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Theorem smueqlem 15834
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 (𝜑𝐴 ⊆ ℕ0)
smueq.b (𝜑𝐵 ⊆ ℕ0)
smueq.n (𝜑𝑁 ∈ ℕ0)
smueq.p 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
smueq.q 𝑄 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ (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 (𝜑𝐴 ⊆ ℕ0)
21adantr 481 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝐴 ⊆ ℕ0)
3 smueq.b . . . . . . . 8 (𝜑𝐵 ⊆ ℕ0)
43adantr 481 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝐵 ⊆ ℕ0)
5 smueq.p . . . . . . 7 𝑃 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
6 elfzouz 13037 . . . . . . . . 9 (𝑘 ∈ (0..^𝑁) → 𝑘 ∈ (ℤ‘0))
76adantl 482 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ (ℤ‘0))
8 nn0uz 12274 . . . . . . . 8 0 = (ℤ‘0)
97, 8syl6eleqr 2929 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℕ0)
109nn0zd 12079 . . . . . . . . 9 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℤ)
1110peano2zd 12084 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ∈ ℤ)
12 smueq.n . . . . . . . . . 10 (𝜑𝑁 ∈ ℕ0)
1312adantr 481 . . . . . . . . 9 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ0)
1413nn0zd 12079 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ)
15 elfzolt2 13042 . . . . . . . . . 10 (𝑘 ∈ (0..^𝑁) → 𝑘 < 𝑁)
1615adantl 482 . . . . . . . . 9 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑘 < 𝑁)
17 nn0ltp1le 12034 . . . . . . . . . 10 ((𝑘 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑘 < 𝑁 ↔ (𝑘 + 1) ≤ 𝑁))
189, 13, 17syl2anc 584 . . . . . . . . 9 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 < 𝑁 ↔ (𝑘 + 1) ≤ 𝑁))
1916, 18mpbid 233 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ≤ 𝑁)
20 eluz2 12243 . . . . . . . 8 (𝑁 ∈ (ℤ‘(𝑘 + 1)) ↔ ((𝑘 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑁))
2111, 14, 19, 20syl3anbrc 1337 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ (ℤ‘(𝑘 + 1)))
222, 4, 5, 9, 21smuval2 15826 . . . . . 6 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ (𝑃𝑁)))
2312, 8syl6eleq 2928 . . . . . . . . . . 11 (𝜑𝑁 ∈ (ℤ‘0))
24 eluzfz2b 12911 . . . . . . . . . . 11 (𝑁 ∈ (ℤ‘0) ↔ 𝑁 ∈ (0...𝑁))
2523, 24sylib 219 . . . . . . . . . 10 (𝜑𝑁 ∈ (0...𝑁))
26 fveq2 6669 . . . . . . . . . . . . . 14 (𝑥 = 0 → (𝑃𝑥) = (𝑃‘0))
2726ineq1d 4192 . . . . . . . . . . . . 13 (𝑥 = 0 → ((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑃‘0) ∩ (0..^𝑁)))
28 fveq2 6669 . . . . . . . . . . . . . 14 (𝑥 = 0 → (𝑄𝑥) = (𝑄‘0))
2928ineq1d 4192 . . . . . . . . . . . . 13 (𝑥 = 0 → ((𝑄𝑥) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁)))
3027, 29eqeq12d 2842 . . . . . . . . . . . 12 (𝑥 = 0 → (((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑄𝑥) ∩ (0..^𝑁)) ↔ ((𝑃‘0) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁))))
3130imbi2d 342 . . . . . . . . . . 11 (𝑥 = 0 → ((𝜑 → ((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑄𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃‘0) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁)))))
32 fveq2 6669 . . . . . . . . . . . . . 14 (𝑥 = 𝑖 → (𝑃𝑥) = (𝑃𝑖))
3332ineq1d 4192 . . . . . . . . . . . . 13 (𝑥 = 𝑖 → ((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑃𝑖) ∩ (0..^𝑁)))
34 fveq2 6669 . . . . . . . . . . . . . 14 (𝑥 = 𝑖 → (𝑄𝑥) = (𝑄𝑖))
3534ineq1d 4192 . . . . . . . . . . . . 13 (𝑥 = 𝑖 → ((𝑄𝑥) ∩ (0..^𝑁)) = ((𝑄𝑖) ∩ (0..^𝑁)))
3633, 35eqeq12d 2842 . . . . . . . . . . . 12 (𝑥 = 𝑖 → (((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑄𝑥) ∩ (0..^𝑁)) ↔ ((𝑃𝑖) ∩ (0..^𝑁)) = ((𝑄𝑖) ∩ (0..^𝑁))))
