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Theorem smueqlem 16524
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 480 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝐴 ⊆ ℕ0)
3 smueq.b . . . . . . . 8 (𝜑 → 𝐵 ⊆ ℕ0)
43adantr 480 . . . . . . 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 13700 . . . . . . . . 9 (𝑘 ∈ (0..^𝑁) → 𝑘 ∈ (ℤ≥‘0))
76adantl 481 . . . . . . . 8 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ (ℤ≥‘0))
8 nn0uz 12918 . . . . . . . 8 ℕ0 = (ℤ≥‘0)
97, 8eleqtrrdi 2850 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℕ0)
109nn0zd 12637 . . . . . . . . 9 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℤ)
1110peano2zd 12723 . . . . . . . 8 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ∈ ℤ)
12 smueq.n . . . . . . . . . 10 (𝜑 → 𝑁 ∈ ℕ0)
1312adantr 480 . . . . . . . . 9 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ0)
1413nn0zd 12637 . . . . . . . 8 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ)
15 elfzolt2 13705 . . . . . . . . . 10 (𝑘 ∈ (0..^𝑁) → 𝑘 < 𝑁)
1615adantl 481 . . . . . . . . 9 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑘 < 𝑁)
17 nn0ltp1le 12674 . . . . . . . . . 10 ((𝑘 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑘 < 𝑁 ↔ (𝑘 + 1) ≤ 𝑁))
189, 13, 17syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 < 𝑁 ↔ (𝑘 + 1) ≤ 𝑁))
1916, 18mpbid 232 . . . . . . . 8 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ≤ 𝑁)
20 eluz2 12882 . . . . . . . 8 (𝑁 ∈ (ℤ≥‘(𝑘 + 1)) ↔ ((𝑘 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑁))
2111, 14, 19, 20syl3anbrc 1342 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ (ℤ≥‘(𝑘 + 1)))
222, 4, 5, 9, 21smuval2 16516 . . . . . 6 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ (𝑃‘𝑁)))
2312, 8eleqtrdi 2849 . . . . . . . . . . 11 (𝜑 → 𝑁 ∈ (ℤ≥‘0))
24 eluzfz2b 13570 . . . . . . . . . . 11 (𝑁 ∈ (ℤ≥‘0) ↔ 𝑁 ∈ (0...𝑁))
2523, 24sylib 218 . . . . . . . . . 10 (𝜑 → 𝑁 ∈ (0...𝑁))
26 fveq2 6907 . . . . . . . . . . . . . 14 (𝑥 = 0 → (𝑃‘𝑥) = (𝑃‘0))
2726ineq1d 4227 . . . . . . . . . . . . 13 (𝑥 = 0 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘0) ∩ (0..^𝑁)))
28 fveq2 6907 . . . . . . . . . . . . . 14 (𝑥 = 0 → (𝑄‘𝑥) = (𝑄‘0))
2928ineq1d 4227 . . . . . . . . . . . . 13 (𝑥 = 0 → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁)))
3027, 29eqeq12d 2751 . . . . . . . . . . . 12 (𝑥 = 0 → (((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁)) ↔ ((𝑃‘0) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁))))
3130imbi2d 340 . . . . . . . . . . 11 (𝑥 = 0 → ((𝜑 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃‘0) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁)))))
32 fveq2 6907 . . . . . . . . . . . . . 14 (𝑥 = 𝑖 → (𝑃‘𝑥) = (𝑃‘𝑖))
3332ineq1d 4227 . . . . . . . . . . . . 13 (𝑥 = 𝑖 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘𝑖) ∩ (0..^𝑁)))
34 fveq2 6907 . . . . . . . . . . . . . 14 (𝑥 = 𝑖 → (𝑄‘𝑥) = (𝑄‘𝑖))
3534ineq1d 4227 . . . . . . . . . . . . 13 (𝑥 = 𝑖 → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)))
3633, 35eqeq12d 2751 . . . . . . . . . . . 12 (𝑥 = 𝑖 → (((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁)) ↔ ((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁))))
3736imbi2d 340 . . . . . . . . . . 11 (𝑥 = 𝑖 → ((𝜑 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)))))
38 fveq2 6907 . . . . . . . . . . . . . 14 (𝑥 = (𝑖 + 1) → (𝑃‘𝑥) = (𝑃‘(𝑖 + 1)))
3938ineq1d 4227 . . . . . . . . . . . . 13 (𝑥 = (𝑖 + 1) → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)))
40 fveq2 6907 . . . . . . . . . . . . . 14 (𝑥 = (𝑖 + 1) → (𝑄‘𝑥) = (𝑄‘(𝑖 + 1)))
