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Theorem bj-finsumval0 34145
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 7024 . 2 (𝐴 FinSum 𝐵) = ( FinSum ‘⟨𝐴, 𝐵⟩)
2 df-bj-finsum 34144 . . . 4 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))))
32a1i 11 . . 3 (𝜑 → 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)))))
4 simpr 485 . . . . . . . . . 10 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → 𝑥 = ⟨𝐴, 𝐵⟩)
54fveq2d 6547 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (1st𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
6 bj-finsumval0.1 . . . . . . . . . . 11 (𝜑𝐴 ∈ CMnd)
76adantr 481 . . . . . . . . . 10 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → 𝐴 ∈ CMnd)
8 bj-finsumval0.3 . . . . . . . . . . . 12 (𝜑𝐵:𝐼⟶(Base‘𝐴))
9 bj-finsumval0.2 . . . . . . . . . . . 12 (𝜑𝐼 ∈ Fin)
10 fex 6860 . . . . . . . . . . . 12 ((𝐵:𝐼⟶(Base‘𝐴) ∧ 𝐼 ∈ Fin) → 𝐵 ∈ V)
118, 9, 10syl2anc 584 . . . . . . . . . . 11 (𝜑𝐵 ∈ V)
1211adantr 481 . . . . . . . . . 10 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → 𝐵 ∈ V)
13 op1stg 7562 . . . . . . . . . 10 ((𝐴 ∈ CMnd ∧ 𝐵 ∈ V) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
147, 12, 13syl2anc 584 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
155, 14eqtrd 2831 . . . . . . . 8 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (1st𝑥) = 𝐴)
164fveq2d 6547 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (2nd𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
17 op2ndg 7563 . . . . . . . . . 10 ((𝐴 ∈ CMnd ∧ 𝐵 ∈ V) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
187, 12, 17syl2anc 584 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
1916, 18eqtrd 2831 . . . . . . . 8 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (2nd𝑥) = 𝐵)
2019dmeqd 5665 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → dom (2nd𝑥) = dom 𝐵)
218fdmd 6396 . . . . . . . . . 10 (𝜑 → dom 𝐵 = 𝐼)
2221adantr 481 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → dom 𝐵 = 𝐼)
2320, 22eqtrd 2831 . . . . . . . 8 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → dom (2nd𝑥) = 𝐼)
24 f1oeq3 6479 . . . . . . . . . . . . . . 15 (dom (2nd𝑥) = 𝐼 → (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ↔ 𝑓:(1...𝑚)–1-1-onto𝐼))
2524biimpd 230 . . . . . . . . . . . . . 14 (dom (2nd𝑥) = 𝐼 → (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) → 𝑓:(1...𝑚)–1-1-onto𝐼))
2625ad2antll 725 . . . . . . . . . . . . 13 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) → 𝑓:(1...𝑚)–1-1-onto𝐼))
2726adantrd 492 . . . . . . . . . . . 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))
2827adantr 481 . . . . . . . . . . 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))
29 eqidd 2796 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 1 = 1)
30 simprl 767 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → (1st𝑥) = 𝐴)
3130fveq2d 6547 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → (+g‘(1st𝑥)) = (+g𝐴))
3231adantrr 713 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (+g‘(1st𝑥)) = (+g𝐴))
33 simprrl 777 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → (2nd𝑥) = 𝐵)
3433adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → (2nd𝑥) = 𝐵)
3534fveq1d 6545 . . . . . . . . . . . . . . . . . . 19 (((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → ((2nd𝑥)‘(𝑓𝑛)) = (𝐵‘(𝑓𝑛)))
3635mpteq2dva 5060 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))) = (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))
3736adantrr 713 . . . . . . . . . . . . . . . . 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))))
3829, 32, 37seqeq123d 13233 . . . . . . . . . . . . . . . 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)))))
39 simpr 485 . . . . . . . . . . . . . . . . . 18 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
40 simprr 769 . . . . . . . . . . . . . . . . . . 19 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → dom (2nd𝑥) = 𝐼)
4140adantr 481 . . . . . . . . . . . . . . . . . 18 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → dom (2nd𝑥) = 𝐼)
4239, 41jca 512 . . . . . . . . . . . . . . . . 17 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼))
43 hashfz1 13561 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ0 → (♯‘(1...𝑚)) = 𝑚)
4443eqcomd 2801 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ℕ0𝑚 = (♯‘(1...𝑚)))
4544ad2antrl 724 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → 𝑚 = (♯‘(1...𝑚)))
46 fzfid 13196 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → (1...𝑚) ∈ Fin)
47 19.8a 2144 . . . . . . . . . . . . . . . . . . . 20 (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) → ∃𝑓 𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥))
4847adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → ∃𝑓 𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥))