3736imbi2d 342 . . . . . . . . . . 11 (𝑥 = 𝑖 → ((𝜑 → ((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑄𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃𝑖) ∩ (0..^𝑁)) = ((𝑄𝑖) ∩ (0..^𝑁)))))
38 fveq2 6669 . . . . . . . . . . . . . 14 (𝑥 = (𝑖 + 1) → (𝑃𝑥) = (𝑃‘(𝑖 + 1)))
3938ineq1d 4192 . . . . . . . . . . . . 13 (𝑥 = (𝑖 + 1) → ((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)))
40 fveq2 6669 . . . . . . . . . . . . . 14 (𝑥 = (𝑖 + 1) → (𝑄𝑥) = (𝑄‘(𝑖 + 1)))
4140ineq1d 4192 . . . . . . . . . . . . 13 (𝑥 = (𝑖 + 1) → ((𝑄𝑥) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))
4239, 41eqeq12d 2842 . . . . . . . . . . . 12 (𝑥 = (𝑖 + 1) → (((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑄𝑥) ∩ (0..^𝑁)) ↔ ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁))))
4342imbi2d 342 . . . . . . . . . . 11 (𝑥 = (𝑖 + 1) → ((𝜑 → ((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑄𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))))
44 fveq2 6669 . . . . . . . . . . . . . 14 (𝑥 = 𝑁 → (𝑃𝑥) = (𝑃𝑁))
4544ineq1d 4192 . . . . . . . . . . . . 13 (𝑥 = 𝑁 → ((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑃𝑁) ∩ (0..^𝑁)))
46 fveq2 6669 . . . . . . . . . . . . . 14 (𝑥 = 𝑁 → (𝑄𝑥) = (𝑄𝑁))
4746ineq1d 4192 . . . . . . . . . . . . 13 (𝑥 = 𝑁 → ((𝑄𝑥) ∩ (0..^𝑁)) = ((𝑄𝑁) ∩ (0..^𝑁)))
4845, 47eqeq12d 2842 . . . . . . . . . . . 12 (𝑥 = 𝑁 → (((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑄𝑥) ∩ (0..^𝑁)) ↔ ((𝑃𝑁) ∩ (0..^𝑁)) = ((𝑄𝑁) ∩ (0..^𝑁))))
4948imbi2d 342 . . . . . . . . . . 11 (𝑥 = 𝑁 → ((𝜑 → ((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑄𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃𝑁) ∩ (0..^𝑁)) = ((𝑄𝑁) ∩ (0..^𝑁)))))
501, 3, 5smup0 15823 . . . . . . . . . . . . . 14 (𝜑 → (𝑃‘0) = ∅)
51 inss1 4209 . . . . . . . . . . . . . . . 16 (𝐵 ∩ (0..^𝑁)) ⊆ 𝐵
5251, 3sstrid 3982 . . . . . . . . . . . . . . 15 (𝜑 → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ0)
53 smueq.q . . . . . . . . . . . . . . 15 𝑄 = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ (𝐵 ∩ (0..^𝑁)))})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
541, 52, 53smup0 15823 . . . . . . . . . . . . . 14 (𝜑 → (𝑄‘0) = ∅)
5550, 54eqtr4d 2864 . . . . . . . . . . . . 13 (𝜑 → (𝑃‘0) = (𝑄‘0))
5655ineq1d 4192 . . . . . . . . . . . 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 7157 . . . . . . . . . . . . . . 15 (((𝑃𝑖) ∩ (0..^𝑁)) = ((𝑄𝑖) ∩ (0..^𝑁)) → (((𝑃𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) = (((𝑄𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)} ∩ (0..^𝑁))))
5958ineq1d 4192 . . . . . . . . . . . . . 14 (((𝑃𝑖) ∩ (0..^𝑁)) = ((𝑄𝑖) ∩ (0..^𝑁)) → ((((𝑃𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)) = ((((𝑄𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)))
601adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝐴 ⊆ ℕ0)
613adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝐵 ⊆ ℕ0)
62 elfzonn0 13077 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℕ0)
6362adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0)
6460, 61, 5, 63smupp1 15824 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑃‘(𝑖 + 1)) = ((𝑃𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)}))
6564ineq1d 4192 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = (((𝑃𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)}) ∩ (0..^𝑁)))
661, 3, 5smupf 15822 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑃:ℕ0⟶𝒫 ℕ0)
67 ffvelrn 6847 . . . . . . . . . . . . . . . . . . 19 ((𝑃:ℕ0⟶𝒫 ℕ0𝑖 ∈ ℕ0) → (𝑃𝑖) ∈ 𝒫 ℕ0)
6866, 62, 67syl2an 595 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑃𝑖) ∈ 𝒫 ℕ0)
6968elpwid 4556 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑃𝑖) ⊆ ℕ0)
