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Theorem smueqlem 15841
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 483 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝐴 ⊆ ℕ0)
3 smueq.b . . . . . . . 8 (𝜑 → 𝐵 ⊆ ℕ0)
43adantr 483 . . . . . . 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 13045 . . . . . . . . 9 (𝑘 ∈ (0..^𝑁) → 𝑘 ∈ (ℤ≥‘0))
76adantl 484 . . . . . . . 8 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ (ℤ≥‘0))
8 nn0uz 12283 . . . . . . . 8 ℕ0 = (ℤ≥‘0)
97, 8eleqtrrdi 2926 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℕ0)
109nn0zd 12088 . . . . . . . . 9 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℤ)
1110peano2zd 12093 . . . . . . . 8 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ∈ ℤ)
12 smueq.n . . . . . . . . . 10 (𝜑 → 𝑁 ∈ ℕ0)
1312adantr 483 . . . . . . . . 9 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ0)
1413nn0zd 12088 . . . . . . . 8 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ)
15 elfzolt2 13050 . . . . . . . . . 10 (𝑘 ∈ (0..^𝑁) → 𝑘 < 𝑁)
1615adantl 484 . . . . . . . . 9 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 < 𝑁)
17 nn0ltp1le 12043 . . . . . . . . . 10 ((𝑘 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑘 < 𝑁 ↔ (𝑘 + 1) ≤ 𝑁))
189, 13, 17syl2anc 586 . . . . . . . . 9 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 < 𝑁 ↔ (𝑘 + 1) ≤ 𝑁))
1916, 18mpbid 234 . . . . . . . 8 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ≤ 𝑁)
20 eluz2 12252 . . . . . . . 8 (𝑁 ∈ (ℤ≥‘(𝑘 + 1)) ↔ ((𝑘 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑁))
2111, 14, 19, 20syl3anbrc 1339 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ (ℤ≥‘(𝑘 + 1)))
222, 4, 5, 9, 21smuval2 15833 . . . . . 6 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ (𝑃‘𝑁)))
2312, 8eleqtrdi 2925 . . . . . . . . . . 11 (𝜑 → 𝑁 ∈ (ℤ≥‘0))
24 eluzfz2b 12919 . . . . . . . . . . 11 (𝑁 ∈ (ℤ≥‘0) ↔ 𝑁 ∈ (0...𝑁))
2523, 24sylib 220 . . . . . . . . . 10 (𝜑 → 𝑁 ∈ (0...𝑁))
26 fveq2 6672 . . . . . . . . . . . . . 14 (𝑥 = 0 → (𝑃‘𝑥) = (𝑃‘0))
2726ineq1d 4190 . . . . . . . . . . . . 13 (𝑥 = 0 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘0) ∩ (0..^𝑁)))
28 fveq2 6672 . . . . . . . . . . . . . 14 (𝑥 = 0 → (𝑄‘𝑥) = (𝑄‘0))
2928ineq1d 4190 . . . . . . . . . . . . 13 (𝑥 = 0 → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁)))
3027, 29eqeq12d 2839 . . . . . . . . . . . 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 6672 . . . . . . . . . . . . . 14 (𝑥 = 𝑖 → (𝑃‘𝑥) = (𝑃‘𝑖))
3332ineq1d 4190 . . . . . . . . . . . . 13 (𝑥 = 𝑖 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘𝑖) ∩ (0..^𝑁)))
34 fveq2 6672 . . . . . . . . . . . . . 14 (𝑥 = 𝑖 → (𝑄‘𝑥) = (𝑄‘𝑖))
3534ineq1d 4190 . . . . . . . . . . . . 13 (𝑥 = 𝑖 → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)))
3633, 35eqeq12d 2839 . . . . . . . . . . . 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 6672 . . . . . . . . . . . . . 14 (𝑥 = (𝑖 + 1) → (𝑃‘𝑥) = (𝑃‘(𝑖 + 1)))
3938ineq1d 4190 . . . . . . . . . . . . 13 (𝑥 = (𝑖 + 1) → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)))
40 fveq2 6672 . . . . . . . . . . . . . 14 (𝑥 = (𝑖 + 1) → (𝑄‘𝑥) = (𝑄‘(𝑖 + 1)))
4140ineq1d 4190 . . . . . . . . . . . . 13 (𝑥 = (𝑖 + 1) → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))
4239, 41eqeq12d 2839 . . . . . . . . . . . 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 6672 . . . . . . . . . . . . . 14 (𝑥 = 𝑁 → (𝑃‘𝑥) = (𝑃‘𝑁))
4544ineq1d 4190 . . . . . . . . . . . . 13 (𝑥 = 𝑁 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘𝑁) ∩ (0..^𝑁)))
46 fveq2 6672 . . . . . . . . . . . . . 14 (𝑥 = 𝑁 → (𝑄‘𝑥) = (𝑄‘𝑁))
4746ineq1d 4190 . . . . . . . . . . . . 13 (𝑥 = 𝑁 → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁)))
4845, 47eqeq12d 2839 . . . . . . . . . . . 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 15830 . . . . . . . . . . . . . 14 (𝜑 → (𝑃‘0) = ∅)
51 inss1 4207 . . . . . . . . . . . . . . . 16 (𝐵 ∩ (0..^𝑁)) ⊆ 𝐵
5251, 3sstrid 3980 . . . . . . . . . . . . . . 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 15830 . . . . . . . . . . . . . 14 (𝜑 → (𝑄‘0) = ∅)
5550, 54eqtr4d 2861 . . . . . . . . . . . . 13 (𝜑 → (𝑃‘0) = (𝑄‘0))
5655ineq1d 4190 . . . . . . . . . . . 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 7165 . . . . . . . . . . . . . . 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 4190 . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝐴 ⊆ ℕ0)
613adantr 483 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝐵 ⊆ ℕ0)
62 elfzonn0 13085 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℕ0)
6362adantl 484 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0)
6460, 61, 5, 63smupp1 15831 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑃‘(𝑖 + 1)) = ((𝑃‘𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)}))
