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Theorem bj-finsumval0 36103
Description: Value of a finite sum. (Contributed by BJ, 9-Jun-2019.) (Proof shortened by AV, 5-May-2021.)
Hypotheses
Ref Expression
bj-finsumval0.1 (𝜑𝐴 ∈ CMnd)
bj-finsumval0.2 (𝜑𝐼 ∈ Fin)
bj-finsumval0.3 (𝜑𝐵:𝐼⟶(Base‘𝐴))
Assertion
Ref Expression
bj-finsumval0 (𝜑 → (𝐴 FinSum 𝐵) = (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)))))
Distinct variable groups:   𝐴,𝑠,𝑓,𝑚,𝑛   𝐵,𝑓,𝑚,𝑛,𝑠   𝑓,𝐼,𝑛   𝜑,𝑓,𝑚,𝑠
Allowed substitution hints:   𝜑(𝑛)   𝐼(𝑚,𝑠)

Proof of Theorem bj-finsumval0
Dummy variables 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 7406 . 2 (𝐴 FinSum 𝐵) = ( FinSum ‘⟨𝐴, 𝐵⟩)
2 df-bj-finsum 36102 . . 3 FinSum = (𝑥 ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))} ↦ (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚))))
3 simpr 486 . . . . . . . . . 10 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → 𝑥 = ⟨𝐴, 𝐵⟩)
43fveq2d 6891 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (1st𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
5 bj-finsumval0.1 . . . . . . . . . . 11 (𝜑𝐴 ∈ CMnd)
65adantr 482 . . . . . . . . . 10 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → 𝐴 ∈ CMnd)
7 bj-finsumval0.3 . . . . . . . . . . . 12 (𝜑𝐵:𝐼⟶(Base‘𝐴))
8 bj-finsumval0.2 . . . . . . . . . . . 12 (𝜑𝐼 ∈ Fin)
97, 8fexd 7223 . . . . . . . . . . 11 (𝜑𝐵 ∈ V)
109adantr 482 . . . . . . . . . 10 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → 𝐵 ∈ V)
11 op1stg 7981 . . . . . . . . . 10 ((𝐴 ∈ CMnd ∧ 𝐵 ∈ V) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
126, 10, 11syl2anc 585 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
134, 12eqtrd 2773 . . . . . . . 8 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (1st𝑥) = 𝐴)
143fveq2d 6891 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (2nd𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
15 op2ndg 7982 . . . . . . . . . 10 ((𝐴 ∈ CMnd ∧ 𝐵 ∈ V) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
166, 10, 15syl2anc 585 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
1714, 16eqtrd 2773 . . . . . . . 8 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (2nd𝑥) = 𝐵)
1817dmeqd 5902 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → dom (2nd𝑥) = dom 𝐵)
197fdmd 6724 . . . . . . . . . 10 (𝜑 → dom 𝐵 = 𝐼)
2019adantr 482 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → dom 𝐵 = 𝐼)
2118, 20eqtrd 2773 . . . . . . . 8 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → dom (2nd𝑥) = 𝐼)
22 f1oeq3 6819 . . . . . . . . . . . . . . 15 (dom (2nd𝑥) = 𝐼 → (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ↔ 𝑓:(1...𝑚)–1-1-onto𝐼))
2322biimpd 228 . . . . . . . . . . . . . 14 (dom (2nd𝑥) = 𝐼 → (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) → 𝑓:(1...𝑚)–1-1-onto𝐼))
2423ad2antll 728 . . . . . . . . . . . . 13 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) → 𝑓:(1...𝑚)–1-1-onto𝐼))
2524adantrd 493 . . . . . . . . . . . 12 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚)) → 𝑓:(1...𝑚)–1-1-onto𝐼))
2625adantr 482 . . . . . . . . . . 11 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚)) → 𝑓:(1...𝑚)–1-1-onto𝐼))
27 eqidd 2734 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 1 = 1)
28 simprl 770 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → (1st𝑥) = 𝐴)
2928fveq2d 6891 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → (+g‘(1st𝑥)) = (+g𝐴))
3029adantrr 716 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (+g‘(1st𝑥)) = (+g𝐴))
31 simprrl 780 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → (2nd𝑥) = 𝐵)
3231adantr 482 . . . . . . . . . . . . . . . . . . . 20 (((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → (2nd𝑥) = 𝐵)
3332fveq1d 6889 . . . . . . . . . . . . . . . . . . 19 (((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → ((2nd𝑥)‘(𝑓𝑛)) = (𝐵‘(𝑓𝑛)))
3433mpteq2dva 5246 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))) = (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))
