Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-finsumval0 Structured version   Visualization version   GIF version

Theorem bj-finsumval0 33450
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 6808 . 2 (𝐴 FinSum 𝐵) = ( FinSum ‘⟨𝐴, 𝐵⟩)
2 df-bj-finsum 33449 . . . 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 479 . . . . . . . . . 10 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → 𝑥 = ⟨𝐴, 𝐵⟩)
54fveq2d 6348 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (1st𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
6 bj-finsumval0.1 . . . . . . . . . . 11 (𝜑𝐴 ∈ CMnd)
76adantr 472 . . . . . . . . . 10 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → 𝐴 ∈ CMnd)
8 bj-finsumval0.3 . . . . . . . . . . . 12 (𝜑𝐵:𝐼⟶(Base‘𝐴))
9 bj-finsumval0.2 . . . . . . . . . . . 12 (𝜑𝐼 ∈ Fin)
10 fex 6645 . . . . . . . . . . . 12 ((𝐵:𝐼⟶(Base‘𝐴) ∧ 𝐼 ∈ Fin) → 𝐵 ∈ V)
118, 9, 10syl2anc 696 . . . . . . . . . . 11 (𝜑𝐵 ∈ V)
1211adantr 472 . . . . . . . . . 10 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → 𝐵 ∈ V)
13 op1stg 7337 . . . . . . . . . 10 ((𝐴 ∈ CMnd ∧ 𝐵 ∈ V) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
147, 12, 13syl2anc 696 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
155, 14eqtrd 2786 . . . . . . . 8 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (1st𝑥) = 𝐴)
164fveq2d 6348 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (2nd𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
17 op2ndg 7338 . . . . . . . . . 10 ((𝐴 ∈ CMnd ∧ 𝐵 ∈ V) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
187, 12, 17syl2anc 696 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
1916, 18eqtrd 2786 . . . . . . . 8 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → (2nd𝑥) = 𝐵)
2019dmeqd 5473 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → dom (2nd𝑥) = dom 𝐵)
21 fdm 6204 . . . . . . . . . . 11 (𝐵:𝐼⟶(Base‘𝐴) → dom 𝐵 = 𝐼)
228, 21syl 17 . . . . . . . . . 10 (𝜑 → dom 𝐵 = 𝐼)
2322adantr 472 . . . . . . . . 9 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → dom 𝐵 = 𝐼)
2420, 23eqtrd 2786 . . . . . . . 8 ((𝜑𝑥 = ⟨𝐴, 𝐵⟩) → dom (2nd𝑥) = 𝐼)
25 f1oeq3 6282 . . . . . . . . . . . . . . 15 (dom (2nd𝑥) = 𝐼 → (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ↔ 𝑓:(1...𝑚)–1-1-onto𝐼))
2625biimpd 219 . . . . . . . . . . . . . 14 (dom (2nd𝑥) = 𝐼 → (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) → 𝑓:(1...𝑚)–1-1-onto𝐼))
2726ad2antll 767 . . . . . . . . . . . . 13 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) → 𝑓:(1...𝑚)–1-1-onto𝐼))
2827adantrd 485 . . . . . . . . . . . 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 472 . . . . . . . . . . 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 2753 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 1 = 1)
31 simprl 811 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → (1st𝑥) = 𝐴)
3231fveq2d 6348 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → (+g‘(1st𝑥)) = (+g𝐴))
3332adantrr 755 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (+g‘(1st𝑥)) = (+g𝐴))
34 simprrl 823 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → (2nd𝑥) = 𝐵)
3534adantr 472 . . . . . . . . . . . . . . . . . . . 20 (((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → (2nd𝑥) = 𝐵)
3635fveq1d 6346 . . . . . . . . . . . . . . . . . . 19 (((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → ((2nd𝑥)‘(𝑓𝑛)) = (𝐵‘(𝑓𝑛)))
3736mpteq2dva 4888 . . . . . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . . 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 12996 . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . 18 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
41 simprr 813 . . . . . . . . . . . . . . . . . . 19 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → dom (2nd𝑥) = 𝐼)
4241adantr 472 . . . . . . . . . . . . . . . . . 18 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → dom (2nd𝑥) = 𝐼)
4340, 42jca 555 . . . . . . . . . . . . . . . . 17 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼))
