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Theorem gsum2dlem2 19880
Description: Lemma for gsum2d 19881. (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 9371 . . 3 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
3 dmfi 9332 . . 3 ((𝐹 supp 0 ) ∈ Fin → dom (𝐹 supp 0 ) ∈ Fin)
42, 3syl 17 . 2 (𝜑 → dom (𝐹 supp 0 ) ∈ Fin)
5 reseq2 5975 . . . . . . . . 9 (𝑥 = ∅ → (𝐴𝑥) = (𝐴 ↾ ∅))
6 res0 5984 . . . . . . . . 9 (𝐴 ↾ ∅) = ∅
75, 6eqtrdi 2786 . . . . . . . 8 (𝑥 = ∅ → (𝐴𝑥) = ∅)
87reseq2d 5980 . . . . . . 7 (𝑥 = ∅ → (𝐹 ↾ (𝐴𝑥)) = (𝐹 ↾ ∅))
9 res0 5984 . . . . . . 7 (𝐹 ↾ ∅) = ∅
108, 9eqtrdi 2786 . . . . . 6 (𝑥 = ∅ → (𝐹 ↾ (𝐴𝑥)) = ∅)
1110oveq2d 7427 . . . . 5 (𝑥 = ∅ → (𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg ∅))
12 mpteq1 5240 . . . . . . 7 (𝑥 = ∅ → (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗 ∈ ∅ ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))
13 mpt0 6691 . . . . . . 7 (𝑗 ∈ ∅ ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = ∅
1412, 13eqtrdi 2786 . . . . . 6 (𝑥 = ∅ → (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = ∅)
1514oveq2d 7427 . . . . 5 (𝑥 = ∅ → (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg ∅))
1611, 15eqeq12d 2746 . . . 4 (𝑥 = ∅ → ((𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) ↔ (𝐺 Σg ∅) = (𝐺 Σg ∅)))
1716imbi2d 339 . . 3 (𝑥 = ∅ → ((𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) ↔ (𝜑 → (𝐺 Σg ∅) = (𝐺 Σg ∅))))
18 reseq2 5975 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
1918reseq2d 5980 . . . . . 6 (𝑥 = 𝑦 → (𝐹 ↾ (𝐴𝑥)) = (𝐹 ↾ (𝐴𝑦)))
2019oveq2d 7427 . . . . 5 (𝑥 = 𝑦 → (𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝐹 ↾ (𝐴𝑦))))
21 mpteq1 5240 . . . . . 6 (𝑥 = 𝑦 → (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))
2221oveq2d 7427 . . . . 5 (𝑥 = 𝑦 → (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))
2320, 22eqeq12d 2746 . . . 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 339 . . 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 5975 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴𝑥) = (𝐴 ↾ (𝑦 ∪ {𝑧})))
2625reseq2d 5980 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 ↾ (𝐴𝑥)) = (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))))
2726oveq2d 7427 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))))
28 mpteq1 5240 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))
2928oveq2d 7427 . . . . 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 2746 . . . 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 339 . . 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 5975 . . . . . . 7 (𝑥 = dom (𝐹 supp 0 ) → (𝐴𝑥) = (𝐴 ↾ dom (𝐹 supp 0 )))
3332reseq2d 5980 . . . . . 6 (𝑥 = dom (𝐹 supp 0 ) → (𝐹 ↾ (𝐴𝑥)) = (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 ))))
3433oveq2d 7427 . . . . 5 (𝑥 = dom (𝐹 supp 0 ) → (𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))))
35 mpteq1 5240 . . . . . 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 7427 . . . . 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 2746 . . . 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 339 . . 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 2731 . . 3 (𝜑 → (𝐺 Σg ∅) = (𝐺 Σg ∅))
40 oveq1 7418 . . . . . 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 2730 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
44 gsum2d.g . . . . . . . . . 10 (𝜑𝐺 ∈ CMnd)
4544adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝐺 ∈ CMnd)
46 gsum2d.a . . . . . . . . . . 11 (𝜑𝐴𝑉)
4746resexd 6027 . . . . . . . . . 10 (𝜑 → (𝐴 ↾ (𝑦 ∪ {𝑧})) ∈ V)
4847adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐴 ↾ (𝑦 ∪ {𝑧})) ∈ V)
49 gsum2d.f . . . . . . . . . . 11 (𝜑𝐹:𝐴𝐵)
50 resss 6005 . . . . . . . . . . 11 (𝐴 ↾ (𝑦 ∪ {𝑧})) ⊆ 𝐴
51 fssres 6756 . . . . . . . . . . 11 ((𝐹:𝐴𝐵 ∧ (𝐴 ↾ (𝑦 ∪ {𝑧})) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))):(𝐴 ↾ (𝑦 ∪ {𝑧}))⟶𝐵)
5249, 50, 51sylancl 584 . . . . . . . . . 10 (𝜑 → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))):(𝐴 ↾ (𝑦 ∪ {𝑧}))⟶𝐵)
5352adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))):(𝐴 ↾ (𝑦 ∪ {𝑧}))⟶𝐵)
5449ffund 6720 . . . . . . . . . . . 12 (𝜑 → Fun 𝐹)
5554funresd 6590 . . . . . . . . . . 11 (𝜑 → Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))))
5655adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))))
