ¬ 𝑧𝑦)
70 disjsn 4672 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
7169, 70sylibr 233 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
7271reseq2d 5937 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐴 ↾ (𝑦 ∩ {𝑧})) = (𝐴 ↾ ∅))
73 resindi 5953 . . . . . . . . . 10 (𝐴 ↾ (𝑦 ∩ {𝑧})) = ((𝐴𝑦) ∩ (𝐴 ↾ {𝑧}))
7472, 73, 63eqtr3g 2799 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((𝐴𝑦) ∩ (𝐴 ↾ {𝑧})) = ∅)
75 resundi 5951 . . . . . . . . . 10 (𝐴 ↾ (𝑦 ∪ {𝑧})) = ((𝐴𝑦) ∪ (𝐴 ↾ {𝑧}))
7675a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐴 ↾ (𝑦 ∪ {𝑧})) = ((𝐴𝑦) ∪ (𝐴 ↾ {𝑧})))
7741, 42, 43, 45, 48, 53, 68, 74, 76gsumsplit 19705 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = ((𝐺 Σg ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴𝑦)))(+g𝐺)(𝐺 Σg ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ {𝑧})))))
78 ssun1 4132 . . . . . . . . . . 11 𝑦 ⊆ (𝑦 ∪ {𝑧})
79 ssres2 5965 . . . . . . . . . . 11 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (𝐴𝑦) ⊆ (𝐴 ↾ (𝑦 ∪ {𝑧})))
80 resabs1 5967 . . . . . . . . . . 11 ((𝐴𝑦) ⊆ (𝐴 ↾ (𝑦 ∪ {𝑧})) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴𝑦)) = (𝐹 ↾ (𝐴𝑦)))
8178, 79, 80mp2b 10 . . . . . . . . . 10 ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴𝑦)) = (𝐹 ↾ (𝐴𝑦))
8281oveq2i 7368 . . . . . . . . 9 (𝐺 Σg ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴𝑦))) = (𝐺 Σg (𝐹 ↾ (𝐴𝑦)))
83 ssun2 4133 . . . . . . . . . . 11 {𝑧} ⊆ (𝑦 ∪ {𝑧})
84 ssres2 5965 . . . . . . . . . . 11 ({𝑧} ⊆ (𝑦 ∪ {𝑧}) → (𝐴 ↾ {𝑧}) ⊆ (𝐴 ↾ (𝑦 ∪ {𝑧})))
85 resabs1 5967 . . . . . . . . . . 11 ((𝐴 ↾ {𝑧}) ⊆ (𝐴 ↾ (𝑦 ∪ {𝑧})) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ {𝑧})) = (𝐹 ↾ (𝐴 ↾ {𝑧})))
8683, 84, 85mp2b 10 . . . . . . . . . 10 ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ {𝑧})) = (𝐹 ↾ (𝐴 ↾ {𝑧}))
8786oveq2i 7368 . . . . . . . . 9 (𝐺 Σg ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ {𝑧}))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))
8882, 87oveq12i 7369 . . . . . . . 8 ((𝐺 Σg ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴𝑦)))(+g𝐺)(𝐺 Σg ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ {𝑧})))) = ((𝐺 Σg (𝐹 ↾ (𝐴𝑦)))(+g𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))
8977, 88eqtrdi 2792 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = ((𝐺 Σg (𝐹 ↾ (𝐴𝑦)))(+g𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))))
90 simprl 769 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑦 ∈ Fin)
91 gsum2d.r . . . . . . . . . . 11 (𝜑 → Rel 𝐴)
92 gsum2d.d . . . . . . . . . . 11 (𝜑𝐷𝑊)
93 gsum2d.s . . . . . . . . . . 11 (𝜑 → dom 𝐴𝐷)
9441, 42, 44, 46, 91, 92, 93, 49, 1gsum2dlem1 19747 . . . . . . . . . 10 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵)
9594ad2antrr 724 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) ∧ 𝑗𝑦) → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵)
96 vex 3449 . . . . . . . . . 10 𝑧 ∈ V
9796a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑧 ∈ V)
98 sneq 4596 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑧 → {𝑗} = {𝑧})
9998imaeq2d 6013 . . . . . . . . . . . . . . 15 (𝑗 = 𝑧 → (𝐴 “ {𝑗}) = (𝐴 “ {𝑧}))
100 oveq1 7364 . . . . . . . . . . . . . . 15 (𝑗 = 𝑧 → (𝑗𝐹𝑘) = (𝑧𝐹𝑘))
10199, 100mpteq12dv 5196 . . . . . . . . . . . . . 14 (𝑗 = 𝑧 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘)))
102101oveq2d 7373 . . . . . . . . . . . . 13 (𝑗 = 𝑧 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))))
103102eleq1d 2822 . . . . . . . . . . . 12 (𝑗 = 𝑧 → ((𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵 ↔ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵))
104103imbi2d 340 . . . . . . . . . . 11 (𝑗 = 𝑧 → ((𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵) ↔ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵)))
