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Theorem bj-finsumval0 32324
 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 6552 . 2 (𝐴 FinSum 𝐵) = ( FinSum ‘⟨𝐴, 𝐵⟩)
2 df-bj-finsum 32323 . . . 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 476 . . . . . . . . . 10 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → 𝑥 = ⟨𝐴, 𝐵⟩)
54fveq2d 6107 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (1st𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
6 bj-finsumval0.1 . . . . . . . . . . 11 (𝜑𝐴 ∈ CMnd)
76adantr 480 . . . . . . . . . 10 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → 𝐴 ∈ CMnd)
8 bj-finsumval0.3 . . . . . . . . . . . 12 (𝜑𝐵:𝐼⟶(Base‘𝐴))
9 bj-finsumval0.2 . . . . . . . . . . . 12 (𝜑𝐼 ∈ Fin)
10 fex 6394 . . . . . . . . . . . 12 ((𝐵:𝐼⟶(Base‘𝐴) ∧ 𝐼 ∈ Fin) → 𝐵 ∈ V)
118, 9, 10syl2anc 691 . . . . . . . . . . 11 (𝜑𝐵 ∈ V)
1211adantr 480 . . . . . . . . . 10 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → 𝐵 ∈ V)
13 op1stg 7071 . . . . . . . . . 10 ((𝐴 ∈ CMnd ∧ 𝐵 ∈ V) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
147, 12, 13syl2anc 691 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
155, 14eqtrd 2644 . . . . . . . 8 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (1st𝑥) = 𝐴)
164fveq2d 6107 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (2nd𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
17 op2ndg 7072 . . . . . . . . . 10 ((𝐴 ∈ CMnd ∧ 𝐵 ∈ V) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
187, 12, 17syl2anc 691 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
1916, 18eqtrd 2644 . . . . . . . 8 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (2nd𝑥) = 𝐵)
2019dmeqd 5248 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → dom (2nd𝑥) = dom 𝐵)
21 fdm 5964 . . . . . . . . . . 11 (𝐵:𝐼⟶(Base‘𝐴) → dom 𝐵 = 𝐼)
228, 21syl 17 . . . . . . . . . 10 (𝜑 → dom 𝐵 = 𝐼)
2322adantr 480 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → dom 𝐵 = 𝐼)
2420, 23eqtrd 2644 . . . . . . . 8 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → dom (2nd𝑥) = 𝐼)
25 f1oeq3 6042 . . . . . . . . . . . . . . 15 (dom (2nd𝑥) = 𝐼 → (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ↔ 𝑓:(1...𝑚)–1-1-onto𝐼))
2625biimpd 218 . . . . . . . . . . . . . 14 (dom (2nd𝑥) = 𝐼 → (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) → 𝑓:(1...𝑚)–1-1-onto𝐼))
2726ad2antll 761 . . . . . . . . . . . . 13 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) → 𝑓:(1...𝑚)–1-1-onto𝐼))
2827adantrd 483 . . . . . . . . . . . 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))
2928adantr 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))
30 eqidd 2611 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 1 = 1)
31 simprl 790 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → (1st𝑥) = 𝐴)
3231fveq2d 6107 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → (+g‘(1st𝑥)) = (+g𝐴))
3332adantrr 749 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (+g‘(1st𝑥)) = (+g𝐴))
34 simprrl 800 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → (2nd𝑥) = 𝐵)
3534adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → (2nd𝑥) = 𝐵)
3635fveq1d 6105 . . . . . . . . . . . . . . . . . . 19 (((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → ((2nd𝑥)‘(𝑓𝑛)) = (𝐵‘(𝑓𝑛)))
3736mpteq2dva 4672 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))) = (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))
3837adantrr 749 . . . . . . . . . . . . . . . . 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))))
3930, 33, 38seqeq123d 12672 . . . . . . . . . . . . . . . 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)))))
40 simpr 476 . . . . . . . . . . . . . . . . . 18 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
41 simprr 792 . . . . . . . . . . . . . . . . . . 19 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → dom (2nd𝑥) = 𝐼)
4241adantr 480 . . . . . . . . . . . . . . . . . 18 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → dom (2nd𝑥) = 𝐼)
4340, 42jca 553 . . . . . . . . . . . . . . . . 17 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼))
44 hashfz1 12996 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ0 → (#‘(1...𝑚)) = 𝑚)
4544eqcomd 2616 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ℕ0𝑚 = (#‘(1...𝑚)))
4645ad2antrl 760 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → 𝑚 = (#‘(1...𝑚)))
47 fzfid 12634 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → (1...𝑚) ∈ Fin)
