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Theorem bj-finsumval0 36656
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 7404 . 2 (𝐴 FinSum 𝐵) = ( FinSum ‘⟨𝐴, 𝐵⟩)
2 df-bj-finsum 36655 . . 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 484 . . . . . . . . . 10 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → 𝑥 = ⟨𝐴, 𝐵⟩)
43fveq2d 6885 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (1st𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
5 bj-finsumval0.1 . . . . . . . . . . 11 (𝜑𝐴 ∈ CMnd)
65adantr 480 . . . . . . . . . 10 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → 𝐴 ∈ CMnd)
7 bj-finsumval0.3 . . . . . . . . . . . 12 (𝜑𝐵:𝐼⟶(Base‘𝐴))
8 bj-finsumval0.2 . . . . . . . . . . . 12 (𝜑𝐼 ∈ Fin)
97, 8fexd 7220 . . . . . . . . . . 11 (𝜑𝐵 ∈ V)
109adantr 480 . . . . . . . . . 10 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → 𝐵 ∈ V)
11 op1stg 7980 . . . . . . . . . 10 ((𝐴 ∈ CMnd ∧ 𝐵 ∈ V) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
126, 10, 11syl2anc 583 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
134, 12eqtrd 2764 . . . . . . . 8 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (1st𝑥) = 𝐴)
143fveq2d 6885 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (2nd𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
15 op2ndg 7981 . . . . . . . . . 10 ((𝐴 ∈ CMnd ∧ 𝐵 ∈ V) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
166, 10, 15syl2anc 583 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
1714, 16eqtrd 2764 . . . . . . . 8 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (2nd𝑥) = 𝐵)
1817dmeqd 5895 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → dom (2nd𝑥) = dom 𝐵)
197fdmd 6718 . . . . . . . . . 10 (𝜑 → dom 𝐵 = 𝐼)
2019adantr 480 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → dom 𝐵 = 𝐼)
2118, 20eqtrd 2764 . . . . . . . 8 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → dom (2nd𝑥) = 𝐼)
22 f1oeq3 6813 . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . 13 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) → 𝑓:(1...𝑚)–1-1-onto𝐼))
2524adantrd 491 . . . . . . . . . . . 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 480 . . . . . . . . . . 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 2725 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 1 = 1)
28 simprl 768 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → (1st𝑥) = 𝐴)
2928fveq2d 6885 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → (+g‘(1st𝑥)) = (+g𝐴))
3029adantrr 714 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (+g‘(1st𝑥)) = (+g𝐴))
31 simprrl 778 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → (2nd𝑥) = 𝐵)
3231adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → (2nd𝑥) = 𝐵)
3332fveq1d 6883 . . . . . . . . . . . . . . . . . . 19 (((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → ((2nd𝑥)‘(𝑓𝑛)) = (𝐵‘(𝑓𝑛)))
3433mpteq2dva 5238 . . . . . . . . . . . . . . . . . 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 714 . . . . . . . . . . . . . . . . 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 13972 . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . . . 18 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → dom (2nd𝑥) = 𝐼)
3837anim1ci 615 . . . . . . . . . . . . . . . . 17 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼))
39 hashfz1 14303 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ0 → (♯‘(1...𝑚)) = 𝑚)
4039eqcomd 2730 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ℕ0𝑚 = (♯‘(1...𝑚)))
4140ad2antrl 725 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → 𝑚 = (♯‘(1...𝑚)))
42 fzfid 13935 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → (1...𝑚) ∈ Fin)
43 19.8a 2166 . . . . . . . . . . . . . . . . . . . 20 (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) → ∃𝑓 𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥))
4443adantr 480 . . . . . . . . . . . . . . . . . . 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 14308 . . . . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → dom (2nd𝑥) = 𝐼)
4847fveq2d 6885 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → (♯‘dom (2nd𝑥)) = (♯‘𝐼))
4941, 46, 483eqtrd 2768 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → 𝑚 = (♯‘𝐼))
5038, 49sylan2 592 . . . . . . . . . . . . . . . 16 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑚 = (♯‘𝐼))
5136, 50fveq12d 6888 . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . 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 451 . . . . . . . . . . . 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 512 . . . . . . . . . 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 726 . . . . . . . . . . . . 13 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → (𝑓:(1...𝑚)–1-1-onto𝐼𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥)))
5958adantr 480 . . . . . . . . . . . 12 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → (𝑓:(1...𝑚)–1-1-onto𝐼𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥)))
6059adantrd 491 . . . . . . . . . . 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 2725 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 1 = 1)
