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Theorem gsum2dlem2 18142
Description: Lemma for gsum2d 18143. (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 8143 . . 3 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
3 dmfi 8107 . . 3 ((𝐹 supp 0 ) ∈ Fin → dom (𝐹 supp 0 ) ∈ Fin)
42, 3syl 17 . 2 (𝜑 → dom (𝐹 supp 0 ) ∈ Fin)
5 reseq2 5299 . . . . . . . . 9 (𝑥 = ∅ → (𝐴𝑥) = (𝐴 ↾ ∅))
6 res0 5308 . . . . . . . . 9 (𝐴 ↾ ∅) = ∅
75, 6syl6eq 2659 . . . . . . . 8 (𝑥 = ∅ → (𝐴𝑥) = ∅)
87reseq2d 5304 . . . . . . 7 (𝑥 = ∅ → (𝐹 ↾ (𝐴𝑥)) = (𝐹 ↾ ∅))
9 res0 5308 . . . . . . 7 (𝐹 ↾ ∅) = ∅
108, 9syl6eq 2659 . . . . . 6 (𝑥 = ∅ → (𝐹 ↾ (𝐴𝑥)) = ∅)
1110oveq2d 6543 . . . . 5 (𝑥 = ∅ → (𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg ∅))
12 mpteq1 4659 . . . . . . 7 (𝑥 = ∅ → (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗 ∈ ∅ ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))
13 mpt0 5920 . . . . . . 7 (𝑗 ∈ ∅ ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = ∅
1412, 13syl6eq 2659 . . . . . 6 (𝑥 = ∅ → (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = ∅)
1514oveq2d 6543 . . . . 5 (𝑥 = ∅ → (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg ∅))
1611, 15eqeq12d 2624 . . . 4 (𝑥 = ∅ → ((𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) ↔ (𝐺 Σg ∅) = (𝐺 Σg ∅)))
1716imbi2d 328 . . 3 (𝑥 = ∅ → ((𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) ↔ (𝜑 → (𝐺 Σg ∅) = (𝐺 Σg ∅))))
18 reseq2 5299 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
1918reseq2d 5304 . . . . . 6 (𝑥 = 𝑦 → (𝐹 ↾ (𝐴𝑥)) = (𝐹 ↾ (𝐴𝑦)))
2019oveq2d 6543 . . . . 5 (𝑥 = 𝑦 → (𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝐹 ↾ (𝐴𝑦))))
21 mpteq1 4659 . . . . . 6 (𝑥 = 𝑦 → (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))
2221oveq2d 6543 . . . . 5 (𝑥 = 𝑦 → (𝐺 Σg (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))
2320, 22eqeq12d 2624 . . . 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 328 . . 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 5299 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴𝑥) = (𝐴 ↾ (𝑦 ∪ {𝑧})))
2625reseq2d 5304 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 ↾ (𝐴𝑥)) = (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))))
2726oveq2d 6543 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))))
28 mpteq1 4659 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑗𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))
2928oveq2d 6543 . . . . 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 2624 . . . 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 328 . . 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 5299 . . . . . . 7 (𝑥 = dom (𝐹 supp 0 ) → (𝐴𝑥) = (𝐴 ↾ dom (𝐹 supp 0 )))
3332reseq2d 5304 . . . . . 6 (𝑥 = dom (𝐹 supp 0 ) → (𝐹 ↾ (𝐴𝑥)) = (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 ))))
3433oveq2d 6543 . . . . 5 (𝑥 = dom (𝐹 supp 0 ) → (𝐺 Σg (𝐹 ↾ (𝐴𝑥))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))))
35 mpteq1 4659 . . . . . 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 6543 . . . . 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 2624 . . . 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 328 . . 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 2610 . . 3 (𝜑 → (𝐺 Σg ∅) = (𝐺 Σg ∅))
40 oveq1 6534 . . . . . 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 2609 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
44 gsum2d.g . . . . . . . . . 10 (𝜑𝐺 ∈ CMnd)
4544adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝐺 ∈ CMnd)
46 gsum2d.a . . . . . . . . . . 11 (𝜑𝐴𝑉)