6564ineq1d 4190 . . . . . . . . . . . . . . . 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 15829 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 𝑃:ℕ0⟶𝒫 ℕ0)
67 ffvelrn 6851 . . . . . . . . . . . . . . . . . . 19 ((𝑃:ℕ0⟶𝒫 ℕ0 ∧ 𝑖 ∈ ℕ0) → (𝑃‘𝑖) ∈ 𝒫 ℕ0)
6866, 62, 67syl2an 597 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑃‘𝑖) ∈ 𝒫 ℕ0)
6968elpwid 4552 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑃‘𝑖) ⊆ ℕ0)
70 ssrab2 4058 . . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ0)
7369, 71, 72sadeq 15823 . . . . . . . . . . . . . . . 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 2858 . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ0)
7660, 75, 53, 63smupp1 15831 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑄‘(𝑖 + 1)) = ((𝑄‘𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))}))
7776ineq1d 4190 . . . . . . . . . . . . . . . 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 15829 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 𝑄:ℕ0⟶𝒫 ℕ0)
79 ffvelrn 6851 . . . . . . . . . . . . . . . . . . 19 ((𝑄:ℕ0⟶𝒫 ℕ0 ∧ 𝑖 ∈ ℕ0) → (𝑄‘𝑖) ∈ 𝒫 ℕ0)
8078, 62, 79syl2an 597 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑄‘𝑖) ∈ 𝒫 ℕ0)
8180elpwid 4552 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑄‘𝑖) ⊆ ℕ0)
82 ssrab2 4058 . . . . . . . . . . . . . . . . . 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 15823 . . . . . . . . . . . . . . . 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 4175 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ (ℕ0 ∩ (0..^𝑁)) → 𝑛 ∈ (0..^𝑁))
8661adantr 483 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝐵 ⊆ ℕ0)
8786sseld 3968 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ 𝐵 → (𝑛 − 𝑖) ∈ ℕ0))
88 elfzo0 13081 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 ∈ (0..^𝑁) ↔ (𝑛 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑛 < 𝑁))
8988simp2bi 1142 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ (0..^𝑁) → 𝑁 ∈ ℕ)
9089adantl 484 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ)
91 elfzonn0 13085 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 ∈ (0..^𝑁) → 𝑛 ∈ ℕ0)
9291adantl 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℕ0)
9392nn0red 11959 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℝ)
9463adantr 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0)
9594nn0red 11959 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑖 ∈ ℝ)
9693, 95resubcld 11070 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛 − 𝑖) ∈ ℝ)
9790nnred 11655 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑁 ∈ ℝ)
9894nn0ge0d 11961 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 0 ≤ 𝑖)
9993, 95subge02d 11234 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (0 ≤ 𝑖 ↔ (𝑛 − 𝑖) ≤ 𝑛))
10098, 99mpbid 234 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛 − 𝑖) ≤ 𝑛)
101 elfzolt2 13050 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 ∈ (0..^𝑁) → 𝑛 < 𝑁)
102101adantl 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 < 𝑁)
10396, 93, 97, 100, 102lelttrd 10800 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛 − 𝑖) < 𝑁)
10490, 103jca 514 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁))
105 elfzo0 13081 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑛 − 𝑖) ∈ (0..^𝑁) ↔ ((𝑛 − 𝑖) ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁))
106 3anass 1091 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑛 − 𝑖) ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁) ↔ ((𝑛 − 𝑖) ∈ ℕ0 ∧ (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁)))
107105, 106bitri 277 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛 − 𝑖) ∈ (0..^𝑁) ↔ ((𝑛 − 𝑖) ∈ ℕ0 ∧ (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁)))
108107baib 538 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑛 − 𝑖) ∈ ℕ0 → ((𝑛 − 𝑖) ∈ (0..^𝑁) ↔ (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁)))
109104, 108syl5ibrcom 249 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ ℕ0 → (𝑛 − 𝑖) ∈ (0..^𝑁)))
11087, 109syld 47 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ 𝐵 → (𝑛 − 𝑖) ∈ (0..^𝑁)))
111110pm4.71rd 565 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ 𝐵 ↔ ((𝑛 − 𝑖) ∈ (0..^𝑁) ∧ (𝑛 − 𝑖) ∈ 𝐵)))
112 ancom 463 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 − 𝑖) ∈ (0..^𝑁) ∧ (𝑛 − 𝑖) ∈ 𝐵) ↔ ((𝑛 − 𝑖) ∈ 𝐵 ∧ (𝑛 − 𝑖) ∈ (0..^𝑁)))
113 elin 4171 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)) ↔ ((𝑛 − 𝑖) ∈ 𝐵 ∧ (𝑛 − 𝑖) ∈ (0..^𝑁)))
114112, 113bitr4i 280 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 − 𝑖) ∈ (0..^𝑁) ∧ (𝑛 − 𝑖) ∈ 𝐵) ↔ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))
115111, 114syl6rbb 290 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)) ↔ (𝑛 − 𝑖) ∈ 𝐵))
116115anbi2d 630 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁))) ↔ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)))
