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Theorem smueqlem 16583
Description: Any element of a sequence multiplication only depends on the values of the argument sequences up to and including that point. (Contributed by Mario Carneiro, 20-Sep-2016.)
Hypotheses
Ref Expression
smueq.a (𝜑 → 𝐴 ⊆ ℕ0)
smueq.b (𝜑 → 𝐵 ⊆ ℕ0)
smueq.n (𝜑 → 𝑁 ∈ ℕ0)
smueq.p 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
smueq.q 𝑄 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ (𝐵 ∩ (0..^𝑁)))})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
Assertion
Ref Expression
smueqlem (𝜑 → ((𝐴 smul 𝐵) ∩ (0..^𝑁)) = (((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∩ (0..^𝑁)))
Distinct variable groups:   𝑚,𝑛,𝑝,𝐴   𝐵,𝑚,𝑛,𝑝   𝑚,𝑁,𝑛,𝑝   𝜑,𝑛
Allowed substitution hints:   𝜑(𝑚, 𝑝)   𝑃(𝑚, 𝑛, 𝑝)   𝑄(𝑚, 𝑛, 𝑝)
Proof of Theorem smueqlem
Dummy variables 𝑘 𝑖 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smueq.a . . . . . . . 8 (𝜑 → 𝐴 ⊆ ℕ0)
21adantr 486 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝐴 ⊆ ℕ0)
3 smueq.b . . . . . . . 8 (𝜑 → 𝐵 ⊆ ℕ0)
43adantr 486 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝐵 ⊆ ℕ0)
5 smueq.p . . . . . . 7 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
6 elfzouz 13722 . . . . . . . . 9 (𝑘 ∈ (0..^𝑁) → 𝑘 ∈ (ℤ≥‘0))
76adantl 487 . . . . . . . 8 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ (ℤ≥‘0))
8 nn0uz 12928 . . . . . . . 8 ℕ0 = (ℤ≥‘0)
97, 8eleqtrrdi 2871 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℕ0)
109nn0zd 12643 . . . . . . . . 9 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℤ)
1110peano2zd 12731 . . . . . . . 8 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ∈ ℤ)
12 smueq.n . . . . . . . . . 10 (𝜑 → 𝑁 ∈ ℕ0)
1312adantr 486 . . . . . . . . 9 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ0)
1413nn0zd 12643 . . . . . . . 8 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ)
15 elfzolt2 13727 . . . . . . . . . 10 (𝑘 ∈ (0..^𝑁) → 𝑘 < 𝑁)
1615adantl 487 . . . . . . . . 9 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 < 𝑁)
17 nn0ltp1le 12682 . . . . . . . . . 10 ((𝑘 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑘 < 𝑁 ↔ (𝑘 + 1) ≤ 𝑁))
189, 13, 17syl2anc 596 . . . . . . . . 9 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 < 𝑁 ↔ (𝑘 + 1) ≤ 𝑁))
1916, 18mpbid 235 . . . . . . . 8 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ≤ 𝑁)
20 eluz2 12896 . . . . . . . 8 (𝑁 ∈ (ℤ≥‘(𝑘 + 1)) ↔ ((𝑘 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑁))
2111, 14, 19, 20syl3anbrc 1362 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ (ℤ≥‘(𝑘 + 1)))
222, 4, 5, 9, 21smuval2 16575 . . . . . 6 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ (𝑃‘𝑁)))
2312, 8eleqtrdi 2870 . . . . . . . . . . 11 (𝜑 → 𝑁 ∈ (ℤ≥‘0))
24 eluzfz2b 13590 . . . . . . . . . . 11 (𝑁 ∈ (ℤ≥‘0) ↔ 𝑁 ∈ (0...𝑁))
2523, 24sylib 221 . . . . . . . . . 10 (𝜑 → 𝑁 ∈ (0...𝑁))
26 fveq2 6879 . . . . . . . . . . . . . 14 (𝑥 = 0 → (𝑃‘𝑥) = (𝑃‘0))
2726ineq1d 4165 . . . . . . . . . . . . 13 (𝑥 = 0 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘0) ∩ (0..^𝑁)))
28 fveq2 6879 . . . . . . . . . . . . . 14 (𝑥 = 0 → (𝑄‘𝑥) = (𝑄‘0))
2928ineq1d 4165 . . . . . . . . . . . . 13 (𝑥 = 0 → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁)))
3027, 29eqeq12d 2776 . . . . . . . . . . . 12 (𝑥 = 0 → (((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁)) ↔ ((𝑃‘0) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁))))
3130imbi2d 343 . . . . . . . . . . 11 (𝑥 = 0 → ((𝜑 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃‘0) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁)))))
32 fveq2 6879 . . . . . . . . . . . . . 14 (𝑥 = 𝑖 → (𝑃‘𝑥) = (𝑃‘𝑖))
3332ineq1d 4165 . . . . . . . . . . . . 13 (𝑥 = 𝑖 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘𝑖) ∩ (0..^𝑁)))
34 fveq2 6879 . . . . . . . . . . . . . 14 (𝑥 = 𝑖 → (𝑄‘𝑥) = (𝑄‘𝑖))