49 hasheqf1oi 13567 . . . . . . . . . . . . . . . . . . 19 ((1...𝑚) ∈ Fin → (∃𝑓 𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) → (♯‘(1...𝑚)) = (♯‘dom (2nd𝑥))))
5046, 48, 49sylc 65 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → (♯‘(1...𝑚)) = (♯‘dom (2nd𝑥)))
51 simprr 769 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → dom (2nd𝑥) = 𝐼)
5251fveq2d 6547 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → (♯‘dom (2nd𝑥)) = (♯‘𝐼))
5345, 50, 523eqtrd 2835 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → 𝑚 = (♯‘𝐼))
5442, 53sylan2 592 . . . . . . . . . . . . . . . 16 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑚 = (♯‘𝐼))
5538, 54fveq12d 6550 . . . . . . . . . . . . . . 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)))
5655eqeq2d 2805 . . . . . . . . . . . . . 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))))
5756biimpd 230 . . . . . . . . . . . . 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))))
5857impancom 452 . . . . . . . . . . . 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))))
5958com12 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))))
6028, 59jcad 513 . . . . . . . . . 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)))))
6124biimprd 249 . . . . . . . . . . . . . 14 (dom (2nd𝑥) = 𝐼 → (𝑓:(1...𝑚)–1-1-onto𝐼𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥)))
6261ad2antll 725 . . . . . . . . . . . . 13 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → (𝑓:(1...𝑚)–1-1-onto𝐼𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥)))
6362adantr 481 . . . . . . . . . . . 12 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → (𝑓:(1...𝑚)–1-1-onto𝐼𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥)))
6463adantrd 492 . . . . . . . . . . 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)))
65 eqidd 2796 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 1 = 1)
66 simpl 483 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → (1st𝑥) = 𝐴)
67 tru 1526 . . . . . . . . . . . . . . . . . . . . 21
6866, 67jctir 521 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → ((1st𝑥) = 𝐴 ∧ ⊤))
6968ad2antrl 724 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → ((1st𝑥) = 𝐴 ∧ ⊤))
70 simpl 483 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑥) = 𝐴 ∧ ⊤) → (1st𝑥) = 𝐴)
7170eqcomd 2801 . . . . . . . . . . . . . . . . . . 19 (((1st𝑥) = 𝐴 ∧ ⊤) → 𝐴 = (1st𝑥))
7269, 71syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝐴 = (1st𝑥))
7372fveq2d 6547 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (+g𝐴) = (+g‘(1st𝑥)))
74 simpl 483 . . . . . . . . . . . . . . . . . . . . . . 23 (((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼) → (2nd𝑥) = 𝐵)
7574eqcomd 2801 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼) → 𝐵 = (2nd𝑥))
7675ad2antll 725 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → 𝐵 = (2nd𝑥))
7776adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → 𝐵 = (2nd𝑥))
7877fveq1d 6545 . . . . . . . . . . . . . . . . . . 19 (((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → (𝐵‘(𝑓𝑛)) = ((2nd𝑥)‘(𝑓𝑛)))
7978adantlrr 717 . . . . . . . . . . . . . . . . . 18 (((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) ∧ 𝑛 ∈ ℕ) → (𝐵‘(𝑓𝑛)) = ((2nd𝑥)‘(𝑓𝑛)))
8079mpteq2dva 5060 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))) = (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))
8165, 73, 80seqeq123d 13233 . . . . . . . . . . . . . . . 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)))))
8263impcom 408 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥))
83 simprr 769 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑚 ∈ ℕ0)
8440ad2antrl 724 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → dom (2nd𝑥) = 𝐼)
8582, 83, 84, 53syl12anc 833 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑚 = (♯‘𝐼))
8685eqcomd 2801 . . . . . . . . . . . . . . . 16 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (♯‘𝐼) = 𝑚)
8781, 86fveq12d 6550 . . . . . . . . . . . . . . 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))
8887eqeq2d 2805 . . . . . . . . . . . . . 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)))
8988biimpd 230 . . . . . . . . . . . . 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)))
9089impancom 452 . . . . . . . . . . . 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)))
9190com12 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)))
9264, 91jcad 513 . . . . . . . . . 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))))
9360, 92impbid 213 . . . . . . . . 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)))))
9493ex 413 . . . . . . . 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))))))