11785, 116sylan2 594 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (ℕ0 ∩ (0..^𝑁))) → ((𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁))) ↔ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)))
118117rabbidva 3480 . . . . . . . . . . . . . . . . . . 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 4278 . . . . . . . . . . . . . . . . . . 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 4278 . . . . . . . . . . . . . . . . . . 19 ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁)) = {𝑛 ∈ (ℕ0 ∩ (0..^𝑁)) ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)}
121118, 119, 1203eqtr4g 2883 . . . . . . . . . . . . . . . . . 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 7174 . . . . . . . . . . . . . . . . 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 4190 . . . . . . . . . . . . . . . 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 2862 . . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . . 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, 125syl5ibr 248 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)) → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁))))
127126expcom 416 . . . . . . . . . . . 12 (𝑖 ∈ (0..^𝑁) → (𝜑 → (((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)) → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))))
128127a2d 29 . . . . . . . . . . 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 13158 . . . . . . . . . 10 (𝑁 ∈ (0...𝑁) → (𝜑 → ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁))))
13025, 129mpcom 38 . . . . . . . . 9 (𝜑 → ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁)))
131130adantr 483 . . . . . . . 8 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁)))
132131eleq2d 2900 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁))))
133 elin 4171 . . . . . . . . 9 (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝑃‘𝑁) ∧ 𝑘 ∈ (0..^𝑁)))
134133rbaib 541 . . . . . . . 8 (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑃‘𝑁)))
135134adantl 484 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑃‘𝑁)))
136 elin 4171 . . . . . . . . 9 (𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝑄‘𝑁) ∧ 𝑘 ∈ (0..^𝑁)))
137136rbaib 541 . . . . . . . 8 (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑄‘𝑁)))
138137adantl 484 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑄‘𝑁)))
139132, 135, 1383bitr3d 311 . . . . . 6 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝑃‘𝑁) ↔ 𝑘 ∈ (𝑄‘𝑁)))
14052adantr 483 . . . . . . . 8 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ0)
1412, 140, 53, 13smupval 15839 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑄‘𝑁) = ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))))
142141eleq2d 2900 . . . . . 6 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝑄‘𝑁) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁)))))
14322, 139, 1423bitrd 307 . . . . 5 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁)))))
144143ex 415 . . . 4 (𝜑 → (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))))))
145144pm5.32rd 580 . . 3 (𝜑 → ((𝑘 ∈ (𝐴 smul 𝐵) ∧ 𝑘 ∈ (0..^𝑁)) ↔ (𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∧ 𝑘 ∈ (0..^𝑁))))
146 elin 4171 . . 3 (𝑘 ∈ ((𝐴 smul 𝐵) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝐴 smul 𝐵) ∧ 𝑘 ∈ (0..^𝑁)))
147 elin 4171 . . 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 316 . 2 (𝜑 → (𝑘 ∈ ((𝐴 smul 𝐵) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∩ (0..^𝑁))))
149148eqrdv 2821 1 (𝜑 → ((𝐴 smul 𝐵) ∩ (0..^𝑁)) = (((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∩ (0..^𝑁)))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  {crab 3144   ∩ cin 3937   ⊆ wss 3938  ∅c0 4293  ifcif 4469  𝒫 cpw 4541   class class class wbr 5068   ↦ cmpt 5148  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160  0cc0 10539  1c1 10540   + caddc 10542   < clt 10677   ≤ cle 10678   − cmin 10872  ℕcn 11640  ℕ0cn0 11900  ℤcz 11984  ℤ≥cuz 12246  ...cfz 12895  ..^cfzo 13036  seqcseq 13372   sadd csad 15771   smul csmu 15772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-xor 1502  df-tru 1540  df-fal 1550  df-had 1594  df-cad 1608  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-disj 5034  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-map 8410  df-pm 8411  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-sup 8908  df-inf 8909  df-oi 8976  df-dju 9332  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-n0 11901  df-xnn0 11971  df-z 11985  df-uz 12247  df-rp 12393  df-fz 12896  df-fzo 13037  df-fl 13165  df-mod 13241  df-seq 13373  df-exp 13433  df-hash 13694  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-clim 14847  df-sum 15045  df-dvds 15610  df-bits 15773  df-sad 15802  df-smu 15827
This theorem is referenced by:  smueq  15842
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