105104, 94chvarvv 2002 . . . . . . . . . 10 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵)
106105adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵)
10741, 43, 45, 90, 95, 97, 69, 106, 102gsumunsn 19737 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = ((𝐺 Σg (𝑗𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))(+g𝐺)(𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘)))))
10898reseq2d 5937 . . . . . . . . . . . . . . 15 (𝑗 = 𝑧 → (𝐴 ↾ {𝑗}) = (𝐴 ↾ {𝑧}))
109108reseq2d 5937 . . . . . . . . . . . . . 14 (𝑗 = 𝑧 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = (𝐹 ↾ (𝐴 ↾ {𝑧})))
110109oveq2d 7373 . . . . . . . . . . . . 13 (𝑗 = 𝑧 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗}))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))
111102, 110eqeq12d 2752 . . . . . . . . . . . 12 (𝑗 = 𝑧 → ((𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗}))) ↔ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))))
112111imbi2d 340 . . . . . . . . . . 11 (𝑗 = 𝑧 → ((𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗})))) ↔ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))))
113 imaexg 7852 . . . . . . . . . . . . . 14 (𝐴𝑉 → (𝐴 “ {𝑗}) ∈ V)
11446, 113syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝐴 “ {𝑗}) ∈ V)
115 vex 3449 . . . . . . . . . . . . . . . 16 𝑗 ∈ V
116 vex 3449 . . . . . . . . . . . . . . . 16 𝑘 ∈ V
117115, 116elimasn 6041 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝐴 “ {𝑗}) ↔ ⟨𝑗, 𝑘⟩ ∈ 𝐴)
118 df-ov 7360 . . . . . . . . . . . . . . . 16 (𝑗𝐹𝑘) = (𝐹‘⟨𝑗, 𝑘⟩)
11949ffvelcdmda 7035 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ 𝐴) → (𝐹‘⟨𝑗, 𝑘⟩) ∈ 𝐵)
120118, 119eqeltrid 2842 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ 𝐴) → (𝑗𝐹𝑘) ∈ 𝐵)
121117, 120sylan2b 594 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (𝐴 “ {𝑗})) → (𝑗𝐹𝑘) ∈ 𝐵)
122121fmpttd 7063 . . . . . . . . . . . . 13 (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)):(𝐴 “ {𝑗})⟶𝐵)
123 funmpt 6539 . . . . . . . . . . . . . . 15 Fun (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))
124123a1i 11 . . . . . . . . . . . . . 14 (𝜑 → Fun (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))
125 rnfi 9279 . . . . . . . . . . . . . . . 16 ((𝐹 supp 0 ) ∈ Fin → ran (𝐹 supp 0 ) ∈ Fin)
1262, 125syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ran (𝐹 supp 0 ) ∈ Fin)
127117biimpi 215 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (𝐴 “ {𝑗}) → ⟨𝑗, 𝑘⟩ ∈ 𝐴)
128115, 116opelrn 5898 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑗, 𝑘⟩ ∈ (𝐹 supp 0 ) → 𝑘 ∈ ran (𝐹 supp 0 ))
129128con3i 154 . . . . . . . . . . . . . . . . . . 19 𝑘 ∈ ran (𝐹 supp 0 ) → ¬ ⟨𝑗, 𝑘⟩ ∈ (𝐹 supp 0 ))
130127, 129anim12i 613 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 )) → (⟨𝑗, 𝑘⟩ ∈ 𝐴 ∧ ¬ ⟨𝑗, 𝑘⟩ ∈ (𝐹 supp 0 )))
131 eldif 3920 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 )) ↔ (𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 )))
132 eldif 3920 . . . . . . . . . . . . . . . . . 18 (⟨𝑗, 𝑘⟩ ∈ (𝐴 ∖ (𝐹 supp 0 )) ↔ (⟨𝑗, 𝑘⟩ ∈ 𝐴 ∧ ¬ ⟨𝑗, 𝑘⟩ ∈ (𝐹 supp 0 )))
133130, 131, 1323imtr4i 291 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 )) → ⟨𝑗, 𝑘⟩ ∈ (𝐴 ∖ (𝐹 supp 0 )))
134 ssidd 3967 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
13559a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑0 ∈ V)
13649, 134, 46, 135suppssr 8127 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ (𝐴 ∖ (𝐹 supp 0 ))) → (𝐹‘⟨𝑗, 𝑘⟩) = 0 )
137118, 136eqtrid 2788 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ (𝐴 ∖ (𝐹 supp 0 ))) → (𝑗𝐹𝑘) = 0 )
138133, 137sylan2 593 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 ))) → (𝑗𝐹𝑘) = 0 )
139138, 114suppss2 8131 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ⊆ ran (𝐹 supp 0 ))