3534ineq1d 4165 . . . . . . . . . . . . 13 (𝑥 = 𝑖 → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)))
3633, 35eqeq12d 2776 . . . . . . . . . . . 12 (𝑥 = 𝑖 → (((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁)) ↔ ((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁))))
3736imbi2d 343 . . . . . . . . . . 11 (𝑥 = 𝑖 → ((𝜑 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)))))
38 fveq2 6879 . . . . . . . . . . . . . 14 (𝑥 = (𝑖 + 1) → (𝑃‘𝑥) = (𝑃‘(𝑖 + 1)))
3938ineq1d 4165 . . . . . . . . . . . . 13 (𝑥 = (𝑖 + 1) → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)))
40 fveq2 6879 . . . . . . . . . . . . . 14 (𝑥 = (𝑖 + 1) → (𝑄‘𝑥) = (𝑄‘(𝑖 + 1)))
4140ineq1d 4165 . . . . . . . . . . . . 13 (𝑥 = (𝑖 + 1) → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))
4239, 41eqeq12d 2776 . . . . . . . . . . . 12 (𝑥 = (𝑖 + 1) → (((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁)) ↔ ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁))))
4342imbi2d 343 . . . . . . . . . . 11 (𝑥 = (𝑖 + 1) → ((𝜑 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))))
44 fveq2 6879 . . . . . . . . . . . . . 14 (𝑥 = 𝑁 → (𝑃‘𝑥) = (𝑃‘𝑁))
4544ineq1d 4165 . . . . . . . . . . . . 13 (𝑥 = 𝑁 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘𝑁) ∩ (0..^𝑁)))
46 fveq2 6879 . . . . . . . . . . . . . 14 (𝑥 = 𝑁 → (𝑄‘𝑥) = (𝑄‘𝑁))
4746ineq1d 4165 . . . . . . . . . . . . 13 (𝑥 = 𝑁 → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁)))
4845, 47eqeq12d 2776 . . . . . . . . . . . 12 (𝑥 = 𝑁 → (((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁)) ↔ ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁))))
4948imbi2d 343 . . . . . . . . . . 11 (𝑥 = 𝑁 → ((𝜑 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁)))))
501, 3, 5smup0 16572 . . . . . . . . . . . . . 14 (𝜑 → (𝑃‘0) = ∅)
51 inss1 4182 . . . . . . . . . . . . . . . 16 (𝐵 ∩ (0..^𝑁)) ⊆ 𝐵
5251, 3sstrid 3942 . . . . . . . . . . . . . . 15 (𝜑 → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ0)
53 smueq.q . . . . . . . . . . . . . . 15 𝑄 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝐴 ∧ (𝑛 − 𝑚) ∈ (𝐵 ∩ (0..^𝑁)))})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
541, 52, 53smup0 16572 . . . . . . . . . . . . . 14 (𝜑 → (𝑄‘0) = ∅)
5550, 54eqtr4d 2798 . . . . . . . . . . . . 13 (𝜑 → (𝑃‘0) = (𝑄‘0))
5655ineq1d 4165 . . . . . . . . . . . 12 (𝜑 → ((𝑃‘0) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁)))
5756a1i 11 . . . . . . . . . . 11 (𝑁 ∈ (ℤ≥‘0) → (𝜑 → ((𝑃‘0) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁))))
58 oveq1 7421 . . . . . . . . . . . . . . 15 (((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)) → (((𝑃‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) = (((𝑄‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))))
5958ineq1d 4165 . . . . . . . . . . . . . 14 (((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)) → ((((𝑃‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)) = ((((𝑄‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)))
601adantr 486 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝐴 ⊆ ℕ0)
613adantr 486 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝐵 ⊆ ℕ0)
62 elfzonn0 13766 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℕ0)
6362adantl 487 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0)
6460, 61, 5, 63smupp1 16573 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑃‘(𝑖 + 1)) = ((𝑃‘𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)}))
6564ineq1d 4165 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = (((𝑃‘𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)}) ∩ (0..^𝑁)))
661, 3, 5smupf 16571 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 𝑃:ℕ0⟶𝒫 ℕ0)
67 ffvelcdm 7075 . . . . . . . . . . . . . . . . . . 19 ((𝑃:ℕ0⟶𝒫 ℕ0 ∧ 𝑖 ∈ ℕ0) → (𝑃‘𝑖) ∈ 𝒫 ℕ0)
6866, 62, 67syl2an 608 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑃‘𝑖) ∈ 𝒫 ℕ0)