48 19.8a 2039 . . . . . . . . . . . . . . . . . . . 20 (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) → ∃𝑓 𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥))
4948adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → ∃𝑓 𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥))
50 hasheqf1oi 13002 . . . . . . . . . . . . . . . . . . 19 ((1...𝑚) ∈ Fin → (∃𝑓 𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) → (#‘(1...𝑚)) = (#‘dom (2nd𝑥))))
5147, 49, 50sylc 63 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → (#‘(1...𝑚)) = (#‘dom (2nd𝑥)))
52 simprr 792 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → dom (2nd𝑥) = 𝐼)
5352fveq2d 6107 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → (#‘dom (2nd𝑥)) = (#‘𝐼))
5446, 51, 533eqtrd 2648 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → 𝑚 = (#‘𝐼))
5543, 54sylan2 490 . . . . . . . . . . . . . . . 16 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑚 = (#‘𝐼))
5639, 55fveq12d 6109 . . . . . . . . . . . . . . 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)))
5756eqeq2d 2620 . . . . . . . . . . . . . 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))))
5857biimpd 218 . . . . . . . . . . . . 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))))
5958impancom 455 . . . . . . . . . . . 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))))
6059com12 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))))
6129, 60jcad 554 . . . . . . . . . 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)))))
6225biimprd 237 . . . . . . . . . . . . . 14 (dom (2nd𝑥) = 𝐼 → (𝑓:(1...𝑚)–1-1-onto𝐼𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥)))
6362ad2antll 761 . . . . . . . . . . . . 13 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → (𝑓:(1...𝑚)–1-1-onto𝐼𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥)))
6463adantr 480 . . . . . . . . . . . 12 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → (𝑓:(1...𝑚)–1-1-onto𝐼𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥)))
6564adantrd 483 . . . . . . . . . . 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)))
66 eqidd 2611 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 1 = 1)
67 simpl 472 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → (1st𝑥) = 𝐴)
68 tru 1479 . . . . . . . . . . . . . . . . . . . . 21
6967, 68jctir 559 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → ((1st𝑥) = 𝐴 ∧ ⊤))
7069ad2antrl 760 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → ((1st𝑥) = 𝐴 ∧ ⊤))
71 simpl 472 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑥) = 𝐴 ∧ ⊤) → (1st𝑥) = 𝐴)
7271eqcomd 2616 . . . . . . . . . . . . . . . . . . 19 (((1st𝑥) = 𝐴 ∧ ⊤) → 𝐴 = (1st𝑥))
7370, 72syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝐴 = (1st𝑥))
7473fveq2d 6107 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (+g𝐴) = (+g‘(1st𝑥)))
75 simpl 472 . . . . . . . . . . . . . . . . . . . . . . 23 (((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼) → (2nd𝑥) = 𝐵)
7675eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼) → 𝐵 = (2nd𝑥))
7776ad2antll 761 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → 𝐵 = (2nd𝑥))
7877adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → 𝐵 = (2nd𝑥))
7978fveq1d 6105 . . . . . . . . . . . . . . . . . . 19 (((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → (𝐵‘(𝑓𝑛)) = ((2nd𝑥)‘(𝑓𝑛)))
8079adantlrr 753 . . . . . . . . . . . . . . . . . 18 (((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) ∧ 𝑛 ∈ ℕ) → (𝐵‘(𝑓𝑛)) = ((2nd𝑥)‘(𝑓𝑛)))
8180mpteq2dva 4672 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))) = (𝑛 ∈ ℕ ↦ ((2nd𝑥)‘(𝑓𝑛))))
8266, 74, 81seqeq123d 12672 . . . . . . . . . . . . . . . 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)))))
8364impcom 445 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥))
84 simprr 792 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑚 ∈ ℕ0)
8541ad2antrl 760 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → dom (2nd𝑥) = 𝐼)
8683, 84, 85, 54syl12anc 1316 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑚 = (#‘𝐼))
8786eqcomd 2616 . . . . . . . . . . . . . . . 16 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (#‘𝐼) = 𝑚)
8882, 87fveq12d 6109 . . . . . . . . . . . . . . 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))
8988eqeq2d 2620 . . . . . . . . . . . . . 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)))