572adantr 479 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹 supp 0 ) ∈ Fin)
5849, 46fexd 7230 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
5942fvexi 6904 . . . . . . . . . . . . 13 0 ∈ V
60 ressuppss 8170 . . . . . . . . . . . . 13 ((𝐹 ∈ V ∧ 0 ∈ V) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ⊆ (𝐹 supp 0 ))
6158, 59, 60sylancl 584 . . . . . . . . . . . 12 (𝜑 → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ⊆ (𝐹 supp 0 ))
6261adantr 479 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ⊆ (𝐹 supp 0 ))
6357, 62ssfid 9269 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ∈ Fin)
6458resexd 6027 . . . . . . . . . . . 12 (𝜑 → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∈ V)
65 isfsupp 9367 . . . . . . . . . . . 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 584 . . . . . . . . . . 11 (𝜑 → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) finSupp 0 ↔ (Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∧ ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ∈ Fin)))
6766adantr 479 . . . . . . . . . 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 709 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) finSupp 0 )
69 simprr 769 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ¬ 𝑧𝑦)
70 disjsn 4714 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
7169, 70sylibr 233 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
7271reseq2d 5980 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐴 ↾ (𝑦 ∩ {𝑧})) = (𝐴 ↾ ∅))
73 resindi 5996 . . . . . . . . . 10 (𝐴 ↾ (𝑦 ∩ {𝑧})) = ((𝐴𝑦) ∩ (𝐴 ↾ {𝑧}))
7472, 73, 63eqtr3g 2793 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((𝐴𝑦) ∩ (𝐴 ↾ {𝑧})) = ∅)
75 resundi 5994 . . . . . . . . . 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 19837 . . . . . . . 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 4171 . . . . . . . . . . 11 𝑦 ⊆ (𝑦 ∪ {𝑧})
79 ssres2 6008 . . . . . . . . . . 11 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (𝐴𝑦) ⊆ (𝐴 ↾ (𝑦 ∪ {𝑧})))
80 resabs1 6010 . . . . . . . . . . 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 7422 . . . . . . . . 9 (𝐺 Σg ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴𝑦))) = (𝐺 Σg (𝐹 ↾ (𝐴𝑦)))
83 ssun2 4172 . . . . . . . . . . 11 {𝑧} ⊆ (𝑦 ∪ {𝑧})
84 ssres2 6008 . . . . . . . . . . 11 ({𝑧} ⊆ (𝑦 ∪ {𝑧}) → (𝐴 ↾ {𝑧}) ⊆ (𝐴 ↾ (𝑦 ∪ {𝑧})))
85 resabs1 6010 . . . . . . . . . . 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 7422 . . . . . . . . 9 (𝐺 Σg ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ {𝑧}))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))
8882, 87oveq12i 7423 . . . . . . . 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 2786 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = ((𝐺 Σg (𝐹 ↾ (𝐴𝑦)))(+g𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))))
90 simprl 767 . . . . . . . . 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 19879 . . . . . . . . . 10 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵)
9594ad2antrr 722 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) ∧ 𝑗𝑦) → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵)
96 vex 3476 . . . . . . . . . 10 𝑧 ∈ V
9796a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑧 ∈ V)
98 sneq 4637 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑧 → {𝑗} = {𝑧})
9998imaeq2d 6058 . . . . . . . . . . . . . . 15 (𝑗 = 𝑧 → (𝐴 “ {𝑗}) = (𝐴 “ {𝑧}))
100 oveq1 7418 . . . . . . . . . . . . . . 15 (𝑗 = 𝑧 → (𝑗𝐹𝑘) = (𝑧𝐹𝑘))
10199, 100mpteq12dv 5238 . . . . . . . . . . . . . 14 (𝑗 = 𝑧 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘)))
102101oveq2d 7427 . . . . . . . . . . . . 13 (𝑗 = 𝑧 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))))
103102eleq1d 2816 . . . . . . . . . . . 12 (𝑗 = 𝑧 → ((𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵 ↔ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵))
104103imbi2d 339 . . . . . . . . . . 11 (𝑗 = 𝑧 → ((𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵) ↔ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵)))
105104, 94chvarvv 2000 . . . . . . . . . 10 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵)
106105adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵)
10741, 43, 45, 90, 95, 97, 69, 106, 102gsumunsn 19869 . . . . . . . 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 5980 . . . . . . . . . . . . . . 15 (𝑗 = 𝑧 → (𝐴 ↾ {𝑗}) = (𝐴 ↾ {𝑧}))