3534adantrr 716 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))) = (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))
3627, 30, 35seqeq123d 13970 . . . . . . . . . . . . . . . 16 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛)))) = seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛)))))
37 simprr 772 . . . . . . . . . . . . . . . . . 18 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → dom (2nd𝑥) = 𝐼)
3837anim1ci 617 . . . . . . . . . . . . . . . . 17 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼))
39 hashfz1 14301 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ0 → (♯‘(1...𝑚)) = 𝑚)
4039eqcomd 2739 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ℕ0𝑚 = (♯‘(1...𝑚)))
4140ad2antrl 727 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → 𝑚 = (♯‘(1...𝑚)))
42 fzfid 13933 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → (1...𝑚) ∈ Fin)
43 19.8a 2175 . . . . . . . . . . . . . . . . . . . 20 (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) → ∃𝑓 𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥))
4443adantr 482 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → ∃𝑓 𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥))
45 hasheqf1oi 14306 . . . . . . . . . . . . . . . . . . 19 ((1...𝑚) ∈ Fin → (∃𝑓 𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) → (♯‘(1...𝑚)) = (♯‘dom (2nd𝑥))))
4642, 44, 45sylc 65 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → (♯‘(1...𝑚)) = (♯‘dom (2nd𝑥)))
47 simprr 772 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → dom (2nd𝑥) = 𝐼)
4847fveq2d 6891 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → (♯‘dom (2nd𝑥)) = (♯‘𝐼))
4941, 46, 483eqtrd 2777 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → 𝑚 = (♯‘𝐼))
5038, 49sylan2 594 . . . . . . . . . . . . . . . 16 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑚 = (♯‘𝐼))
5136, 50fveq12d 6894 . . . . . . . . . . . . . . 15 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚) = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)))
5251eqeq2d 2744 . . . . . . . . . . . . . 14 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚) ↔ 𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼))))
5352biimpd 228 . . . . . . . . . . . . 13 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚) → 𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼))))
5453impancom 453 . . . . . . . . . . . 12 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚)) → ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → 𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼))))
5554com12 32 . . . . . . . . . . 11 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚)) → 𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼))))
5626, 55jcad 514 . . . . . . . . . 10 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚)) → (𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)))))
5722biimprd 247 . . . . . . . . . . . . . 14 (dom (2nd𝑥) = 𝐼 → (𝑓:(1...𝑚)–1-1-onto𝐼𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥)))
5857ad2antll 728 . . . . . . . . . . . . 13 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → (𝑓:(1...𝑚)–1-1-onto𝐼𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥)))
5958adantr 482 . . . . . . . . . . . 12 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → (𝑓:(1...𝑚)–1-1-onto𝐼𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥)))
6059adantrd 493 . . . . . . . . . . 11 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → ((𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼))) → 𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥)))
61 eqidd 2734 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 1 = 1)
62 simpl 484 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → (1st𝑥) = 𝐴)
63 tru 1546 . . . . . . . . . . . . . . . . . . . . 21
6462, 63jctir 522 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → ((1st𝑥) = 𝐴 ∧ ⊤))
6564ad2antrl 727 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → ((1st𝑥) = 𝐴 ∧ ⊤))