70 ssrab2 4060 . . . . . . . . . . . . . . . . . 18 {𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)} ⊆ ℕ0
7170a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑁)) → {𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)} ⊆ ℕ0)
7212adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ0)
7369, 71, 72sadeq 15816 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑁)) → (((𝑃𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)}) ∩ (0..^𝑁)) = ((((𝑃𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)))
7465, 73eqtrd 2861 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((((𝑃𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)))
7552adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ0)
7660, 75, 53, 63smupp1 15824 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑄‘(𝑖 + 1)) = ((𝑄𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))}))
7776ineq1d 4192 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)) = (((𝑄𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))}) ∩ (0..^𝑁)))
781, 52, 53smupf 15822 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑄:ℕ0⟶𝒫 ℕ0)
79 ffvelrn 6847 . . . . . . . . . . . . . . . . . . 19 ((𝑄:ℕ0⟶𝒫 ℕ0𝑖 ∈ ℕ0) → (𝑄𝑖) ∈ 𝒫 ℕ0)
8078, 62, 79syl2an 595 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑄𝑖) ∈ 𝒫 ℕ0)
8180elpwid 4556 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑄𝑖) ⊆ ℕ0)
82 ssrab2 4060 . . . . . . . . . . . . . . . . . 18 {𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ⊆ ℕ0
8382a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑁)) → {𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ⊆ ℕ0)
8481, 83, 72sadeq 15816 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑁)) → (((𝑄𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))}) ∩ (0..^𝑁)) = ((((𝑄𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ∩ (0..^𝑁))) ∩ (0..^𝑁)))
85 elinel2 4177 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ (ℕ0 ∩ (0..^𝑁)) → 𝑛 ∈ (0..^𝑁))
8661adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝐵 ⊆ ℕ0)
8786sseld 3970 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛𝑖) ∈ 𝐵 → (𝑛𝑖) ∈ ℕ0))
88 elfzo0 13073 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 ∈ (0..^𝑁) ↔ (𝑛 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑛 < 𝑁))
8988simp2bi 1140 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ (0..^𝑁) → 𝑁 ∈ ℕ)
9089adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ)
91 elfzonn0 13077 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 ∈ (0..^𝑁) → 𝑛 ∈ ℕ0)
9291adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℕ0)
9392nn0red 11950 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℝ)
9463adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0)
9594nn0red 11950 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑖 ∈ ℝ)
9693, 95resubcld 11062 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛𝑖) ∈ ℝ)
9790nnred 11647 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑁 ∈ ℝ)
9894nn0ge0d 11952 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 0 ≤ 𝑖)
9993, 95subge02d 11226 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (0 ≤ 𝑖 ↔ (𝑛𝑖) ≤ 𝑛))
10098, 99mpbid 233 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛𝑖) ≤ 𝑛)
101 elfzolt2 13042 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 ∈ (0..^𝑁) → 𝑛 < 𝑁)
102101adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 < 𝑁)
10396, 93, 97, 100, 102lelttrd 10792 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛𝑖) < 𝑁)
10490, 103jca 512 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑁 ∈ ℕ ∧ (𝑛𝑖) < 𝑁))
105 elfzo0 13073 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑛𝑖) ∈ (0..^𝑁) ↔ ((𝑛𝑖) ∈ ℕ0𝑁 ∈ ℕ ∧ (𝑛𝑖) < 𝑁))