47 resexg 5349 . . . . . . . . . . 11 (𝐴𝑉 → (𝐴 ↾ (𝑦 ∪ {𝑧})) ∈ V)
4846, 47syl 17 . . . . . . . . . 10 (𝜑 → (𝐴 ↾ (𝑦 ∪ {𝑧})) ∈ V)
4948adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐴 ↾ (𝑦 ∪ {𝑧})) ∈ V)
50 gsum2d.f . . . . . . . . . . 11 (𝜑𝐹:𝐴𝐵)
51 resss 5329 . . . . . . . . . . 11 (𝐴 ↾ (𝑦 ∪ {𝑧})) ⊆ 𝐴
52 fssres 5968 . . . . . . . . . . 11 ((𝐹:𝐴𝐵 ∧ (𝐴 ↾ (𝑦 ∪ {𝑧})) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))):(𝐴 ↾ (𝑦 ∪ {𝑧}))⟶𝐵)
5350, 51, 52sylancl 692 . . . . . . . . . 10 (𝜑 → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))):(𝐴 ↾ (𝑦 ∪ {𝑧}))⟶𝐵)
5453adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))):(𝐴 ↾ (𝑦 ∪ {𝑧}))⟶𝐵)
55 ffun 5947 . . . . . . . . . . . . 13 (𝐹:𝐴𝐵 → Fun 𝐹)
5650, 55syl 17 . . . . . . . . . . . 12 (𝜑 → Fun 𝐹)
57 funres 5829 . . . . . . . . . . . 12 (Fun 𝐹 → Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))))
5856, 57syl 17 . . . . . . . . . . 11 (𝜑 → Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))))
5958adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))))
602adantr 479 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹 supp 0 ) ∈ Fin)
61 fex 6372 . . . . . . . . . . . . . 14 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
6250, 46, 61syl2anc 690 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
63 fvex 6098 . . . . . . . . . . . . . 14 (0g𝐺) ∈ V
6442, 63eqeltri 2683 . . . . . . . . . . . . 13 0 ∈ V
65 ressuppss 7179 . . . . . . . . . . . . 13 ((𝐹 ∈ V ∧ 0 ∈ V) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ⊆ (𝐹 supp 0 ))
6662, 64, 65sylancl 692 . . . . . . . . . . . 12 (𝜑 → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ⊆ (𝐹 supp 0 ))
6766adantr 479 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ⊆ (𝐹 supp 0 ))
68 ssfi 8043 . . . . . . . . . . 11 (((𝐹 supp 0 ) ∈ Fin ∧ ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ⊆ (𝐹 supp 0 )) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ∈ Fin)
6960, 67, 68syl2anc 690 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ∈ Fin)
70 resexg 5349 . . . . . . . . . . . . 13 (𝐹 ∈ V → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∈ V)
7162, 70syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∈ V)
72 isfsupp 8140 . . . . . . . . . . . 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)))
7371, 64, 72sylancl 692 . . . . . . . . . . 11 (𝜑 → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) finSupp 0 ↔ (Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∧ ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ∈ Fin)))
7473adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) finSupp 0 ↔ (Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∧ ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ∈ Fin)))
7559, 69, 74mpbir2and 958 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) finSupp 0 )
76 simprr 791 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ¬ 𝑧𝑦)
77 disjsn 4191 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
7876, 77sylibr 222 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
7978reseq2d 5304 . . . . . . . . . 10 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐴 ↾ (𝑦 ∩ {𝑧})) = (𝐴 ↾ ∅))
80 resindi 5319 . . . . . . . . . 10 (𝐴 ↾ (𝑦 ∩ {𝑧})) = ((𝐴𝑦) ∩ (𝐴 ↾ {𝑧}))
8179, 80, 63eqtr3g 2666 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((𝐴𝑦) ∩ (𝐴 ↾ {𝑧})) = ∅)
82 resundi 5317 . . . . . . . . . 10 (𝐴 ↾ (𝑦 ∪ {𝑧})) = ((𝐴𝑦) ∪ (𝐴 ↾ {𝑧}))
8382a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐴 ↾ (𝑦 ∪ {𝑧})) = ((𝐴𝑦) ∪ (𝐴 ↾ {𝑧})))
8441, 42, 43, 45, 49, 54, 75, 81, 83gsumsplit 18100 . . . . . . . 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})))))