9089biimpd 218 . . . . . . . . . . . . 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)))
9190impancom 455 . . . . . . . . . . . 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)))
9291com12 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)))
9365, 92jcad 554 . . . . . . . . . 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))))
9461, 93impbid 201 . . . . . . . . 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)))))
9594ex 449 . . . . . . . 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))))))
9615, 19, 24, 95syl12anc 1316 . . . . . . 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))))))
9796imp 444 . . . . . 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)))))
9897exbidv 1837 . . . . 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)))))
9998rexbidva 3031 . . . 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)))))
10099iotabidv 5789 . . 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)))))
101 eleq1 2676 . . . . . . . . . 10 (𝑡 = 𝐼 → (𝑡 ∈ Fin ↔ 𝐼 ∈ Fin))
102 feq2 5940 . . . . . . . . . 10 (𝑡 = 𝐼 → (𝐵:𝑡⟶(Base‘𝐴) ↔ 𝐵:𝐼⟶(Base‘𝐴)))
103101, 102anbi12d 743 . . . . . . . . 9 (𝑡 = 𝐼 → ((𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴)) ↔ (𝐼 ∈ Fin ∧ 𝐵:𝐼⟶(Base‘𝐴))))
104103ceqsexgv 3305 . . . . . . . 8 (𝐼 ∈ Fin → (∃𝑡(𝑡 = 𝐼 ∧ (𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴))) ↔ (𝐼 ∈ Fin ∧ 𝐵:𝐼⟶(Base‘𝐴))))
1059, 104syl 17 . . . . . . 7 (𝜑 → (∃𝑡(𝑡 = 𝐼 ∧ (𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴))) ↔ (𝐼 ∈ Fin ∧ 𝐵:𝐼⟶(Base‘𝐴))))
1069, 8, 105mpbir2and 959 . . . . . 6 (𝜑 → ∃𝑡(𝑡 = 𝐼 ∧ (𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴))))
107 exsimpr 1784 . . . . . 6 (∃𝑡(𝑡 = 𝐼 ∧ (𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴))) → ∃𝑡(𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴)))
108106, 107syl 17 . . . . 5 (𝜑 → ∃𝑡(𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴)))
109 df-rex 2902 . . . . 5 (∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴) ↔ ∃𝑡(𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴)))
110108, 109sylibr 223 . . . 4 (𝜑 → ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴))
111 eleq1 2676 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 ∈ CMnd ↔ 𝐴 ∈ CMnd))
112 fveq2 6103 . . . . . . . . 9 (𝑦 = 𝐴 → (Base‘𝑦) = (Base‘𝐴))
113112feq3d 5945 . . . . . . . 8 (𝑦 = 𝐴 → (𝑧:𝑡⟶(Base‘𝑦) ↔ 𝑧:𝑡⟶(Base‘𝐴)))
114113rexbidv 3034 . . . . . . 7 (𝑦 = 𝐴 → (∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦) ↔ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴)))
115111, 114anbi12d 743 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦)) ↔ (𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴))))
116 feq1 5939 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧:𝑡⟶(Base‘𝐴) ↔ 𝐵:𝑡⟶(Base‘𝐴)))
117116rexbidv 3034 . . . . . . 7 (𝑧 = 𝐵 → (∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴) ↔ ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴)))
118117anbi2d 736 . . . . . 6 (𝑧 = 𝐵 → ((𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴)) ↔ (𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴))))
119115, 118opelopabg 4918 . . . . 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))))
1206, 11, 119syl2anc 691 . . . 4 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))} ↔ (𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴))))
1216, 110, 120mpbir2and 959 . . 3 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))})
122 iotaex 5785 . . . 4 (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(#‘𝐼)))) ∈ V
123122a1i 11 . . 3 (𝜑 → (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(#‘𝐼)))) ∈ V)
1243, 100, 121, 123fvmptd 6197 . 2 (𝜑 → ( FinSum ‘⟨𝐴, 𝐵⟩) = (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(#‘𝐼)))))
1251, 124syl5eq 2656 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 195   ∧ wa 383   = wceq 1475  ⊤wtru 1476  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173  ⟨cop 4131  {copab 4642   ↦ cmpt 4643  dom cdm 5038  ℩cio 5766  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Fincfn 7841  1c1 9816  ℕcn 10897  ℕ0cn0 11169  ...cfz 12197  seqcseq 12663  #chash 12979  Basecbs 15695  +gcplusg 15768  CMndccmn 18016   FinSum cfinsum 32322 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664  df-hash 12980  df-bj-finsum 32323 This theorem is referenced by: (None)
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