106 3anass 1089 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑛𝑖) ∈ ℕ0𝑁 ∈ ℕ ∧ (𝑛𝑖) < 𝑁) ↔ ((𝑛𝑖) ∈ ℕ0 ∧ (𝑁 ∈ ℕ ∧ (𝑛𝑖) < 𝑁)))
107105, 106bitri 276 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛𝑖) ∈ (0..^𝑁) ↔ ((𝑛𝑖) ∈ ℕ0 ∧ (𝑁 ∈ ℕ ∧ (𝑛𝑖) < 𝑁)))
108107baib 536 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑛𝑖) ∈ ℕ0 → ((𝑛𝑖) ∈ (0..^𝑁) ↔ (𝑁 ∈ ℕ ∧ (𝑛𝑖) < 𝑁)))
109104, 108syl5ibrcom 248 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛𝑖) ∈ ℕ0 → (𝑛𝑖) ∈ (0..^𝑁)))
11087, 109syld 47 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛𝑖) ∈ 𝐵 → (𝑛𝑖) ∈ (0..^𝑁)))
111110pm4.71rd 563 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛𝑖) ∈ 𝐵 ↔ ((𝑛𝑖) ∈ (0..^𝑁) ∧ (𝑛𝑖) ∈ 𝐵)))
112 ancom 461 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛𝑖) ∈ (0..^𝑁) ∧ (𝑛𝑖) ∈ 𝐵) ↔ ((𝑛𝑖) ∈ 𝐵 ∧ (𝑛𝑖) ∈ (0..^𝑁)))
113 elin 4173 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)) ↔ ((𝑛𝑖) ∈ 𝐵 ∧ (𝑛𝑖) ∈ (0..^𝑁)))
114112, 113bitr4i 279 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛𝑖) ∈ (0..^𝑁) ∧ (𝑛𝑖) ∈ 𝐵) ↔ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))
115111, 114syl6rbb 289 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)) ↔ (𝑛𝑖) ∈ 𝐵))
116115anbi2d 628 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁))) ↔ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)))
11785, 116sylan2 592 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (ℕ0 ∩ (0..^𝑁))) → ((𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁))) ↔ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)))
118117rabbidva 3484 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑁)) → {𝑛 ∈ (ℕ0 ∩ (0..^𝑁)) ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} = {𝑛 ∈ (ℕ0 ∩ (0..^𝑁)) ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)})
119 inrab2 4280 . . . . . . . . . . . . . . . . . . 19 ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ∩ (0..^𝑁)) = {𝑛 ∈ (ℕ0 ∩ (0..^𝑁)) ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))}
120 inrab2 4280 . . . . . . . . . . . . . . . . . . 19 ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)} ∩ (0..^𝑁)) = {𝑛 ∈ (ℕ0 ∩ (0..^𝑁)) ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)}
121118, 119, 1203eqtr4g 2886 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑁)) → ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ∩ (0..^𝑁)) = ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)} ∩ (0..^𝑁)))
122121oveq2d 7166 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑁)) → (((𝑄𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ∩ (0..^𝑁))) = (((𝑄𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)} ∩ (0..^𝑁))))
123122ineq1d 4192 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑁)) → ((((𝑄𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ∩ (0..^𝑁))) ∩ (0..^𝑁)) = ((((𝑄𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)))
12477, 84, 1233eqtrd 2865 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((((𝑄𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)))
12574, 124eqeq12d 2842 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (0..^𝑁)) → (((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)) ↔ ((((𝑃𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)) = ((((𝑄𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁))))
12659, 125syl5ibr 247 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → (((𝑃𝑖) ∩ (0..^𝑁)) = ((𝑄𝑖) ∩ (0..^𝑁)) → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁))))
127126expcom 414 . . . . . . . . . . . 12 (𝑖 ∈ (0..^𝑁) → (𝜑 → (((𝑃𝑖) ∩ (0..^𝑁)) = ((𝑄𝑖) ∩ (0..^𝑁)) → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))))