85 ssun1 3737 . . . . . . . . . . 11 𝑦 ⊆ (𝑦 ∪ {𝑧})
86 ssres2 5332 . . . . . . . . . . 11 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (𝐴𝑦) ⊆ (𝐴 ↾ (𝑦 ∪ {𝑧})))
87 resabs1 5334 . . . . . . . . . . 11 ((𝐴𝑦) ⊆ (𝐴 ↾ (𝑦 ∪ {𝑧})) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴𝑦)) = (𝐹 ↾ (𝐴𝑦)))
8885, 86, 87mp2b 10 . . . . . . . . . 10 ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴𝑦)) = (𝐹 ↾ (𝐴𝑦))
8988oveq2i 6538 . . . . . . . . 9 (𝐺 Σg ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴𝑦))) = (𝐺 Σg (𝐹 ↾ (𝐴𝑦)))
90 ssun2 3738 . . . . . . . . . . 11 {𝑧} ⊆ (𝑦 ∪ {𝑧})
91 ssres2 5332 . . . . . . . . . . 11 ({𝑧} ⊆ (𝑦 ∪ {𝑧}) → (𝐴 ↾ {𝑧}) ⊆ (𝐴 ↾ (𝑦 ∪ {𝑧})))
92 resabs1 5334 . . . . . . . . . . 11 ((𝐴 ↾ {𝑧}) ⊆ (𝐴 ↾ (𝑦 ∪ {𝑧})) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ {𝑧})) = (𝐹 ↾ (𝐴 ↾ {𝑧})))
9390, 91, 92mp2b 10 . . . . . . . . . 10 ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ {𝑧})) = (𝐹 ↾ (𝐴 ↾ {𝑧}))
9493oveq2i 6538 . . . . . . . . 9 (𝐺 Σg ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ {𝑧}))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))
9589, 94oveq12i 6539 . . . . . . . 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}))))
9684, 95syl6eq 2659 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = ((𝐺 Σg (𝐹 ↾ (𝐴𝑦)))(+g𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))))
97 simprl 789 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑦 ∈ Fin)
98 gsum2d.r . . . . . . . . . . 11 (𝜑 → Rel 𝐴)
99 gsum2d.d . . . . . . . . . . 11 (𝜑𝐷𝑊)
100 gsum2d.s . . . . . . . . . . 11 (𝜑 → dom 𝐴𝐷)
10141, 42, 44, 46, 98, 99, 100, 50, 1gsum2dlem1 18141 . . . . . . . . . 10 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵)
102101ad2antrr 757 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) ∧ 𝑗𝑦) → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵)
103 vex 3175 . . . . . . . . . 10 𝑧 ∈ V
104103a1i 11 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑧 ∈ V)
105 sneq 4134 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑧 → {𝑗} = {𝑧})
106105imaeq2d 5372 . . . . . . . . . . . . . . 15 (𝑗 = 𝑧 → (𝐴 “ {𝑗}) = (𝐴 “ {𝑧}))
107 oveq1 6534 . . . . . . . . . . . . . . 15 (𝑗 = 𝑧 → (𝑗𝐹𝑘) = (𝑧𝐹𝑘))
108106, 107mpteq12dv 4657 . . . . . . . . . . . . . 14 (𝑗 = 𝑧 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘)))
109108oveq2d 6543 . . . . . . . . . . . . 13 (𝑗 = 𝑧 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))))
110109eleq1d 2671 . . . . . . . . . . . 12 (𝑗 = 𝑧 → ((𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵 ↔ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵))
111110imbi2d 328 . . . . . . . . . . 11 (𝑗 = 𝑧 → ((𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵) ↔ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵)))
112111, 101chvarv 2250 . . . . . . . . . 10 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵)
113112adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵)
11441, 43, 45, 97, 102, 104, 76, 113, 109gsumunsn 18131 . . . . . . . 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)))))
115105reseq2d 5304 . . . . . . . . . . . . . . 15 (𝑗 = 𝑧 → (𝐴 ↾ {𝑗}) = (𝐴 ↾ {𝑧}))
116115reseq2d 5304 . . . . . . . . . . . . . 14 (𝑗 = 𝑧 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = (𝐹 ↾ (𝐴 ↾ {𝑧})))
117116oveq2d 6543 . . . . . . . . . . . . 13 (𝑗 = 𝑧 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗}))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))
118109, 117eqeq12d 2624 . . . . . . . . . . . 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})))))