44 hashfz1 13320 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ0 → (♯‘(1...𝑚)) = 𝑚)
4544eqcomd 2758 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ℕ0𝑚 = (♯‘(1...𝑚)))
4645ad2antrl 766 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → 𝑚 = (♯‘(1...𝑚)))
47 fzfid 12958 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → (1...𝑚) ∈ Fin)
48 19.8a 2191 . . . . . . . . . . . . . . . . . . . 20 (𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) → ∃𝑓 𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥))
4948adantr 472 . . . . . . . . . . . . . . . . . . 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 13326 . . . . . . . . . . . . . . . . . . 19 ((1...𝑚) ∈ Fin → (∃𝑓 𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) → (♯‘(1...𝑚)) = (♯‘dom (2nd𝑥))))
5147, 49, 50sylc 65 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → (♯‘(1...𝑚)) = (♯‘dom (2nd𝑥)))
52 simprr 813 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → dom (2nd𝑥) = 𝐼)
5352fveq2d 6348 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → (♯‘dom (2nd𝑥)) = (♯‘𝐼))
5446, 51, 533eqtrd 2790 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd𝑥) = 𝐼)) → 𝑚 = (♯‘𝐼))
5543, 54sylan2 492 . . . . . . . . . . . . . . . 16 ((𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥) ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑚 = (♯‘𝐼))
5639, 55fveq12d 6350 . . . . . . . . . . . . . . 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 2762 . . . . . . . . . . . . . 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 219 . . . . . . . . . . . . 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 556 . . . . . . . . . 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 238 . . . . . . . . . . . . . 14 (dom (2nd𝑥) = 𝐼 → (𝑓:(1...𝑚)–1-1-onto𝐼𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥)))
6362ad2antll 767 . . . . . . . . . . . . 13 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → (𝑓:(1...𝑚)–1-1-onto𝐼𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥)))
6463adantr 472 . . . . . . . . . . . 12 ((((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → (𝑓:(1...𝑚)–1-1-onto𝐼𝑓:(1...𝑚)–1-1-onto→dom (2nd𝑥)))
6564adantrd 485 . . . . . . . . . . 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 2753 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 1 = 1)
67 simpl 474 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → (1st𝑥) = 𝐴)
68 tru 1628 . . . . . . . . . . . . . . . . . . . . 21
6967, 68jctir 562 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) → ((1st𝑥) = 𝐴 ∧ ⊤))
7069ad2antrl 766 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → ((1st𝑥) = 𝐴 ∧ ⊤))
71 simpl 474 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑥) = 𝐴 ∧ ⊤) → (1st𝑥) = 𝐴)
7271eqcomd 2758 . . . . . . . . . . . . . . . . . . 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 6348 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (+g𝐴) = (+g‘(1st𝑥)))
75 simpl 474 . . . . . . . . . . . . . . . . . . . . . . 23 (((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼) → (2nd𝑥) = 𝐵)
7675eqcomd 2758 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼) → 𝐵 = (2nd𝑥))
7776ad2antll 767 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) → 𝐵 = (2nd𝑥))
7877adantr 472 . . . . . . . . . . . . . . . . . . . 20 (((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → 𝐵 = (2nd𝑥))
7978fveq1d 6346 . . . . . . . . . . . . . . . . . . 19 (((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ ((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → (𝐵‘(𝑓𝑛)) = ((2nd𝑥)‘(𝑓𝑛)))
8079adantlrr 759 . . . . . . . . . . . . . . . . . 18 (((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) ∧ 𝑛 ∈ ℕ) → (𝐵‘(𝑓𝑛)) = ((2nd𝑥)‘(𝑓𝑛)))
8180mpteq2dva 4888 . . . . . . . . . . . . . . . . 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 12996 . . . . . . . . . . . . . . . 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 813 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑚 ∈ ℕ0)