66 simpl 484 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑥) = 𝐴 ∧ ⊤) → (1st𝑥) = 𝐴)
6766eqcomd 2739 . . . . . . . . . . . . . . . . . . 19 (((1st𝑥) = 𝐴 ∧ ⊤) → 𝐴 = (1st𝑥))
6865, 67syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝐴 = (1st𝑥))
6968fveq2d 6891 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (+g𝐴) = (+g‘(1st𝑥)))
70 simpl 484 . . . . . . . . . . . . . . . . . . . . . . 23 (((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼) → (2nd𝑥) = 𝐵)
7170eqcomd 2739 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼) → 𝐵 = (2nd𝑥))
7271ad2antll 728 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → 𝐵 = (2nd𝑥))
7372adantr 482 . . . . . . . . . . . . . . . . . . . 20 (((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → 𝐵 = (2nd𝑥))
7473fveq1d 6889 . . . . . . . . . . . . . . . . . . 19 (((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → (𝐵‘(𝑓𝑛)) = ((2nd𝑥)‘(𝑓𝑛)))
7574adantlrr 720 . . . . . . . . . . . . . . . . . 18 (((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) ∧ 𝑛 ∈ ℕ) → (𝐵‘(𝑓𝑛)) = ((2nd𝑥)‘(𝑓𝑛)))
7675mpteq2dva 5246 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))) = (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))
7761, 69, 76seqeq123d 13970 . . . . . . . . . . . . . . . 16 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛)))) = seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛)))))
7859impcom 409 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥))
79 simprr 772 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑚 ∈ ℕ0)
8037ad2antrl 727 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → dom (2nd𝑥) = 𝐼)
8178, 79, 80, 49syl12anc 836 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑚 = (♯‘𝐼))
8281eqcomd 2739 . . . . . . . . . . . . . . . 16 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (♯‘𝐼) = 𝑚)
8377, 82fveq12d 6894 . . . . . . . . . . . . . . 15 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)) = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚))
8483eqeq2d 2744 . . . . . . . . . . . . . 14 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)) ↔ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚)))
8584biimpd 228 . . . . . . . . . . . . 13 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)) → 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚)))
8685impancom 453 . . . . . . . . . . . 12 ((𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼))) → ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚)))
8786com12 32 . . . . . . . . . . 11 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → ((𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼))) → 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚)))
8860, 87jcad 514 . . . . . . . . . 10 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → ((𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼))) → (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚))))
8956, 88impbid 211 . . . . . . . . 9 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)))))
9089ex 414 . . . . . . . 8 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → (𝑚 ∈ ℕ0 → ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼))))))
9113, 17, 21, 90syl12anc 836 . . . . . . 7 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (𝑚 ∈ ℕ0 → ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼))))))
9291imp 408 . . . . . 6 (((𝜑𝑥 = ⟨𝐴, 𝐵⟩) ∧ 𝑚 ∈ ℕ0) → ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)))))
9392exbidv 1925 . . . . 5 (((𝜑𝑥 = ⟨𝐴, 𝐵⟩) ∧ 𝑚 ∈ ℕ0) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)))))
9493rexbidva 3177 . . . 4 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (∃𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚)) ↔ ∃𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)))))
9594iotabidv 6523 . . 3 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ 𝑠 = (seq1((+g‘(1st𝑥)), (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))‘𝑚))) = (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)))))
96 eleq1 2822 . . . . . . . . . 10 (𝑡 = 𝐼 → (𝑡 ∈ Fin ↔ 𝐼 ∈ Fin))