119118imbi2d 328 . . . . . . . . . . 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}))))))
120 imaexg 6973 . . . . . . . . . . . . . 14 (𝐴𝑉 → (𝐴 “ {𝑗}) ∈ V)
12146, 120syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝐴 “ {𝑗}) ∈ V)
122 vex 3175 . . . . . . . . . . . . . . . 16 𝑗 ∈ V
123 vex 3175 . . . . . . . . . . . . . . . 16 𝑘 ∈ V
124122, 123elimasn 5396 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝐴 “ {𝑗}) ↔ ⟨𝑗, 𝑘⟩ ∈ 𝐴)
125 df-ov 6530 . . . . . . . . . . . . . . . 16 (𝑗𝐹𝑘) = (𝐹‘⟨𝑗, 𝑘⟩)
12650ffvelrnda 6252 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ 𝐴) → (𝐹‘⟨𝑗, 𝑘⟩) ∈ 𝐵)
127125, 126syl5eqel 2691 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ 𝐴) → (𝑗𝐹𝑘) ∈ 𝐵)
128124, 127sylan2b 490 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (𝐴 “ {𝑗})) → (𝑗𝐹𝑘) ∈ 𝐵)
129 eqid 2609 . . . . . . . . . . . . . 14 (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))
130128, 129fmptd 6277 . . . . . . . . . . . . 13 (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)):(𝐴 “ {𝑗})⟶𝐵)
131 funmpt 5826 . . . . . . . . . . . . . . 15 Fun (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))
132131a1i 11 . . . . . . . . . . . . . 14 (𝜑 → Fun (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))
133 rnfi 8110 . . . . . . . . . . . . . . . 16 ((𝐹 supp 0 ) ∈ Fin → ran (𝐹 supp 0 ) ∈ Fin)
1342, 133syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ran (𝐹 supp 0 ) ∈ Fin)
135124biimpi 204 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (𝐴 “ {𝑗}) → ⟨𝑗, 𝑘⟩ ∈ 𝐴)
136122, 123opelrn 5265 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑗, 𝑘⟩ ∈ (𝐹 supp 0 ) → 𝑘 ∈ ran (𝐹 supp 0 ))
137136con3i 148 . . . . . . . . . . . . . . . . . . 19 𝑘 ∈ ran (𝐹 supp 0 ) → ¬ ⟨𝑗, 𝑘⟩ ∈ (𝐹 supp 0 ))
138135, 137anim12i 587 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 )) → (⟨𝑗, 𝑘⟩ ∈ 𝐴 ∧ ¬ ⟨𝑗, 𝑘⟩ ∈ (𝐹 supp 0 )))
139 eldif 3549 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 )) ↔ (𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 )))
140 eldif 3549 . . . . . . . . . . . . . . . . . 18 (⟨𝑗, 𝑘⟩ ∈ (𝐴 ∖ (𝐹 supp 0 )) ↔ (⟨𝑗, 𝑘⟩ ∈ 𝐴 ∧ ¬ ⟨𝑗, 𝑘⟩ ∈ (𝐹 supp 0 )))
141138, 139, 1403imtr4i 279 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 )) → ⟨𝑗, 𝑘⟩ ∈ (𝐴 ∖ (𝐹 supp 0 )))
142 ssid 3586 . . . . . . . . . . . . . . . . . . . 20 (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )
143142a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
14464a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑0 ∈ V)
14550, 143, 46, 144suppssr 7191 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ (𝐴 ∖ (𝐹 supp 0 ))) → (𝐹‘⟨𝑗, 𝑘⟩) = 0 )
146125, 145syl5eq 2655 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ (𝐴 ∖ (𝐹 supp 0 ))) → (𝑗𝐹𝑘) = 0 )
147141, 146sylan2 489 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 ))) → (𝑗𝐹𝑘) = 0 )
148147, 121suppss2 7194 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ⊆ ran (𝐹 supp 0 ))
149 ssfi 8043 . . . . . . . . . . . . . . 15 ((ran (𝐹 supp 0 ) ∈ Fin ∧ ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ⊆ ran (𝐹 supp 0 )) → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ∈ Fin)
150134, 148, 149syl2anc 690 . . . . . . . . . . . . . 14 (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ∈ Fin)
151 mptexg 6367 . . . . . . . . . . . . . . . 16 ((𝐴 “ {𝑗}) ∈ V → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∈ V)
152121, 151syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∈ V)
153 isfsupp 8140 . . . . . . . . . . . . . . 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)))