6968elpwid 4566 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑃‘𝑖) ⊆ ℕ0)
70 ssrab2 4028 . . . . . . . . . . . . . . . . . 18 {𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ⊆ ℕ0
7170a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → {𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ⊆ ℕ0)
7212adantr 486 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ0)
7369, 71, 72sadeq 16565 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (((𝑃‘𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)}) ∩ (0..^𝑁)) = ((((𝑃‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)))
7465, 73eqtrd 2795 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((((𝑃‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)))
7552adantr 486 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ0)
7660, 75, 53, 63smupp1 16573 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑄‘(𝑖 + 1)) = ((𝑄‘𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))}))
7776ineq1d 4165 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)) = (((𝑄‘𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))}) ∩ (0..^𝑁)))
781, 52, 53smupf 16571 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 𝑄:ℕ0⟶𝒫 ℕ0)
79 ffvelcdm 7075 . . . . . . . . . . . . . . . . . . 19 ((𝑄:ℕ0⟶𝒫 ℕ0 ∧ 𝑖 ∈ ℕ0) → (𝑄‘𝑖) ∈ 𝒫 ℕ0)
8078, 62, 79syl2an 608 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑄‘𝑖) ∈ 𝒫 ℕ0)
8180elpwid 4566 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑄‘𝑖) ⊆ ℕ0)
82 ssrab2 4028 . . . . . . . . . . . . . . . . . 18 {𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ⊆ ℕ0
8382a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → {𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ⊆ ℕ0)
8481, 83, 72sadeq 16565 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (((𝑄‘𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))}) ∩ (0..^𝑁)) = ((((𝑄‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ∩ (0..^𝑁))) ∩ (0..^𝑁)))
85 elinel2 4148 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ (ℕ0 ∩ (0..^𝑁)) → 𝑛 ∈ (0..^𝑁))
8661adantr 486 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝐵 ⊆ ℕ0)
8786sseld 3930 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ 𝐵 → (𝑛 − 𝑖) ∈ ℕ0))
88 elfzo0 13759 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 ∈ (0..^𝑁) ↔ (𝑛 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑛 < 𝑁))
8988simp2bi 1164 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ (0..^𝑁) → 𝑁 ∈ ℕ)
9089adantl 487 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ)
91 elfzonn0 13766 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 ∈ (0..^𝑁) → 𝑛 ∈ ℕ0)
9291adantl 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℕ0)
9392nn0red 12593 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℝ)
9463adantr 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0)
9594nn0red 12593 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑖 ∈ ℝ)
9693, 95resubcld 11669 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛 − 𝑖) ∈ ℝ)
9790nnred 12275 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑁 ∈ ℝ)
9894nn0ge0d 12595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 0 ≤ 𝑖)
9993, 95subge02d 11833 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (0 ≤ 𝑖 ↔ (𝑛 − 𝑖) ≤ 𝑛))
10098, 99mpbid 235 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛 − 𝑖) ≤ 𝑛)
101 elfzolt2 13727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 ∈ (0..^𝑁) → 𝑛 < 𝑁)
102101adantl 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 < 𝑁)
10396, 93, 97, 100, 102lelttrd 11395 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛 − 𝑖) < 𝑁)
10490, 103jca 521 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁))
105 elfzo0 13759 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑛 − 𝑖) ∈ (0..^𝑁) ↔ ((𝑛 − 𝑖) ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁))
