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Theorem smueqlem 15136
 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 12415 . . . . . . . . 9 (𝑘 ∈ (0..^𝑁) → 𝑘 ∈ (ℤ‘0))
76adantl 482 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ (ℤ‘0))
8 nn0uz 11666 . . . . . . . 8 0 = (ℤ‘0)
97, 8syl6eleqr 2709 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℕ0)
109nn0zd 11424 . . . . . . . . 9 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ ℤ)
1110peano2zd 11429 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ∈ ℤ)
12 smueq.n . . . . . . . . . 10 (𝜑𝑁 ∈ ℕ0)
1312adantr 481 . . . . . . . . 9 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ0)
1413nn0zd 11424 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ)
15 elfzolt2 12420 . . . . . . . . . 10 (𝑘 ∈ (0..^𝑁) → 𝑘 < 𝑁)
1615adantl 482 . . . . . . . . 9 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑘 < 𝑁)
17 nn0ltp1le 11379 . . . . . . . . . 10 ((𝑘 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑘 < 𝑁 ↔ (𝑘 + 1) ≤ 𝑁))
189, 13, 17syl2anc 692 . . . . . . . . 9 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 < 𝑁 ↔ (𝑘 + 1) ≤ 𝑁))
1916, 18mpbid 222 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ≤ 𝑁)
20 eluz2 11637 . . . . . . . 8 (𝑁 ∈ (ℤ‘(𝑘 + 1)) ↔ ((𝑘 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑁))
2111, 14, 19, 20syl3anbrc 1244 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑁 ∈ (ℤ‘(𝑘 + 1)))
222, 4, 5, 9, 21smuval2 15128 . . . . . 6 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ (𝑃𝑁)))
2312, 8syl6eleq 2708 . . . . . . . . . . 11 (𝜑𝑁 ∈ (ℤ‘0))
24 eluzfz2b 12292 . . . . . . . . . . 11 (𝑁 ∈ (ℤ‘0) ↔ 𝑁 ∈ (0...𝑁))
2523, 24sylib 208 . . . . . . . . . 10 (𝜑𝑁 ∈ (0...𝑁))
26 fveq2 6148 . . . . . . . . . . . . . 14 (𝑥 = 0 → (𝑃𝑥) = (𝑃‘0))
2726ineq1d 3791 . . . . . . . . . . . . 13 (𝑥 = 0 → ((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑃‘0) ∩ (0..^𝑁)))
28 fveq2 6148 . . . . . . . . . . . . . 14 (𝑥 = 0 → (𝑄𝑥) = (𝑄‘0))
2928ineq1d 3791 . . . . . . . . . . . . 13 (𝑥 = 0 → ((𝑄𝑥) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁)))
3027, 29eqeq12d 2636 . . . . . . . . . . . 12 (𝑥 = 0 → (((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑄𝑥) ∩ (0..^𝑁)) ↔ ((𝑃‘0) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁))))
3130imbi2d 330 . . . . . . . . . . 11 (𝑥 = 0 → ((𝜑 → ((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑄𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃‘0) ∩ (0..^𝑁)) = ((𝑄‘0) ∩ (0..^𝑁)))))
32 fveq2 6148 . . . . . . . . . . . . . 14 (𝑥 = 𝑖 → (𝑃𝑥) = (𝑃𝑖))
3332ineq1d 3791 . . . . . . . . . . . . 13 (𝑥 = 𝑖 → ((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑃𝑖) ∩ (0..^𝑁)))
34 fveq2 6148 . . . . . . . . . . . . . 14 (𝑥 = 𝑖 → (𝑄𝑥) = (𝑄𝑖))
3534ineq1d 3791 . . . . . . . . . . . . 13 (𝑥 = 𝑖 → ((𝑄𝑥) ∩ (0..^𝑁)) = ((𝑄𝑖) ∩ (0..^𝑁)))
3633, 35eqeq12d 2636 . . . . . . . . . . . 12 (𝑥 = 𝑖 → (((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑄𝑥) ∩ (0..^𝑁)) ↔ ((𝑃𝑖) ∩ (0..^𝑁)) = ((𝑄𝑖) ∩ (0..^𝑁))))
3736imbi2d 330 . . . . . . . . . . 11 (𝑥 = 𝑖 → ((𝜑 → ((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑄𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃𝑖) ∩ (0..^𝑁)) = ((𝑄𝑖) ∩ (0..^𝑁)))))
38 fveq2 6148 . . . . . . . . . . . . . 14 (𝑥 = (𝑖 + 1) → (𝑃𝑥) = (𝑃‘(𝑖 + 1)))
3938ineq1d 3791 . . . . . . . . . . . . 13 (𝑥 = (𝑖 + 1) → ((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)))
40 fveq2 6148 . . . . . . . . . . . . . 14 (𝑥 = (𝑖 + 1) → (𝑄𝑥) = (𝑄‘(𝑖 + 1)))
4140ineq1d 3791 . . . . . . . . . . . . 13 (𝑥 = (𝑖 + 1) → ((𝑄𝑥) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))
4239, 41eqeq12d 2636 . . . . . . . . . . . 12 (𝑥 = (𝑖 + 1) → (((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑄𝑥) ∩ (0..^𝑁)) ↔ ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁))))