140126, 139ssfid 9211 . . . . . . . . . . . . . 14 (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ∈ Fin)
141114mptexd 7174 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∈ V)
142 isfsupp 9309 . . . . . . . . . . . . . . 15 (((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∈ V ∧ 0 ∈ V) → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) finSupp 0 ↔ (Fun (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∧ ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ∈ Fin)))
143141, 59, 142sylancl 586 . . . . . . . . . . . . . 14 (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) finSupp 0 ↔ (Fun (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∧ ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ∈ Fin)))
144124, 140, 143mpbir2and 711 . . . . . . . . . . . . 13 (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) finSupp 0 )
145 2ndconst 8033 . . . . . . . . . . . . . 14 (𝑗 ∈ V → (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))):({𝑗} × (𝐴 “ {𝑗}))–1-1-onto→(𝐴 “ {𝑗}))
146115, 145mp1i 13 . . . . . . . . . . . . 13 (𝜑 → (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))):({𝑗} × (𝐴 “ {𝑗}))–1-1-onto→(𝐴 “ {𝑗}))
14741, 42, 44, 114, 122, 144, 146gsumf1o 19693 . . . . . . . . . . . 12 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))))))
148 1st2nd2 7960 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
149 xp1st 7953 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (1st𝑥) ∈ {𝑗})
150 elsni 4603 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑥) ∈ {𝑗} → (1st𝑥) = 𝑗)
151149, 150syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (1st𝑥) = 𝑗)
152151opeq1d 4836 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → ⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑗, (2nd𝑥)⟩)
153148, 152eqtrd 2776 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → 𝑥 = ⟨𝑗, (2nd𝑥)⟩)
154153fveq2d 6846 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (𝐹𝑥) = (𝐹‘⟨𝑗, (2nd𝑥)⟩))
155 df-ov 7360 . . . . . . . . . . . . . . . 16 (𝑗𝐹(2nd𝑥)) = (𝐹‘⟨𝑗, (2nd𝑥)⟩)
156154, 155eqtr4di 2794 . . . . . . . . . . . . . . 15 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (𝐹𝑥) = (𝑗𝐹(2nd𝑥)))
157156mpteq2ia 5208 . . . . . . . . . . . . . 14 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹𝑥)) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝑗𝐹(2nd𝑥)))
15849feqmptd 6910 . . . . . . . . . . . . . . . 16 (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
159158reseq1d 5936 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = ((𝑥𝐴 ↦ (𝐹𝑥)) ↾ (𝐴 ↾ {𝑗})))
160 resss 5962 . . . . . . . . . . . . . . . . 17 (𝐴 ↾ {𝑗}) ⊆ 𝐴
161 resmpt 5991 . . . . . . . . . . . . . . . . 17 ((𝐴 ↾ {𝑗}) ⊆ 𝐴 → ((𝑥𝐴 ↦ (𝐹𝑥)) ↾ (𝐴 ↾ {𝑗})) = (𝑥 ∈ (𝐴 ↾ {𝑗}) ↦ (𝐹𝑥)))
162160, 161ax-mp 5 . . . . . . . . . . . . . . . 16 ((𝑥𝐴 ↦ (𝐹𝑥)) ↾ (𝐴 ↾ {𝑗})) = (𝑥 ∈ (𝐴 ↾ {𝑗}) ↦ (𝐹𝑥))
163 ressn 6237 . . . . . . . . . . . . . . . . 17 (𝐴 ↾ {𝑗}) = ({𝑗} × (𝐴 “ {𝑗}))
164163mpteq1i 5201 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝐴 ↾ {𝑗}) ↦ (𝐹𝑥)) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹𝑥))
165162, 164eqtri 2764 . . . . . . . . . . . . . . 15 ((𝑥𝐴 ↦ (𝐹𝑥)) ↾ (𝐴 ↾ {𝑗})) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹𝑥))
166159, 165eqtrdi 2792 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹𝑥)))
167 xp2nd 7954 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (2nd𝑥) ∈ (𝐴 “ {𝑗}))
168167adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗}))) → (2nd𝑥) ∈ (𝐴 “ {𝑗}))
169 fo2nd 7942 . . . . . . . . . . . . . . . . . . 19 2nd :V–onto→V
170 fof 6756 . . . . . . . . . . . . . . . . . . 19 (2nd :V–onto→V → 2nd :V⟶V)
171169, 170mp1i 13 . . . . . . . . . . . . . . . . . 18 (𝜑 → 2nd :V⟶V)
172171feqmptd 6910 . . . . . . . . . . . . . . . . 17 (𝜑 → 2nd = (𝑥 ∈ V ↦ (2nd𝑥)))