8541ad2antrl 766 . . . . . . . . . . . . . . . . . 18 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → dom (2nd𝑥) = 𝐼)
8683, 84, 85, 54syl12anc 1471 . . . . . . . . . . . . . . . . 17 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑚 = (♯‘𝐼))
8786eqcomd 2758 . . . . . . . . . . . . . . . 16 ((𝑓:(1...𝑚)–1-1-onto𝐼 ∧ (((1st𝑥) = 𝐴 ∧ ((2nd𝑥) = 𝐵 ∧ dom (2nd𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (♯‘𝐼) = 𝑚)
8882, 87fveq12d 6350 . . . . . . . . . . . . . . 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 2762 . . . . . . . . . . . . . 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 219 . . . . . . . . . . . . 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 556 . . . . . . . . . 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 202 . . . . . . . . 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 1471 . . . . . . 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 1991 . . . . 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 3179 . . . 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 6025 . . 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 2819 . . . . . . . . . 10 (𝑡 = 𝐼 → (𝑡 ∈ Fin ↔ 𝐼 ∈ Fin))
102 feq2 6180 . . . . . . . . . 10 (𝑡 = 𝐼 → (𝐵:𝑡⟶(Base‘𝐴) ↔ 𝐵:𝐼⟶(Base‘𝐴)))
103101, 102anbi12d 749 . . . . . . . . 9 (𝑡 = 𝐼 → ((𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴)) ↔ (𝐼 ∈ Fin ∧ 𝐵:𝐼⟶(Base‘𝐴))))
104103ceqsexgv 3466 . . . . . . . 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 995 . . . . . 6 (𝜑 → ∃𝑡(𝑡 = 𝐼 ∧ (𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴))))
107 exsimpr 1937 . . . . . 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 3048 . . . . 5 (∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴) ↔ ∃𝑡(𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴)))
110108, 109sylibr 224 . . . 4 (𝜑 → ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴))
111 eleq1 2819 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 ∈ CMnd ↔ 𝐴 ∈ CMnd))
112 fveq2 6344 . . . . . . . . 9 (𝑦 = 𝐴 → (Base‘𝑦) = (Base‘𝐴))
113112feq3d 6185 . . . . . . . 8 (𝑦 = 𝐴 → (𝑧:𝑡⟶(Base‘𝑦) ↔ 𝑧:𝑡⟶(Base‘𝐴)))
114113rexbidv 3182 . . . . . . 7 (𝑦 = 𝐴 → (∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦) ↔ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴)))
115111, 114anbi12d 749 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦)) ↔ (𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴))))
116 feq1 6179 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧:𝑡⟶(Base‘𝐴) ↔ 𝐵:𝑡⟶(Base‘𝐴)))
117116rexbidv 3182 . . . . . . 7 (𝑧 = 𝐵 → (∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴) ↔ ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴)))
118117anbi2d 742 . . . . . 6 (𝑧 = 𝐵 → ((𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴)) ↔ (𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴))))
119115, 118opelopabg 5135 . . . . 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 696 . . . 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 995 . . 3 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))})
122 iotaex 6021 . . . 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 6442 . 2 (𝜑 → ( FinSum ‘⟨𝐴, 𝐵⟩) = (℩𝑠𝑚 ∈ ℕ0𝑓(𝑓:(1...𝑚)–1-1-onto𝐼𝑠 = (seq1((+g𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓𝑛))))‘(♯‘𝐼)))))
1251, 124syl5eq 2798 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 196  wa 383   = wceq 1624  wtru 1625  wex 1845  wcel 2131  wrex 3043  Vcvv 3332  cop 4319  {copab 4856  cmpt 4873  dom cdm 5258  cio 6002  wf 6037  1-1-ontowf1o 6040  cfv 6041  (class class class)co 6805  1st c1st 7323  2nd c2nd 7324  Fincfn 8113  1c1 10121  cn 11204  0cn0 11476  ...cfz 12511  seqcseq 12987  chash 13303  Basecbs 16051  +gcplusg 16135  CMndccmn 18385   FinSum cfinsum 33448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8947  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-n0 11477  df-z 11562  df-uz 11872  df-fz 12512  df-seq 12988  df-hash 13304  df-bj-finsum 33449
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator