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Theorem smu01lem 15667
Description: Lemma for smu01 15668 and smu02 15669. (Contributed by Mario Carneiro, 19-Sep-2016.)
Hypotheses
Ref Expression
smu01lem.1 (𝜑𝐴 ⊆ ℕ0)
smu01lem.2 (𝜑𝐵 ⊆ ℕ0)
smu01lem.3 ((𝜑 ∧ (𝑘 ∈ ℕ0𝑛 ∈ ℕ0)) → ¬ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵))
Assertion
Ref Expression
smu01lem (𝜑 → (𝐴 smul 𝐵) = ∅)
Distinct variable groups:   𝑘,𝑛,𝐴   𝐵,𝑘,𝑛   𝜑,𝑘,𝑛

Proof of Theorem smu01lem
Dummy variables 𝑚 𝑝 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smu01lem.1 . . . . . 6 (𝜑𝐴 ⊆ ℕ0)
2 smu01lem.2 . . . . . 6 (𝜑𝐵 ⊆ ℕ0)
3 smucl 15666 . . . . . 6 ((𝐴 ⊆ ℕ0𝐵 ⊆ ℕ0) → (𝐴 smul 𝐵) ⊆ ℕ0)
41, 2, 3syl2anc 584 . . . . 5 (𝜑 → (𝐴 smul 𝐵) ⊆ ℕ0)
54sseld 3888 . . . 4 (𝜑 → (𝑘 ∈ (𝐴 smul 𝐵) → 𝑘 ∈ ℕ0))
6 noel 4216 . . . . . . 7 ¬ 𝑘 ∈ ∅
7 peano2nn0 11785 . . . . . . . . . 10 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
8 fveqeq2 6547 . . . . . . . . . . . 12 (𝑥 = 0 → ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) = ∅ ↔ (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘0) = ∅))
98imbi2d 342 . . . . . . . . . . 11 (𝑥 = 0 → ((𝜑 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) = ∅) ↔ (𝜑 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘0) = ∅)))
10 fveqeq2 6547 . . . . . . . . . . . 12 (𝑥 = 𝑘 → ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) = ∅ ↔ (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) = ∅))
1110imbi2d 342 . . . . . . . . . . 11 (𝑥 = 𝑘 → ((𝜑 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) = ∅) ↔ (𝜑 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) = ∅)))
12 fveqeq2 6547 . . . . . . . . . . . 12 (𝑥 = (𝑘 + 1) → ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) = ∅ ↔ (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) = ∅))
1312imbi2d 342 . . . . . . . . . . 11 (𝑥 = (𝑘 + 1) → ((𝜑 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑥) = ∅) ↔ (𝜑 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) = ∅)))
14 eqid 2795 . . . . . . . . . . . 12 seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
151, 2, 14smup0 15661 . . . . . . . . . . 11 (𝜑 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘0) = ∅)
16 oveq1 7023 . . . . . . . . . . . . . 14 ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) = ∅ → ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)}) = (∅ sadd {𝑛 ∈ ℕ0 ∣ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)}))
171adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ0) → 𝐴 ⊆ ℕ0)
182adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ0) → 𝐵 ⊆ ℕ0)
19 simpr 485 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
2017, 18, 14, 19smupp1 15662 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ0) → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) = ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)}))
21 smu01lem.3 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑘 ∈ ℕ0𝑛 ∈ ℕ0)) → ¬ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵))
2221anassrs 468 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → ¬ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵))
2322ralrimiva 3149 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ ℕ0) → ∀𝑛 ∈ ℕ0 ¬ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵))
24 rabeq0 4258 . . . . . . . . . . . . . . . . . 18 ({𝑛 ∈ ℕ0 ∣ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)} = ∅ ↔ ∀𝑛 ∈ ℕ0 ¬ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵))
2523, 24sylibr 235 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ0) → {𝑛 ∈ ℕ0 ∣ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)} = ∅)
2625oveq2d 7032 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ0) → (∅ sadd {𝑛 ∈ ℕ0 ∣ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)}) = (∅ sadd ∅))
27 0ss 4270 . . . . . . . . . . . . . . . . 17 ∅ ⊆ ℕ0
28 sadid1 15650 . . . . . . . . . . . . . . . . 17 (∅ ⊆ ℕ0 → (∅ sadd ∅) = ∅)
2927, 28mp1i 13 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ0) → (∅ sadd ∅) = ∅)
3026, 29eqtr2d 2832 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ0) → ∅ = (∅ sadd {𝑛 ∈ ℕ0 ∣ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)}))
3120, 30eqeq12d 2810 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ0) → ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) = ∅ ↔ ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) sadd {𝑛 ∈ ℕ0 ∣ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)}) = (∅ sadd {𝑛 ∈ ℕ0 ∣ (𝑘𝐴 ∧ (𝑛𝑘) ∈ 𝐵)})))
3216, 31syl5ibr 247 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ0) → ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) = ∅ → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) = ∅))
3332expcom 414 . . . . . . . . . . . 12 (𝑘 ∈ ℕ0 → (𝜑 → ((seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) = ∅ → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) = ∅)))
3433a2d 29 . . . . . . . . . . 11 (𝑘 ∈ ℕ0 → ((𝜑 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘𝑘) = ∅) → (𝜑 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) = ∅)))
359, 11, 13, 13, 15, 34nn0ind 11926 . . . . . . . . . 10 ((𝑘 + 1) ∈ ℕ0 → (𝜑 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) = ∅))
367, 35syl 17 . . . . . . . . 9 (𝑘 ∈ ℕ0 → (𝜑 → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) = ∅))
3736impcom 408 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) = ∅)
3837eleq2d 2868 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → (𝑘 ∈ (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)) ↔ 𝑘 ∈ ∅))
396, 38mtbiri 328 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → ¬ 𝑘 ∈ (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1)))
4017, 18, 14, 19smuval 15663 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → (𝑘 ∈ (𝐴 smul 𝐵) ↔ 𝑘 ∈ (seq0((𝑝 ∈ 𝒫 ℕ0, 𝑚 ∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚𝐴 ∧ (𝑛𝑚) ∈ 𝐵)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1))))
4139, 40mtbird 326 . . . . 5 ((𝜑𝑘 ∈ ℕ0) → ¬ 𝑘 ∈ (𝐴 smul 𝐵))
4241ex 413 . . . 4 (𝜑 → (𝑘 ∈ ℕ0 → ¬ 𝑘 ∈ (𝐴 smul 𝐵)))
435, 42syld 47 . . 3 (𝜑 → (𝑘 ∈ (𝐴 smul 𝐵) → ¬ 𝑘 ∈ (𝐴 smul 𝐵)))
4443pm2.01d 191 . 2 (𝜑 → ¬ 𝑘 ∈ (𝐴 smul 𝐵))
4544eq0rdv 4277 1 (𝜑 → (𝐴 smul 𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1522  wcel 2081  wral 3105  {crab 3109  wss 3859  c0 4211  ifcif 4381  𝒫 cpw 4453  cmpt 5041  cfv 6225  (class class class)co 7016  cmpo 7018  0cc0 10383  1c1 10384   + caddc 10386  cmin 10717  0cn0 11745  seqcseq 13219   sadd csad 15602   smul csmu 15603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-xor 1497  df-tru 1525  df-had 1576  df-cad 1589  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-2o 7954  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-n0 11746  df-z 11830  df-uz 12094  df-fz 12743  df-seq 13220  df-sad 15633  df-smu 15658
This theorem is referenced by:  smu01  15668  smu02  15669
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