9515, 19, 23, 94syl12anc 833 . . . . . . 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))))))
9695imp 407 . . . . . 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)))))
9796exbidv 1899 . . . . 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)))))
9897rexbidva 3259 . . . 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)))))
9998iotabidv 6215 . . 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)))))
100 eleq1 2870 . . . . . . . . . 10 (𝑡 = 𝐼 → (𝑡 ∈ Fin ↔ 𝐼 ∈ Fin))
101 feq2 6369 . . . . . . . . . 10 (𝑡 = 𝐼 → (𝐵:𝑡⟶(Base‘𝐴) ↔ 𝐵:𝐼⟶(Base‘𝐴)))
102100, 101anbi12d 630 . . . . . . . . 9 (𝑡 = 𝐼 → ((𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴)) ↔ (𝐼 ∈ Fin ∧ 𝐵:𝐼⟶(Base‘𝐴))))
103102ceqsexgv 3586 . . . . . . . 8 (𝐼 ∈ Fin → (∃𝑡(𝑡 = 𝐼 ∧ (𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴))) ↔ (𝐼 ∈ Fin ∧ 𝐵:𝐼⟶(Base‘𝐴))))
1049, 103syl 17 . . . . . . 7 (𝜑 → (∃𝑡(𝑡 = 𝐼 ∧ (𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴))) ↔ (𝐼 ∈ Fin ∧ 𝐵:𝐼⟶(Base‘𝐴))))
1059, 8, 104mpbir2and 709 . . . . . 6 (𝜑 → ∃𝑡(𝑡 = 𝐼 ∧ (𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴))))
106 exsimpr 1851 . . . . . 6 (∃𝑡(𝑡 = 𝐼 ∧ (𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴))) → ∃𝑡(𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴)))
107105, 106syl 17 . . . . 5 (𝜑 → ∃𝑡(𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴)))
108 df-rex 3111 . . . . 5 (∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴) ↔ ∃𝑡(𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴)))
109107, 108sylibr 235 . . . 4 (𝜑 → ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴))
110 eleq1 2870 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 ∈ CMnd ↔ 𝐴 ∈ CMnd))
111 fveq2 6543 . . . . . . . . 9 (𝑦 = 𝐴 → (Base‘𝑦) = (Base‘𝐴))
112111feq3d 6374 . . . . . . . 8 (𝑦 = 𝐴 → (𝑧:𝑡⟶(Base‘𝑦) ↔ 𝑧:𝑡⟶(Base‘𝐴)))
113112rexbidv 3260 . . . . . . 7 (𝑦 = 𝐴 → (∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦) ↔ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴)))
114110, 113anbi12d 630 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦)) ↔ (𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴))))
115 feq1 6368 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧:𝑡⟶(Base‘𝐴) ↔ 𝐵:𝑡⟶(Base‘𝐴)))
116115rexbidv 3260 . . . . . . 7 (𝑧 = 𝐵 → (∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴) ↔ ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴)))
117116anbi2d 628 . . . . . 6 (𝑧 = 𝐵 → ((𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴)) ↔ (𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴))))
118114, 117opelopabg 5320 . . . . 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))))
1196, 11, 118syl2anc 584 . . . 4 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))} ↔ (𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴))))
1206, 109, 119mpbir2and 709 . . 3 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))})
121 iotaex 6211 . . . 4 (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)))) ∈ V
122121a1i 11 . . 3 (𝜑 → (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)))) ∈ V)
1233, 99, 120, 122fvmptd 6646 . 2 (𝜑 → ( FinSum ‘⟨𝐴, 𝐵⟩) = (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)))))
1241, 123syl5eq 2843 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 207  wa 396   = wceq 1522  wtru 1523  wex 1761  wcel 2081  wrex 3106  Vcvv 3437  cop 4482  {copab 5028  cmpt 5045  dom cdm 5448  cio 6192  wf 6226  1-1-ontowf1o 6229  cfv 6230  (class class class)co 7021  1st c1st 7548  2nd c2nd 7549  Fincfn 8362  1c1 10389  cn 11491  0cn0 11750  ...cfz 12747  seqcseq 13224  chash 13545  Basecbs 16317  +gcplusg 16399  CMndccmn 18638   FinSum cfinsum 34143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324  ax-cnex 10444  ax-resscn 10445  ax-1cn 10446  ax-icn 10447  ax-addcl 10448  ax-addrcl 10449  ax-mulcl 10450  ax-mulrcl 10451  ax-mulcom 10452  ax-addass 10453  ax-mulass 10454  ax-distr 10455  ax-i2m1 10456  ax-1ne0 10457  ax-1rid 10458  ax-rnegex 10459  ax-rrecex 10460  ax-cnre 10461  ax-pre-lttri 10462  ax-pre-lttrn 10463  ax-pre-ltadd 10464  ax-pre-mulgt0 10465
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-int 4787  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-riota 6982  df-ov 7024  df-oprab 7025  df-mpo 7026  df-om 7442  df-1st 7550  df-2nd 7551  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-1o 7958  df-er 8144  df-en 8363  df-dom 8364  df-sdom 8365  df-fin 8366  df-card 9219  df-pnf 10528  df-mnf 10529  df-xr 10530  df-ltxr 10531  df-le 10532  df-sub 10724  df-neg 10725  df-nn 11492  df-n0 11751  df-z 11835  df-uz 12099  df-fz 12748  df-seq 13225  df-hash 13546  df-bj-finsum 34144
This theorem is referenced by: (None)
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