4342imbi2d 330 . . . . . . . . . . 11 (𝑥 = (𝑖 + 1) → ((𝜑 → ((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑄𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))))
44 fveq2 6148 . . . . . . . . . . . . . 14 (𝑥 = 𝑁 → (𝑃𝑥) = (𝑃𝑁))
4544ineq1d 3791 . . . . . . . . . . . . 13 (𝑥 = 𝑁 → ((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑃𝑁) ∩ (0..^𝑁)))
46 fveq2 6148 . . . . . . . . . . . . . 14 (𝑥 = 𝑁 → (𝑄𝑥) = (𝑄𝑁))
4746ineq1d 3791 . . . . . . . . . . . . 13 (𝑥 = 𝑁 → ((𝑄𝑥) ∩ (0..^𝑁)) = ((𝑄𝑁) ∩ (0..^𝑁)))
4845, 47eqeq12d 2636 . . . . . . . . . . . 12 (𝑥 = 𝑁 → (((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑄𝑥) ∩ (0..^𝑁)) ↔ ((𝑃𝑁) ∩ (0..^𝑁)) = ((𝑄𝑁) ∩ (0..^𝑁))))
4948imbi2d 330 . . . . . . . . . . 11 (𝑥 = 𝑁 → ((𝜑 → ((𝑃𝑥) ∩ (0..^𝑁)) = ((𝑄𝑥) ∩ (0..^𝑁))) ↔ (𝜑 → ((𝑃𝑁) ∩ (0..^𝑁)) = ((𝑄𝑁) ∩ (0..^𝑁)))))
501, 3, 5smup0 15125 . . . . . . . . . . . . . 14 (𝜑 → (𝑃‘0) = ∅)
51 inss1 3811 . . . . . . . . . . . . . . . 16 (𝐵 ∩ (0..^𝑁)) ⊆ 𝐵
5251, 3syl5ss 3594 . . . . . . . . . . . . . . 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 15125 . . . . . . . . . . . . . 14 (𝜑 → (𝑄‘0) = ∅)
5550, 54eqtr4d 2658 . . . . . . . . . . . . 13 (𝜑 → (𝑃‘0) = (𝑄‘0))
5655ineq1d 3791 . . . . . . . . . . . 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 6611 . . . . . . . . . . . . . . 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 3791 . . . . . . . . . . . . . 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 12453 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℕ0)
6362adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0)
6460, 61, 5, 63smupp1 15126 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑃‘(𝑖 + 1)) = ((𝑃𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)}))
6564ineq1d 3791 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = (((𝑃𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)}) ∩ (0..^𝑁)))
661, 3, 5smupf 15124 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑃:ℕ0⟶𝒫 ℕ0)
67 ffvelrn 6313 . . . . . . . . . . . . . . . . . . 19 ((𝑃:ℕ0⟶𝒫 ℕ0𝑖 ∈ ℕ0) → (𝑃𝑖) ∈ 𝒫 ℕ0)
6866, 62, 67syl2an 494 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑃𝑖) ∈ 𝒫 ℕ0)
6968elpwid 4141 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑃𝑖) ⊆ ℕ0)
70 ssrab2 3666 . . . . . . . . . . . . . . . . . 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 15118 . . . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . . . 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 15126 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑄‘(𝑖 + 1)) = ((𝑄𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))}))
7776ineq1d 3791 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)) = (((𝑄𝑖) sadd {𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))}) ∩ (0..^𝑁)))
781, 52, 53smupf 15124 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑄:ℕ0⟶𝒫 ℕ0)
79 ffvelrn 6313 . . . . . . . . . . . . . . . . . . 19 ((𝑄:ℕ0⟶𝒫 ℕ0𝑖 ∈ ℕ0) → (𝑄𝑖) ∈ 𝒫 ℕ0)
8078, 62, 79syl2an 494 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑄𝑖) ∈ 𝒫 ℕ0)
8180elpwid 4141 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 ∈ (0..^𝑁)) → (𝑄𝑖) ⊆ ℕ0)
82 ssrab2 3666 . . . . . . . . . . . . . . . . . 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 15118 . . . . . . . . . . . . . . . 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 inss2 3812 . . . . . . . . . . . . . . . . . . . . . 22 (ℕ0 ∩ (0..^𝑁)) ⊆ (0..^𝑁)
8685sseli 3579 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ (ℕ0 ∩ (0..^𝑁)) → 𝑛 ∈ (0..^𝑁))
8761adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝐵 ⊆ ℕ0)
8887sseld 3582 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛𝑖) ∈ 𝐵 → (𝑛𝑖) ∈ ℕ0))
89 elfzo0 12449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 ∈ (0..^𝑁) ↔ (𝑛 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑛 < 𝑁))
9089simp2bi 1075 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ (0..^𝑁) → 𝑁 ∈ ℕ)
9190adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ)
92 elfzonn0 12453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 ∈ (0..^𝑁) → 𝑛 ∈ ℕ0)
9392adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℕ0)
9493nn0red 11296 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℝ)
9563adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0)
9695nn0red 11296 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑖 ∈ ℝ)
9794, 96resubcld 10402 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛𝑖) ∈ ℝ)
9891nnred 10979 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑁 ∈ ℝ)
9995nn0ge0d 11298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 0 ≤ 𝑖)
10094, 96subge02d 10563 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (0 ≤ 𝑖 ↔ (𝑛𝑖) ≤ 𝑛))
10199, 100mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛𝑖) ≤ 𝑛)
102 elfzolt2 12420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 ∈ (0..^𝑁) → 𝑛 < 𝑁)
103102adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 < 𝑁)
10497, 94, 98, 101, 103lelttrd 10139 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛𝑖) < 𝑁)
10591, 104jca 554 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → (𝑁 ∈ ℕ ∧ (𝑛𝑖) < 𝑁))
106 elfzo0 12449 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑛𝑖) ∈ (0..^𝑁) ↔ ((𝑛𝑖) ∈ ℕ0𝑁 ∈ ℕ ∧ (𝑛𝑖) < 𝑁))
107 3anass 1040 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑛𝑖) ∈ ℕ0𝑁 ∈ ℕ ∧ (𝑛𝑖) < 𝑁) ↔ ((𝑛𝑖) ∈ ℕ0 ∧ (𝑁 ∈ ℕ ∧ (𝑛𝑖) < 𝑁)))
108106, 107bitri 264 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑛𝑖) ∈ (0..^𝑁) ↔ ((𝑛𝑖) ∈ ℕ0 ∧ (𝑁 ∈ ℕ ∧ (𝑛𝑖) < 𝑁)))
109108baib 943 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑛𝑖) ∈ ℕ0 → ((𝑛𝑖) ∈ (0..^𝑁) ↔ (𝑁 ∈ ℕ ∧ (𝑛𝑖) < 𝑁)))
110105, 109syl5ibrcom 237 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛𝑖) ∈ ℕ0 → (𝑛𝑖) ∈ (0..^𝑁)))
11188, 110syld 47 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛𝑖) ∈ 𝐵 → (𝑛𝑖) ∈ (0..^𝑁)))
112111pm4.71rd 666 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛𝑖) ∈ 𝐵 ↔ ((𝑛𝑖) ∈ (0..^𝑁) ∧ (𝑛𝑖) ∈ 𝐵)))
113 ancom 466 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛𝑖) ∈ (0..^𝑁) ∧ (𝑛𝑖) ∈ 𝐵) ↔ ((𝑛𝑖) ∈ 𝐵 ∧ (𝑛𝑖) ∈ (0..^𝑁)))
114 elin 3774 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)) ↔ ((𝑛𝑖) ∈ 𝐵 ∧ (𝑛𝑖) ∈ (0..^𝑁)))
115113, 114bitr4i 267 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑛𝑖) ∈ (0..^𝑁) ∧ (𝑛𝑖) ∈ 𝐵) ↔ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))
116112, 115syl6rbb 277 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)) ↔ (𝑛𝑖) ∈ 𝐵))
117116anbi2d 739 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (0..^𝑁)) → ((𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁))) ↔ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)))
11886, 117sylan2 491 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑖 ∈ (0..^𝑁)) ∧ 𝑛 ∈ (ℕ0 ∩ (0..^𝑁))) → ((𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁))) ↔ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)))
119118rabbidva 3176 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (0..^𝑁)) → {𝑛 ∈ (ℕ0 ∩ (0..^𝑁)) ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} = {𝑛 ∈ (ℕ0 ∩ (0..^𝑁)) ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)})
120 inrab2 3876 . . . . . . . . . . . . . . . . . . 19 ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ∩ (0..^𝑁)) = {𝑛 ∈ (ℕ0 ∩ (0..^𝑁)) ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))}
121 inrab2 3876 . . . . . . . . . . . . . . . . . . 19 ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)} ∩ (0..^𝑁)) = {𝑛 ∈ (ℕ0 ∩ (0..^𝑁)) ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)}