106 3anass 1111 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑛 − 𝑖) ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁) ↔ ((𝑛 − 𝑖) ∈ ℕ0 ∧ (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁)))
107105, 106bitri 278 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛 − 𝑖) ∈ (0..^𝑁) ↔ ((𝑛 − 𝑖) ∈ ℕ0 ∧ (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁)))
108107baib 545 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑛 − 𝑖) ∈ ℕ0 → ((𝑛 − 𝑖) ∈ (0..^𝑁) ↔ (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁)))
109104, 108syl5ibrcom 250 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ ℕ0 → (𝑛 − 𝑖) ∈ (0..^𝑁)))
11087, 109syld 48 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ 𝐵 → (𝑛 − 𝑖) ∈ (0..^𝑁)))
111110pm4.71rd 572 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ 𝐵 ↔ ((𝑛 − 𝑖) ∈ (0..^𝑁) ∧ (𝑛 − 𝑖) ∈ 𝐵)))
112 ancom 466 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 − 𝑖) ∈ (0..^𝑁) ∧ (𝑛 − 𝑖) ∈ 𝐵) ↔ ((𝑛 − 𝑖) ∈ 𝐵 ∧ (𝑛 − 𝑖) ∈ (0..^𝑁)))
113 elin 3915 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)) ↔ ((𝑛 − 𝑖) ∈ 𝐵 ∧ (𝑛 − 𝑖) ∈ (0..^𝑁)))
114112, 113bitr4i 281 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 − 𝑖) ∈ (0..^𝑁) ∧ (𝑛 − 𝑖) ∈ 𝐵) ↔ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))
115111, 114bitr2di 291 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)) ↔ (𝑛 − 𝑖) ∈ 𝐵))
116115anbi2d 642 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁))) ↔ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)))
11785, 116sylan2 605 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (ℕ0 ∩ (0..^𝑁))) → ((𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁))) ↔ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)))
118117rabbidva 3418 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → {𝑛 ∈ (ℕ0 ∩ (0..^𝑁)) ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} = {𝑛 ∈ (ℕ0 ∩ (0..^𝑁)) ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)})
119 inrab2 4263 . . . . . . . . . . . . . . . . . . 19 ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ∩ (0..^𝑁)) = {𝑛 ∈ (ℕ0 ∩ (0..^𝑁)) ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))}
120 inrab2 4263 . . . . . . . . . . . . . . . . . . 19 ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁)) = {𝑛 ∈ (ℕ0 ∩ (0..^𝑁)) ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)}
121118, 119, 1203eqtr4g 2820 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ∩ (0..^𝑁)) = ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁)))
122121oveq2d 7430 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (((𝑄‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ∩ (0..^𝑁))) = (((𝑄‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))))
123122ineq1d 4165 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((((𝑄‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ∩ (0..^𝑁))) ∩ (0..^𝑁)) = ((((𝑄‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)))
12477, 84, 1233eqtrd 2799 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((((𝑄‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)))
12574, 124eqeq12d 2776 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)) ↔ ((((𝑃‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)) = ((((𝑄‘𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁))))
12659, 125imbitrrid 249 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)) → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁))))
127126expcom 419 . . . . . . . . . . . 12 (𝑖 ∈ (0..^𝑁) → (𝜑 → (((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)) → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))))
128127a2d 30 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑁) → ((𝜑 → ((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁))) → (𝜑 → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))))