109108reseq2d 5980 . . . . . . . . . . . . . 14 (𝑗 = 𝑧 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = (𝐹 ↾ (𝐴 ↾ {𝑧})))
110109oveq2d 7427 . . . . . . . . . . . . 13 (𝑗 = 𝑧 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗}))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))
111102, 110eqeq12d 2746 . . . . . . . . . . . 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 339 . . . . . . . . . . 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 7908 . . . . . . . . . . . . . 14 (𝐴𝑉 → (𝐴 “ {𝑗}) ∈ V)
11446, 113syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝐴 “ {𝑗}) ∈ V)
115 vex 3476 . . . . . . . . . . . . . . . 16 𝑗 ∈ V
116 vex 3476 . . . . . . . . . . . . . . . 16 𝑘 ∈ V
117115, 116elimasn 6087 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝐴 “ {𝑗}) ↔ ⟨𝑗, 𝑘⟩ ∈ 𝐴)
118 df-ov 7414 . . . . . . . . . . . . . . . 16 (𝑗𝐹𝑘) = (𝐹‘⟨𝑗, 𝑘⟩)
11949ffvelcdmda 7085 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ 𝐴) → (𝐹‘⟨𝑗, 𝑘⟩) ∈ 𝐵)
120118, 119eqeltrid 2835 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ 𝐴) → (𝑗𝐹𝑘) ∈ 𝐵)
121117, 120sylan2b 592 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (𝐴 “ {𝑗})) → (𝑗𝐹𝑘) ∈ 𝐵)
122121fmpttd 7115 . . . . . . . . . . . . 13 (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)):(𝐴 “ {𝑗})⟶𝐵)
123 funmpt 6585 . . . . . . . . . . . . . . 15 Fun (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))
124123a1i 11 . . . . . . . . . . . . . 14 (𝜑 → Fun (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))
125 rnfi 9337 . . . . . . . . . . . . . . . 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 5941 . . . . . . . . . . . . . . . . . . . 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 611 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 )) → (⟨𝑗, 𝑘⟩ ∈ 𝐴 ∧ ¬ ⟨𝑗, 𝑘⟩ ∈ (𝐹 supp 0 )))
131 eldif 3957 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 )) ↔ (𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 )))
132 eldif 3957 . . . . . . . . . . . . . . . . . 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 4004 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
13559a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑0 ∈ V)
13649, 134, 46, 135suppssr 8183 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ (𝐴 ∖ (𝐹 supp 0 ))) → (𝐹‘⟨𝑗, 𝑘⟩) = 0 )
137118, 136eqtrid 2782 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ (𝐴 ∖ (𝐹 supp 0 ))) → (𝑗𝐹𝑘) = 0 )
138133, 137sylan2 591 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 ))) → (𝑗𝐹𝑘) = 0 )
139138, 114suppss2 8187 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ⊆ ran (𝐹 supp 0 ))
140126, 139ssfid 9269 . . . . . . . . . . . . . 14 (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ∈ Fin)
141114mptexd 7227 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∈ V)
142 isfsupp 9367 . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . 14 (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) finSupp 0 ↔ (Fun (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∧ ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ∈ Fin)))
144124, 140, 143mpbir2and 709 . . . . . . . . . . . . 13 (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) finSupp 0 )
145 2ndconst 8089 . . . . . . . . . . . . . 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 19825 . . . . . . . . . . . 12 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))))))
148 1st2nd2 8016 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
149 xp1st 8009 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (1st𝑥) ∈ {𝑗})
150 elsni 4644 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑥) ∈ {𝑗} → (1st𝑥) = 𝑗)
151149, 150syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (1st𝑥) = 𝑗)
152151opeq1d 4878 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → ⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑗, (2nd𝑥)⟩)
153148, 152eqtrd 2770 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → 𝑥 = ⟨𝑗, (2nd𝑥)⟩)
154153fveq2d 6894 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (𝐹𝑥) = (𝐹‘⟨𝑗, (2nd𝑥)⟩))
155 df-ov 7414 . . . . . . . . . . . . . . . 16 (𝑗𝐹(2nd𝑥)) = (𝐹‘⟨𝑗, (2nd𝑥)⟩)
156154, 155eqtr4di 2788 . . . . . . . . . . . . . . 15 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (𝐹𝑥) = (𝑗𝐹(2nd𝑥)))
157156mpteq2ia 5250 . . . . . . . . . . . . . 14 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹𝑥)) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝑗𝐹(2nd𝑥)))
15849feqmptd 6959 . . . . . . . . . . . . . . . 16 (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
159158reseq1d 5979 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = ((𝑥𝐴 ↦ (𝐹𝑥)) ↾ (𝐴 ↾ {𝑗})))
160 resss 6005 . . . . . . . . . . . . . . . . 17 (𝐴 ↾ {𝑗}) ⊆ 𝐴
161 resmpt 6036 . . . . . . . . . . . . . . . . 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 6283 . . . . . . . . . . . . . . . . 17 (𝐴 ↾ {𝑗}) = ({𝑗} × (𝐴 “ {𝑗}))
164163mpteq1i 5243 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝐴 ↾ {𝑗}) ↦ (𝐹𝑥)) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹𝑥))
165162, 164eqtri 2758 . . . . . . . . . . . . . . 15 ((𝑥𝐴 ↦ (𝐹𝑥)) ↾ (𝐴 ↾ {𝑗})) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹𝑥))
166159, 165eqtrdi 2786 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹𝑥)))
167 xp2nd 8010 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (2nd𝑥) ∈ (𝐴 “ {𝑗}))
168167adantl 480 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗}))) → (2nd𝑥) ∈ (𝐴 “ {𝑗}))
169 fo2nd 7998 . . . . . . . . . . . . . . . . . . 19 2nd :V–onto→V
170 fof 6804 . . . . . . . . . . . . . . . . . . 19 (2nd :V–onto→V → 2nd :V⟶V)
171169, 170mp1i 13 . . . . . . . . . . . . . . . . . 18 (𝜑 → 2nd :V⟶V)
172171feqmptd 6959 . . . . . . . . . . . . . . . . 17 (𝜑 → 2nd = (𝑥 ∈ V ↦ (2nd𝑥)))
173172reseq1d 5979 . . . . . . . . . . . . . . . 16 (𝜑 → (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))) = ((𝑥 ∈ V ↦ (2nd𝑥)) ↾ ({𝑗} × (𝐴 “ {𝑗}))))
174 ssv 4005 . . . . . . . . . . . . . . . . 17 ({𝑗} × (𝐴 “ {𝑗})) ⊆ V
175 resmpt 6036 . . . . . . . . . . . . . . . . 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 2786 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (2nd𝑥)))
178 eqidd 2731 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))
179 oveq2 7419 . . . . . . . . . . . . . . 15 (𝑘 = (2nd𝑥) → (𝑗𝐹𝑘) = (𝑗𝐹(2nd𝑥)))
180168, 177, 178, 179fmptco 7128 . . . . . . . . . . . . . 14 (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗})))) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝑗𝐹(2nd𝑥))))
181157, 166, 1803eqtr4a 2796 . . . . . . . . . . . . 13 (𝜑 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗})))))
182181oveq2d 7427 . . . . . . . . . . . 12 (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗}))) = (𝐺 Σg ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))))))
183147, 182eqtr4d 2773 . . . . . . . . . . 11 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗}))))
184112, 183chvarvv 2000 . . . . . . . . . 10 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))
185184adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))
186185oveq2d 7427 . . . . . . . 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 2770 . . . . . . 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 2746 . . . . . 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, 188imbitrrid 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 412 . . . 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 9167 . 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 394   = wceq 1539  wcel 2104  Vcvv 3472  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4321  {csn 4627  cop 4633   class class class wbr 5147  cmpt 5230   × cxp 5673  dom cdm 5675  ran crn 5676  cres 5677  cima 5678  ccom 5679  Rel wrel 5680  Fun wfun 6536  wf 6538  ontowfo 6540  1-1-ontowf1o 6541  cfv 6542  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976   supp csupp 8148  Fincfn 8941   finSupp cfsupp 9363  Basecbs 17148  +gcplusg 17201  0gc0g 17389   Σg cgsu 17390  CMndccmn 19689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  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-iin 4999  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-fzo 13632  df-seq 13971  df-hash 14295  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-0g 17391  df-gsum 17392  df-mre 17534  df-mrc 17535  df-acs 17537  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18706  df-mulg 18987  df-cntz 19222  df-cmn 19691
This theorem is referenced by:  gsum2d  19881
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