122119, 120, 1213eqtr4g 2680 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (0..^𝑁)) → ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ (𝐵 ∩ (0..^𝑁)))} ∩ (0..^𝑁)) = ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)} ∩ (0..^𝑁)))
123122oveq2d 6620 . . . . . . . . . . . . . . . . 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))))
124123ineq1d 3791 . . . . . . . . . . . . . . . 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)))
12577, 84, 1243eqtrd 2659 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑁)) → ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((((𝑄𝑖) ∩ (0..^𝑁)) sadd ({𝑛 ∈ ℕ0 ∣ (𝑖𝐴 ∧ (𝑛𝑖) ∈ 𝐵)} ∩ (0..^𝑁))) ∩ (0..^𝑁)))
12674, 125eqeq12d 2636 . . . . . . . . . . . . . 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))))
12759, 126syl5ibr 236 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (0..^𝑁)) → (((𝑃𝑖) ∩ (0..^𝑁)) = ((𝑄𝑖) ∩ (0..^𝑁)) → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁))))
128127expcom 451 . . . . . . . . . . . 12 (𝑖 ∈ (0..^𝑁) → (𝜑 → (((𝑃𝑖) ∩ (0..^𝑁)) = ((𝑄𝑖) ∩ (0..^𝑁)) → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))))
129128a2d 29 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑁) → ((𝜑 → ((𝑃𝑖) ∩ (0..^𝑁)) = ((𝑄𝑖) ∩ (0..^𝑁))) → (𝜑 → ((𝑃‘(𝑖 + 1)) ∩ (0..^𝑁)) = ((𝑄‘(𝑖 + 1)) ∩ (0..^𝑁)))))
13031, 37, 43, 49, 57, 129fzind2 12526 . . . . . . . . . 10 (𝑁 ∈ (0...𝑁) → (𝜑 → ((𝑃𝑁) ∩ (0..^𝑁)) = ((𝑄𝑁) ∩ (0..^𝑁))))
13125, 130mpcom 38 . . . . . . . . 9 (𝜑 → ((𝑃𝑁) ∩ (0..^𝑁)) = ((𝑄𝑁) ∩ (0..^𝑁)))
132131adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → ((𝑃𝑁) ∩ (0..^𝑁)) = ((𝑄𝑁) ∩ (0..^𝑁)))
133132eleq2d 2684 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑃𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ ((𝑄𝑁) ∩ (0..^𝑁))))
134 elin 3774 . . . . . . . . 9 (𝑘 ∈ ((𝑃𝑁) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝑃𝑁) ∧ 𝑘 ∈ (0..^𝑁)))
135134rbaib 946 . . . . . . . 8 (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ ((𝑃𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑃𝑁)))
136135adantl 482 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑃𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑃𝑁)))
137 elin 3774 . . . . . . . . 9 (𝑘 ∈ ((𝑄𝑁) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝑄𝑁) ∧ 𝑘 ∈ (0..^𝑁)))
138137rbaib 946 . . . . . . . 8 (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ ((𝑄𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑄𝑁)))
139138adantl 482 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ ((𝑄𝑁) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (𝑄𝑁)))
140133, 136, 1393bitr3d 298 . . . . . 6 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝑃𝑁) ↔ 𝑘 ∈ (𝑄𝑁)))
14152adantr 481 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝐵 ∩ (0..^𝑁)) ⊆ ℕ0)
1422, 141, 53, 13smupval 15134 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑄𝑁) = ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))))
143142eleq2d 2684 . . . . . 6 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝑄𝑁) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁)))))
14422, 140, 1433bitrd 294 . . . . 5 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁)))))
145144ex 450 . . . 4 (𝜑 → (𝑘 ∈ (0..^𝑁) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))))))
146145pm5.32rd 671 . . 3 (𝜑 → ((𝑘 ∈ (𝐴 smul 𝐵) ∧ 𝑘 ∈ (0..^𝑁)) ↔ (𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∧ 𝑘 ∈ (0..^𝑁))))
147 elin 3774 . . 3 (𝑘 ∈ ((𝐴 smul 𝐵) ∩ (0..^𝑁)) ↔ (𝑘 ∈ (𝐴 smul 𝐵) ∧ 𝑘 ∈ (0..^𝑁)))
148 elin 3774 . . 3 (𝑘 ∈ (((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∩ (0..^𝑁)) ↔ (𝑘 ∈ ((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∧ 𝑘 ∈ (0..^𝑁)))
149146, 147, 1483bitr4g 303 . 2 (𝜑 → (𝑘 ∈ ((𝐴 smul 𝐵) ∩ (0..^𝑁)) ↔ 𝑘 ∈ (((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∩ (0..^𝑁))))
150149eqrdv 2619 1 (𝜑 → ((𝐴 smul 𝐵) ∩ (0..^𝑁)) = (((𝐴 ∩ (0..^𝑁)) smul (𝐵 ∩ (0..^𝑁))) ∩ (0..^𝑁)))