173172reseq1d 5936 . . . . . . . . . . . . . . . 16 (𝜑 → (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))) = ((𝑥 ∈ V ↦ (2nd𝑥)) ↾ ({𝑗} × (𝐴 “ {𝑗}))))
174 ssv 3968 . . . . . . . . . . . . . . . . 17 ({𝑗} × (𝐴 “ {𝑗})) ⊆ V
175 resmpt 5991 . . . . . . . . . . . . . . . . 17 (({𝑗} × (𝐴 “ {𝑗})) ⊆ V → ((𝑥 ∈ V ↦ (2nd𝑥)) ↾ ({𝑗} × (𝐴 “ {𝑗}))) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (2nd𝑥)))
176174, 175ax-mp 5 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ V ↦ (2nd𝑥)) ↾ ({𝑗} × (𝐴 “ {𝑗}))) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (2nd𝑥))
177173, 176eqtrdi 2792 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (2nd𝑥)))
178 eqidd 2737 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))
179 oveq2 7365 . . . . . . . . . . . . . . 15 (𝑘 = (2nd𝑥) → (𝑗𝐹𝑘) = (𝑗𝐹(2nd𝑥)))
180168, 177, 178, 179fmptco 7075 . . . . . . . . . . . . . 14 (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗})))) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝑗𝐹(2nd𝑥))))
181157, 166, 1803eqtr4a 2802 . . . . . . . . . . . . 13 (𝜑 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗})))))
182181oveq2d 7373 . . . . . . . . . . . 12 (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗}))) = (𝐺 Σg ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))))))
183147, 182eqtr4d 2779 . . . . . . . . . . 11 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗}))))
184112, 183chvarvv 2002 . . . . . . . . . 10 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))
185184adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))
186185oveq2d 7373 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((𝐺 Σg (𝑗𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))(+g𝐺)(𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘)))) = ((𝐺 Σg (𝑗𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))(+g𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))))
187107, 186eqtrd 2776 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = ((𝐺 Σg (𝑗𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))(+g𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))))
18889, 187eqeq12d 2752 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) ↔ ((𝐺 Σg (𝐹 ↾ (𝐴𝑦)))(+g𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))) = ((𝐺 Σg (𝑗𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))(+g𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))))
18940, 188syl5ibr 245 . . . . 5 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((𝐺 Σg (𝐹 ↾ (𝐴𝑦))) = (𝐺 Σg (𝑗𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))))
190189expcom 414 . . . 4 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝜑 → ((𝐺 Σg (𝐹 ↾ (𝐴𝑦))) = (𝐺 Σg (𝑗𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))))
191190a2d 29 . . 3 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴𝑦))) = (𝐺 Σg (𝑗𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) → (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))))
19217, 24, 31, 38, 39, 191findcard2s 9109 . 2 (dom (𝐹 supp 0 ) ∈ Fin → (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))))
1934, 192mpcom 38 1 (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  cdif 3907  cun 3908  cin 3909  wss 3910  c0 4282  {csn 4586  cop 4592   class class class wbr 5105  cmpt 5188   × cxp 5631  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  ccom 5637  Rel wrel 5638  Fun wfun 6490  wf 6492  ontowfo 6494  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920   supp csupp 8092  Fincfn 8883   finSupp cfsupp 9305  Basecbs 17083  +gcplusg 17133  0gc0g 17321   Σg cgsu 17322  CMndccmn 19562
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564
This theorem is referenced by:  gsum2d  19749
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