4140ineq1d 4227 . . . . . . . . . . . . 13 (𝑥 = (𝑖 + 1) → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))
4239, 41eqeq12d 2751 . . . . . . . . . . . 12 (𝑥 = (𝑖 + 1) → (((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁)) ↔ ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁))))
4342imbi2d 340 . . . . . . . . . . 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 6907 . . . . . . . . . . . . . 14 (𝑥 = 𝑁 → (𝑃‘𝑥) = (𝑃‘𝑁))
4544ineq1d 4227 . . . . . . . . . . . . 13 (𝑥 = 𝑁 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑃‘𝑁) ∩ (0..^𝑁)))
46 fveq2 6907 . . . . . . . . . . . . . 14 (𝑥 = 𝑁 → (𝑄‘𝑥) = (𝑄‘𝑁))
4746ineq1d 4227 . . . . . . . . . . . . 13 (𝑥 = 𝑁 → ((𝑄‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁)))
4845, 47eqeq12d 2751 . . . . . . . . . . . 12 (𝑥 = 𝑁 → (((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁)) ↔ ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁))))
4948imbi2d 340 . . . . . . . . . . 11 (𝑥 = 𝑁 → ((𝜑 → ((𝑃‘𝑥) ∩ (0..^𝑁)) = ((𝑄‘𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁)))))
501, 3, 5smup0 16513 . . . . . . . . . . . . . 14 (𝜑 → (𝑃‘0) = ∅)
51 inss1 4245 . . . . . . . . . . . . . . . 16 (𝐵 ∩ (0..^𝑁)) ⊆ 𝐵
5251, 3sstrid 4007 . . . . . . . . . . . . . . 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 16513 . . . . . . . . . . . . . 14 (𝜑 → (𝑄‘0) = ∅)
5550, 54eqtr4d 2778 . . . . . . . . . . . . 13 (𝜑 → (𝑃‘0) = (𝑄‘0))
5655ineq1d 4227 . . . . . . . . . . . 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 7438 . . . . . . . . . . . . . . 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 4227 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝐴 ⊆ ℕ0)
613adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝐵 ⊆ ℕ0)
62 elfzonn0 13744 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℕ0)
6362adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0)
6460, 61, 5, 63smupp1 16514 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑃‘(𝑖 + 1)) = ((𝑃‘𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)}))
6564ineq1d 4227 . . . . . . . . . . . . . . . 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 16512 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 𝑃:ℕ0⟶𝒫 ℕ0)
67 ffvelcdm 7101 . . . . . . . . . . . . . . . . . . 19 ((𝑃:ℕ0⟶𝒫 ℕ0 ∧ 𝑖 ∈ ℕ0) → (𝑃‘𝑖) ∈ 𝒫 ℕ0)
6866, 62, 67syl2an 596 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑃‘𝑖) ∈ 𝒫 ℕ0)
6968elpwid 4614 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑃‘𝑖) ⊆ ℕ0)
70 ssrab2 4090 . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ0)
7369, 71, 72sadeq 16506 . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ0)
7660, 75, 53, 63smupp1 16514 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑄‘(𝑖 + 1)) = ((𝑄‘𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))}))
7776ineq1d 4227 . . . . . . . . . . . . . . . 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 16512 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 𝑄:ℕ0⟶𝒫 ℕ0)
79 ffvelcdm 7101 . . . . . . . . . . . . . . . . . . 19 ((𝑄:ℕ0⟶𝒫 ℕ0 ∧ 𝑖 ∈ ℕ0) → (𝑄‘𝑖) ∈ 𝒫 ℕ0)
8078, 62, 79syl2an 596 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑄‘𝑖) ∈ 𝒫 ℕ0)
8180elpwid 4614 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (𝑄‘𝑖) ⊆ ℕ0)
82 ssrab2 4090 . . . . . . . . . . . . . . . . . 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 16506 . . . . . . . . . . . . . . . 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 4212 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ (ℕ0 ∩ (0..^𝑁)) → 𝑛 ∈ (0..^𝑁))
8661adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝐵 ⊆ ℕ0)