62 simpl 482 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → (1st𝑥) = 𝐴)
63 tru 1537 . . . . . . . . . . . . . . . . . . . . 21
6462, 63jctir 520 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → ((1st𝑥) = 𝐴 ∧ ⊤))
6564ad2antrl 725 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → ((1st𝑥) = 𝐴 ∧ ⊤))
66 simpl 482 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑥) = 𝐴 ∧ ⊤) → (1st𝑥) = 𝐴)
6766eqcomd 2730 . . . . . . . . . . . . . . . . . . 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 6885 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (+g𝐴) = (+g‘(1st𝑥)))
70 simpl 482 . . . . . . . . . . . . . . . . . . . . . . 23 (((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼) → (2nd𝑥) = 𝐵)
7170eqcomd 2730 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼) → 𝐵 = (2nd𝑥))
7271ad2antll 726 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → 𝐵 = (2nd𝑥))
7372adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → 𝐵 = (2nd𝑥))
7473fveq1d 6883 . . . . . . . . . . . . . . . . . . 19 (((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → (𝐵‘(𝑓𝑛)) = ((2nd𝑥)‘(𝑓𝑛)))
7574adantlrr 718 . . . . . . . . . . . . . . . . . 18 (((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) ∧ 𝑛 ∈ ℕ) → (𝐵‘(𝑓𝑛)) = ((2nd𝑥)‘(𝑓𝑛)))
7675mpteq2dva 5238 . . . . . . . . . . . . . . . . 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 13972 . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑚 ∈ ℕ0)
8037ad2antrl 725 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → dom (2nd𝑥) = 𝐼)
8178, 79, 80, 49syl12anc 834 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑚 = (♯‘𝐼))
8281eqcomd 2730 . . . . . . . . . . . . . . . 16 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (♯‘𝐼) = 𝑚)
8377, 82fveq12d 6888 . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . 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 451 . . . . . . . . . . . 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 512 . . . . . . . . . 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 412 . . . . . . . 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 834 . . . . . . 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 406 . . . . . 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 1916 . . . . 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 3168 . . . 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 6517 . . 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 2813 . . . . . . . . . 10 (𝑡 = 𝐼 → (𝑡 ∈ Fin ↔ 𝐼 ∈ Fin))
97 feq2 6689 . . . . . . . . . 10 (𝑡 = 𝐼 → (𝐵:𝑡⟶(Base‘𝐴) ↔ 𝐵:𝐼⟶(Base‘𝐴)))
9896, 97anbi12d 630 . . . . . . . . 9 (𝑡 = 𝐼 → ((𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴)) ↔ (𝐼 ∈ Fin ∧ 𝐵:𝐼⟶(Base‘𝐴))))
9998ceqsexgv 3634 . . . . . . . 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 710 . . . . . 6 (𝜑 → ∃𝑡(𝑡 = 𝐼 ∧ (𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴))))
102 exsimpr 1864 . . . . . 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 3063 . . . . 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 2813 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 ∈ CMnd ↔ 𝐴 ∈ CMnd))
107 fveq2 6881 . . . . . . . . 9 (𝑦 = 𝐴 → (Base‘𝑦) = (Base‘𝐴))
108107feq3d 6694 . . . . . . . 8 (𝑦 = 𝐴 → (𝑧:𝑡⟶(Base‘𝑦) ↔ 𝑧:𝑡⟶(Base‘𝐴)))
109108rexbidv 3170 . . . . . . 7 (𝑦 = 𝐴 → (∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦) ↔ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴)))
110106, 109anbi12d 630 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦)) ↔ (𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴))))
111 feq1 6688 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧:𝑡⟶(Base‘𝐴) ↔ 𝐵:𝑡⟶(Base‘𝐴)))
112111rexbidv 3170 . . . . . . 7 (𝑧 = 𝐵 → (∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴) ↔ ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴)))
113112anbi2d 628 . . . . . 6 (𝑧 = 𝐵 → ((𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴)) ↔ (𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴))))
114110, 113opelopabg 5528 . . . . 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 583 . . . 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 710 . . 3 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))})
117 iotaex 6506 . . . 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 6996 . 2 (𝜑 → ( FinSum ‘⟨𝐴, 𝐵⟩) = (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)))))
1201, 119eqtrid 2776 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 395   = wceq 1533  wtru 1534  wex 1773  wcel 2098  wrex 3062  Vcvv 3466  cop 4626  {copab 5200  cmpt 5221  dom cdm 5666  cio 6483  wf 6529  1-1-ontowf1o 6532  cfv 6533  (class class class)co 7401  1st c1st 7966  2nd c2nd 7967  Fincfn 8935  1c1 11107  cn 12209  0cn0 12469  ...cfz 13481  seqcseq 13963  chash 14287  Basecbs 17143  +gcplusg 17196  CMndccmn 19690   FinSum cfinsum 36654
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-card 9930  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-n0 12470  df-z 12556  df-uz 12820  df-fz 13482  df-seq 13964  df-hash 14288  df-bj-finsum 36655
This theorem is referenced by: (None)
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