12931, 37, 43, 49, 57, 128fzind2 13847 . . . . . . . . . 10 (𝑁 ∈ (0...𝑁) → (𝜑 → ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁))))
13025, 129mpcom 39 . . . . . . . . 9 (𝜑 → ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁)))
131130adantr 486 . . . . . . . 8 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁)))
132131eleq2d 2846 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁))))
133 elin 3915 . . . . . . . . 9 (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝑃‘𝑁) ∧ 𝑘 ∈ (0..^𝑁)))
134133rbaib 548 . . . . . . . 8 (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑃‘𝑁)))
135134adantl 487 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑃‘𝑁)))
136 elin 3915 . . . . . . . . 9 (𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝑄‘𝑁) ∧ 𝑘 ∈ (0..^𝑁)))
137136rbaib 548 . . . . . . . 8 (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑄‘𝑁)))
138137adantl 487 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑄‘𝑁)))
139132, 135, 1383bitr3d 312 . . . . . 6 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝑃‘𝑁) ↔ 𝑘 ∈ (𝑄‘𝑁)))
14052adantr 486 . . . . . . . 8 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ0)
1412, 140, 53, 13smupval 16581 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑄‘𝑁) = ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))))
142141eleq2d 2846 . . . . . 6 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝑄‘𝑁) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁)))))
14322, 139, 1423bitrd 308 . . . . 5 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁)))))
144143ex 418 . . . 4 (𝜑 → (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))))))
145144pm5.32rd 589 . . 3 (𝜑 → ((𝑘 ∈ (𝐴 smul 𝐵) ∧ 𝑘 ∈ (0..^𝑁)) ↔ (𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∧ 𝑘 ∈ (0..^𝑁))))
146 elin 3915 . . 3 (𝑘 ∈ ((𝐴 smul 𝐵) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝐴 smul 𝐵) ∧ 𝑘 ∈ (0..^𝑁)))
147 elin 3915 . . 3 (𝑘 ∈ (((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∩ (0..^𝑁)) ↔ (𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∧ 𝑘 ∈ (0..^𝑁)))
148145, 146, 1473bitr4g 317 . 2 (𝜑 → (𝑘 ∈ ((𝐴 smul 𝐵) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∩ (0..^𝑁))))
149148eqrdv 2758 1 (𝜑 → ((𝐴 smul 𝐵) ∩ (0..^𝑁)) = (((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∩ (0..^𝑁)))
Colors of variables:    wff setvar class
This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {crab 3412   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  ifcif 4482  𝒫 cpw 4557   class class class wbr 5103   ↦ cmpt 5186  ⟶wf 6529  ‘cfv 6533  (class class class)co 7414   ∈ cmpo 7416  0cc0 11127  1c1 11128   + caddc 11130   < clt 11270   ≤ cle 11271   − cmin 11468  ℕcn 12260  ℕ0cn0 12531  ℤcz 12618  ℤ≥cuz 12890  ...cfz 13564  ..^cfzo 13712  seqcseq 14068   sadd csad 16513   smul csmu 16514
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-inf2 9623  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-pre-sup 11205
This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-xor 1542  df-tru 1573  df-fal 1583  df-had 1624  df-cad 1640  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-disj 5071  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8458  df-2o 8459  df-oadd 8462  df-er 8699  df-map 8831  df-pm 8832  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9415  df-inf 9416  df-oi 9485  df-dju 9909  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-div 11899  df-nn 12261  df-2 12330  df-3 12331  df-n0 12532  df-xnn0 12605  df-z 12619  df-uz 12891  df-rp 13046  df-fz 13565  df-fzo 13713  df-fl 13856  df-mod 13934  df-seq 14069  df-exp 14129  df-hash 14398  df-cj 15189  df-re 15190  df-im 15191  df-sqrt 15325  df-abs 15326  df-clim 15578  df-sum 15777  df-dvds 16346  df-bits 16515  df-sad 16544  df-smu 16569
This theorem is used by:  smueq  16584
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