8786sseld 3994 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ 𝐵 → (𝑛 − 𝑖) ∈ ℕ0))
88 elfzo0 13737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 ∈ (0..^𝑁) ↔ (𝑛 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑛 < 𝑁))
8988simp2bi 1145 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ (0..^𝑁) → 𝑁 ∈ ℕ)
9089adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ)
91 elfzonn0 13744 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 ∈ (0..^𝑁) → 𝑛 ∈ ℕ0)
9291adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℕ0)
9392nn0red 12586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℝ)
9463adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0)
9594nn0red 12586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑖 ∈ ℝ)
9693, 95resubcld 11689 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛 − 𝑖) ∈ ℝ)
9790nnred 12279 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑁 ∈ ℝ)
9894nn0ge0d 12588 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 0 ≤ 𝑖)
9993, 95subge02d 11853 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (0 ≤ 𝑖 ↔ (𝑛 − 𝑖) ≤ 𝑛))
10098, 99mpbid 232 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛 − 𝑖) ≤ 𝑛)
101 elfzolt2 13705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 ∈ (0..^𝑁) → 𝑛 < 𝑁)
102101adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 < 𝑁)
10396, 93, 97, 100, 102lelttrd 11417 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛 − 𝑖) < 𝑁)
10490, 103jca 511 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁))
105 elfzo0 13737 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑛 − 𝑖) ∈ (0..^𝑁) ↔ ((𝑛 − 𝑖) ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁))
106 3anass 1094 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑛 − 𝑖) ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁) ↔ ((𝑛 − 𝑖) ∈ ℕ0 ∧ (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁)))
107105, 106bitri 275 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛 − 𝑖) ∈ (0..^𝑁) ↔ ((𝑛 − 𝑖) ∈ ℕ0 ∧ (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁)))
108107baib 535 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑛 − 𝑖) ∈ ℕ0 → ((𝑛 − 𝑖) ∈ (0..^𝑁) ↔ (𝑁 ∈ ℕ ∧ (𝑛 − 𝑖) < 𝑁)))
109104, 108syl5ibrcom 247 . . . . . . . . . . . . . . . . . . . . . . . . 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 562 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛 − 𝑖) ∈ 𝐵 ↔ ((𝑛 − 𝑖) ∈ (0..^𝑁) ∧ (𝑛 − 𝑖) ∈ 𝐵)))
112 ancom 460 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 − 𝑖) ∈ (0..^𝑁) ∧ (𝑛 − 𝑖) ∈ 𝐵) ↔ ((𝑛 − 𝑖) ∈ 𝐵 ∧ (𝑛 − 𝑖) ∈ (0..^𝑁)))
113 elin 3979 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)) ↔ ((𝑛 − 𝑖) ∈ 𝐵 ∧ (𝑛 − 𝑖) ∈ (0..^𝑁)))
114112, 113bitr4i 278 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛 − 𝑖) ∈ (0..^𝑁) ∧ (𝑛 − 𝑖) ∈ 𝐵) ↔ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁)))
115111, 114bitr2di 288 . . . . . . . . . . . . . . . . . . . . . 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 593 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (ℕ0 ∩ (0..^𝑁))) → ((𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ (𝐵 ∩ (0..^𝑁))) ↔ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)))
118117rabbidva 3440 . . . . . . . . . . . . . . . . . . 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 4323 . . . . . . . . . . . . . . . . . . 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 4323 . . . . . . . . . . . . . . . . . . 19 ({𝑛 ∈ ℕ0 ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)} ∩ (0..^𝑁)) = {𝑛 ∈ (ℕ0 ∩ (0..^𝑁)) ∣ (𝑖 ∈ 𝐴 ∧ (𝑛 − 𝑖) ∈ 𝐵)}