154152, 64, 153sylancl 692 . . . . . . . . . . . . . 14 (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) finSupp 0 ↔ (Fun (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∧ ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ∈ Fin)))
155132, 150, 154mpbir2and 958 . . . . . . . . . . . . 13 (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) finSupp 0 )
156 2ndconst 7131 . . . . . . . . . . . . . 14 (𝑗 ∈ V → (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))):({𝑗} × (𝐴 “ {𝑗}))–1-1-onto→(𝐴 “ {𝑗}))
157122, 156mp1i 13 . . . . . . . . . . . . 13 (𝜑 → (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))):({𝑗} × (𝐴 “ {𝑗}))–1-1-onto→(𝐴 “ {𝑗}))
15841, 42, 44, 121, 130, 155, 157gsumf1o 18089 . . . . . . . . . . . 12 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))))))
159 1st2nd2 7074 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
160 xp1st 7067 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (1st𝑥) ∈ {𝑗})
161 elsni 4141 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑥) ∈ {𝑗} → (1st𝑥) = 𝑗)
162160, 161syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (1st𝑥) = 𝑗)
163162opeq1d 4340 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → ⟨(1st𝑥), (2nd𝑥)⟩ = ⟨𝑗, (2nd𝑥)⟩)
164159, 163eqtrd 2643 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → 𝑥 = ⟨𝑗, (2nd𝑥)⟩)
165164fveq2d 6092 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (𝐹𝑥) = (𝐹‘⟨𝑗, (2nd𝑥)⟩))
166 df-ov 6530 . . . . . . . . . . . . . . . 16 (𝑗𝐹(2nd𝑥)) = (𝐹‘⟨𝑗, (2nd𝑥)⟩)
167165, 166syl6eqr 2661 . . . . . . . . . . . . . . 15 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (𝐹𝑥) = (𝑗𝐹(2nd𝑥)))
168167mpteq2ia 4662 . . . . . . . . . . . . . 14 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹𝑥)) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝑗𝐹(2nd𝑥)))
16950feqmptd 6144 . . . . . . . . . . . . . . . 16 (𝜑𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
170169reseq1d 5303 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = ((𝑥𝐴 ↦ (𝐹𝑥)) ↾ (𝐴 ↾ {𝑗})))
171 resss 5329 . . . . . . . . . . . . . . . . 17 (𝐴 ↾ {𝑗}) ⊆ 𝐴
172 resmpt 5356 . . . . . . . . . . . . . . . . 17 ((𝐴 ↾ {𝑗}) ⊆ 𝐴 → ((𝑥𝐴 ↦ (𝐹𝑥)) ↾ (𝐴 ↾ {𝑗})) = (𝑥 ∈ (𝐴 ↾ {𝑗}) ↦ (𝐹𝑥)))
173171, 172ax-mp 5 . . . . . . . . . . . . . . . 16 ((𝑥𝐴 ↦ (𝐹𝑥)) ↾ (𝐴 ↾ {𝑗})) = (𝑥 ∈ (𝐴 ↾ {𝑗}) ↦ (𝐹𝑥))
174 ressn 5574 . . . . . . . . . . . . . . . . 17 (𝐴 ↾ {𝑗}) = ({𝑗} × (𝐴 “ {𝑗}))
175 mpteq1 4659 . . . . . . . . . . . . . . . . 17 ((𝐴 ↾ {𝑗}) = ({𝑗} × (𝐴 “ {𝑗})) → (𝑥 ∈ (𝐴 ↾ {𝑗}) ↦ (𝐹𝑥)) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹𝑥)))
176174, 175ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (𝐴 ↾ {𝑗}) ↦ (𝐹𝑥)) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹𝑥))
177173, 176eqtri 2631 . . . . . . . . . . . . . . 15 ((𝑥𝐴 ↦ (𝐹𝑥)) ↾ (𝐴 ↾ {𝑗})) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹𝑥))
178170, 177syl6eq 2659 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹𝑥)))
179 xp2nd 7068 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (2nd𝑥) ∈ (𝐴 “ {𝑗}))
180179adantl 480 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗}))) → (2nd𝑥) ∈ (𝐴 “ {𝑗}))
181 fo2nd 7058 . . . . . . . . . . . . . . . . . . 19 2nd :V–onto→V
182 fof 6013 . . . . . . . . . . . . . . . . . . 19 (2nd :V–onto→V → 2nd :V⟶V)
183181, 182mp1i 13 . . . . . . . . . . . . . . . . . 18 (𝜑 → 2nd :V⟶V)
184183feqmptd 6144 . . . . . . . . . . . . . . . . 17 (𝜑 → 2nd = (𝑥 ∈ V ↦ (2nd𝑥)))
185184reseq1d 5303 . . . . . . . . . . . . . . . 16 (𝜑 → (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))) = ((𝑥 ∈ V ↦ (2nd𝑥)) ↾ ({𝑗} × (𝐴 “ {𝑗}))))