128127a2d 29 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑁) → ((𝜑 → ((𝑃𝑖) ∩ (0..^𝑁)) = ((𝑄𝑖) ∩ (0..^𝑁))) → (𝜑 → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))))
12931, 37, 43, 49, 57, 128fzind2 13150 . . . . . . . . . 10 (𝑁 ∈ (0...𝑁) → (𝜑 → ((𝑃𝑁) ∩ (0..^𝑁)) = ((𝑄𝑁) ∩ (0..^𝑁))))
13025, 129mpcom 38 . . . . . . . . 9 (𝜑 → ((𝑃𝑁) ∩ (0..^𝑁)) = ((𝑄𝑁) ∩ (0..^𝑁)))
131130adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → ((𝑃𝑁) ∩ (0..^𝑁)) = ((𝑄𝑁) ∩ (0..^𝑁)))
132131eleq2d 2903 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑃𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ ((𝑄𝑁) ∩ (0..^𝑁))))
133 elin 4173 . . . . . . . . 9 (𝑘 ∈ ((𝑃𝑁) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝑃𝑁) ∧ 𝑘 ∈ (0..^𝑁)))
134133rbaib 539 . . . . . . . 8 (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ ((𝑃𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑃𝑁)))
135134adantl 482 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑃𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑃𝑁)))
136 elin 4173 . . . . . . . . 9 (𝑘 ∈ ((𝑄𝑁) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝑄𝑁) ∧ 𝑘 ∈ (0..^𝑁)))
137136rbaib 539 . . . . . . . 8 (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ ((𝑄𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑄𝑁)))
138137adantl 482 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑄𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑄𝑁)))
139132, 135, 1383bitr3d 310 . . . . . 6 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝑃𝑁) ↔ 𝑘 ∈ (𝑄𝑁)))
14052adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ0)
1412, 140, 53, 13smupval 15832 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑄𝑁) = ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))))
142141eleq2d 2903 . . . . . 6 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝑄𝑁) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁)))))
14322, 139, 1423bitrd 306 . . . . 5 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁)))))
144143ex 413 . . . 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 4173 . . 3 (𝑘 ∈ ((𝐴 smul 𝐵) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝐴 smul 𝐵) ∧ 𝑘 ∈ (0..^𝑁)))
147 elin 4173 . . 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 315 . 2 (𝜑 → (𝑘 ∈ ((𝐴 smul 𝐵) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∩ (0..^𝑁))))
149148eqrdv 2824 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  {crab 3147  cin 3939  wss 3940  c0 4295  ifcif 4470  𝒫 cpw 4542   class class class wbr 5063  cmpt 5143  wf 6350  cfv 6354  (class class class)co 7150  cmpo 7152  0cc0 10531  1c1 10532   + caddc 10534   < clt 10669  cle 10670  cmin 10864  cn 11632  0cn0 11891  cz 11975  cuz 12237  ...cfz 12887  ..^cfzo 13028  seqcseq 13364   sadd csad 15764   smul csmu 15765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-inf2 9098  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-xor 1498  df-tru 1533  df-fal 1543  df-had 1587  df-cad 1601  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-disj 5029  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8284  df-map 8403  df-pm 8404  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-inf 8901  df-oi 8968  df-dju 9324  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-xnn0 11962  df-z 11976  df-uz 12238  df-rp 12385  df-fz 12888  df-fzo 13029  df-fl 13157  df-mod 13233  df-seq 13365  df-exp 13425  df-hash 13686  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-clim 14840  df-sum 15038  df-dvds 15603  df-bits 15766  df-sad 15795  df-smu 15820
This theorem is referenced by:  smueq  15835
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