121118, 119, 1203eqtr4g 2800 . . . . . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . . . . . 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 4227 . . . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . . . . 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 2751 . . . . . . . . . . . . . 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 246 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (((𝑃‘𝑖) ∩ (0..^𝑁)) = ((𝑄‘𝑖) ∩ (0..^𝑁)) → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁))))
127126expcom 413 . . . . . . . . . . . 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 13821 . . . . . . . . . 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 480 . . . . . . . 8 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → ((𝑃‘𝑁) ∩ (0..^𝑁)) = ((𝑄‘𝑁) ∩ (0..^𝑁)))
132131eleq2d 2825 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁))))
133 elin 3979 . . . . . . . . 9 (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝑃‘𝑁) ∧ 𝑘 ∈ (0..^𝑁)))
134133rbaib 538 . . . . . . . 8 (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑃‘𝑁)))
135134adantl 481 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑃‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑃‘𝑁)))
136 elin 3979 . . . . . . . . 9 (𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝑄‘𝑁) ∧ 𝑘 ∈ (0..^𝑁)))
137136rbaib 538 . . . . . . . 8 (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑄‘𝑁)))
138137adantl 481 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑄‘𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑄‘𝑁)))
139132, 135, 1383bitr3d 309 . . . . . 6 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝑃‘𝑁) ↔ 𝑘 ∈ (𝑄‘𝑁)))
14052adantr 480 . . . . . . . 8 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ0)
1412, 140, 53, 13smupval 16522 . . . . . . 7 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑄‘𝑁) = ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))))
142141eleq2d 2825 . . . . . 6 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝑄‘𝑁) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁)))))
14322, 139, 1423bitrd 305 . . . . 5 ((𝜑 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁)))))
144143ex 412 . . . 4 (𝜑 → (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))))))
145144pm5.32rd 578 . . 3 (𝜑 → ((𝑘 ∈ (𝐴 smul 𝐵) ∧ 𝑘 ∈ (0..^𝑁)) ↔ (𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∧ 𝑘 ∈ (0..^𝑁))))
146 elin 3979 . . 3 (𝑘 ∈ ((𝐴 smul 𝐵) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝐴 smul 𝐵) ∧ 𝑘 ∈ (0..^𝑁)))
147 elin 3979 . . 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 314 . 2 (𝜑 → (𝑘 ∈ ((𝐴 smul 𝐵) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∩ (0..^𝑁))))
149148eqrdv 2733 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  {crab 3433   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  ifcif 4531  𝒫 cpw 4605   class class class wbr 5148   ↦ cmpt 5231  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  0cc0 11153  1c1 11154   + caddc 11156   < clt 11293   ≤ cle 11294   − cmin 11490  ℕcn 12264  ℕ0cn0 12524  ℤcz 12611  ℤ≥cuz 12876  ...cfz 13544  ..^cfzo 13691  seqcseq 14039   sadd csad 16454   smul csmu 16455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1509  df-tru 1540  df-fal 1550  df-had 1591  df-cad 1604  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-disj 5116  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-sum 15720  df-dvds 16288  df-bits 16456  df-sad 16485  df-smu 16510
This theorem is referenced by:  smueq  16525
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