186 ssv 3587 . . . . . . . . . . . . . . . . 17 ({𝑗} × (𝐴 “ {𝑗})) ⊆ V
187 resmpt 5356 . . . . . . . . . . . . . . . . 17 (({𝑗} × (𝐴 “ {𝑗})) ⊆ V → ((𝑥 ∈ V ↦ (2nd𝑥)) ↾ ({𝑗} × (𝐴 “ {𝑗}))) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (2nd𝑥)))
188186, 187ax-mp 5 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ V ↦ (2nd𝑥)) ↾ ({𝑗} × (𝐴 “ {𝑗}))) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (2nd𝑥))
189185, 188syl6eq 2659 . . . . . . . . . . . . . . 15 (𝜑 → (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (2nd𝑥)))
190 eqidd 2610 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))
191 oveq2 6535 . . . . . . . . . . . . . . 15 (𝑘 = (2nd𝑥) → (𝑗𝐹𝑘) = (𝑗𝐹(2nd𝑥)))
192180, 189, 190, 191fmptco 6288 . . . . . . . . . . . . . 14 (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗})))) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝑗𝐹(2nd𝑥))))
193168, 178, 1923eqtr4a 2669 . . . . . . . . . . . . 13 (𝜑 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗})))))
194193oveq2d 6543 . . . . . . . . . . . 12 (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗}))) = (𝐺 Σg ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))))))
195158, 194eqtr4d 2646 . . . . . . . . . . 11 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗}))))
196119, 195chvarv 2250 . . . . . . . . . 10 (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))
197196adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))
198197oveq2d 6543 . . . . . . . 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})))))
199114, 198eqtrd 2643 . . . . . . 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})))))
20096, 199eqeq12d 2624 . . . . . 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}))))))
20140, 200syl5ibr 234 . . . . 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)))))))
202201expcom 449 . . . 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))))))))
203202a2d 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))))))))
20417, 24, 31, 38, 39, 203findcard2s 8064 . 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)))))))
2054, 204mpcom 37 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 194  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  cdif 3536  cun 3537  cin 3538  wss 3539  c0 3873  {csn 4124  cop 4130   class class class wbr 4577  cmpt 4637   × cxp 5026  dom cdm 5028  ran crn 5029  cres 5030  cima 5031  ccom 5032  Rel wrel 5033  Fun wfun 5784  wf 5786  ontowfo 5788  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527  1st c1st 7035  2nd c2nd 7036   supp csupp 7160  Fincfn 7819   finSupp cfsupp 8136  Basecbs 15644  +gcplusg 15717  0gc0g 15872   Σg cgsu 15873  CMndccmn 17965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-inf2 8399  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6773  df-om 6936  df-1st 7037  df-2nd 7038  df-supp 7161  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-oadd 7429  df-er 7607  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-fsupp 8137  df-oi 8276  df-card 8626  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-nn 10871  df-2 10929  df-n0 11143  df-z 11214  df-uz 11523  df-fz 12156  df-fzo 12293  df-seq 12622  df-hash 12938  df-ndx 15647  df-slot 15648  df-base 15649  df-sets 15650  df-ress 15651  df-plusg 15730  df-0g 15874  df-gsum 15875  df-mre 16018  df-mrc 16019  df-acs 16021  df-mgm 17014  df-sgrp 17056  df-mnd 17067  df-submnd 17108  df-mulg 17313  df-cntz 17522  df-cmn 17967
This theorem is referenced by:  gsum2d  18143
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