97 feq2 6695 . . . . . . . . . 10 (𝑡 = 𝐼 → (𝐵:𝑡⟶(Base‘𝐴) ↔ 𝐵:𝐼⟶(Base‘𝐴)))
9896, 97anbi12d 632 . . . . . . . . 9 (𝑡 = 𝐼 → ((𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴)) ↔ (𝐼 ∈ Fin ∧ 𝐵:𝐼⟶(Base‘𝐴))))
9998ceqsexgv 3640 . . . . . . . 8 (𝐼 ∈ Fin → (∃𝑡(𝑡 = 𝐼 ∧ (𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴))) ↔ (𝐼 ∈ Fin ∧ 𝐵:𝐼⟶(Base‘𝐴))))
1008, 99syl 17 . . . . . . 7 (𝜑 → (∃𝑡(𝑡 = 𝐼 ∧ (𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴))) ↔ (𝐼 ∈ Fin ∧ 𝐵:𝐼⟶(Base‘𝐴))))
1018, 7, 100mpbir2and 712 . . . . . 6 (𝜑 → ∃𝑡(𝑡 = 𝐼 ∧ (𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴))))
102 exsimpr 1873 . . . . . 6 (∃𝑡(𝑡 = 𝐼 ∧ (𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴))) → ∃𝑡(𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴)))
103101, 102syl 17 . . . . 5 (𝜑 → ∃𝑡(𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴)))
104 df-rex 3072 . . . . 5 (∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴) ↔ ∃𝑡(𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴)))
105103, 104sylibr 233 . . . 4 (𝜑 → ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴))
106 eleq1 2822 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 ∈ CMnd ↔ 𝐴 ∈ CMnd))
107 fveq2 6887 . . . . . . . . 9 (𝑦 = 𝐴 → (Base‘𝑦) = (Base‘𝐴))
108107feq3d 6700 . . . . . . . 8 (𝑦 = 𝐴 → (𝑧:𝑡⟶(Base‘𝑦) ↔ 𝑧:𝑡⟶(Base‘𝐴)))
109108rexbidv 3179 . . . . . . 7 (𝑦 = 𝐴 → (∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦) ↔ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴)))
110106, 109anbi12d 632 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦)) ↔ (𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴))))
111 feq1 6694 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧:𝑡⟶(Base‘𝐴) ↔ 𝐵:𝑡⟶(Base‘𝐴)))
112111rexbidv 3179 . . . . . . 7 (𝑧 = 𝐵 → (∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴) ↔ ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴)))
113112anbi2d 630 . . . . . 6 (𝑧 = 𝐵 → ((𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴)) ↔ (𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴))))
114110, 113opelopabg 5536 . . . . 5 ((𝐴 ∈ CMnd ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))} ↔ (𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴))))
1155, 9, 114syl2anc 585 . . . 4 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))} ↔ (𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴))))
1165, 105, 115mpbir2and 712 . . 3 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))})
117 iotaex 6512 . . . 4 (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)))) ∈ V
118117a1i 11 . . 3 (𝜑 → (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)))) ∈ V)
1192, 95, 116, 118fvmptd2 7001 . 2 (𝜑 → ( FinSum ‘⟨𝐴, 𝐵⟩) = (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)))))
1201, 119eqtrid 2785 1 (𝜑 → (𝐴 FinSum 𝐵) = (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wtru 1543  wex 1782  wcel 2107  wrex 3071  Vcvv 3475  cop 4632  {copab 5208  cmpt 5229  dom cdm 5674  cio 6489  wf 6535  1-1-ontowf1o 6538  cfv 6539  (class class class)co 7403  1st c1st 7967  2nd c2nd 7968  Fincfn 8934  1c1 11106  cn 12207  0cn0 12467  ...cfz 13479  seqcseq 13961  chash 14285  Basecbs 17139  +gcplusg 17192  CMndccmn 19640   FinSum cfinsum 36101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5283  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-int 4949  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-tr 5264  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6296  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7359  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8260  df-wrecs 8291  df-recs 8365  df-rdg 8404  df-1o 8460  df-er 8698  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-card 9929  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11441  df-neg 11442  df-nn 12208  df-n0 12468  df-z 12554  df-uz 12818  df-fz 13480  df-seq 13962  df-hash 14286  df-bj-finsum 36102
This theorem is referenced by: (None)
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