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Mirrors > Home > ILE Home > Th. List > nnm0r | GIF version |
Description: Multiplication with zero. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
nnm0r | ⊢ (𝐴 ∈ ω → (∅ ·o 𝐴) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5698 | . . 3 ⊢ (𝑥 = ∅ → (∅ ·o 𝑥) = (∅ ·o ∅)) | |
2 | 1 | eqeq1d 2103 | . 2 ⊢ (𝑥 = ∅ → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o ∅) = ∅)) |
3 | oveq2 5698 | . . 3 ⊢ (𝑥 = 𝑦 → (∅ ·o 𝑥) = (∅ ·o 𝑦)) | |
4 | 3 | eqeq1d 2103 | . 2 ⊢ (𝑥 = 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝑦) = ∅)) |
5 | oveq2 5698 | . . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ·o 𝑥) = (∅ ·o suc 𝑦)) | |
6 | 5 | eqeq1d 2103 | . 2 ⊢ (𝑥 = suc 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o suc 𝑦) = ∅)) |
7 | oveq2 5698 | . . 3 ⊢ (𝑥 = 𝐴 → (∅ ·o 𝑥) = (∅ ·o 𝐴)) | |
8 | 7 | eqeq1d 2103 | . 2 ⊢ (𝑥 = 𝐴 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝐴) = ∅)) |
9 | 0elon 4243 | . . 3 ⊢ ∅ ∈ On | |
10 | om0 6259 | . . 3 ⊢ (∅ ∈ On → (∅ ·o ∅) = ∅) | |
11 | 9, 10 | ax-mp 7 | . 2 ⊢ (∅ ·o ∅) = ∅ |
12 | oveq1 5697 | . . . 4 ⊢ ((∅ ·o 𝑦) = ∅ → ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅)) | |
13 | oa0 6258 | . . . . 5 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅) | |
14 | 9, 13 | ax-mp 7 | . . . 4 ⊢ (∅ +o ∅) = ∅ |
15 | 12, 14 | syl6eq 2143 | . . 3 ⊢ ((∅ ·o 𝑦) = ∅ → ((∅ ·o 𝑦) +o ∅) = ∅) |
16 | peano1 4437 | . . . . 5 ⊢ ∅ ∈ ω | |
17 | nnmsuc 6278 | . . . . 5 ⊢ ((∅ ∈ ω ∧ 𝑦 ∈ ω) → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅)) | |
18 | 16, 17 | mpan 416 | . . . 4 ⊢ (𝑦 ∈ ω → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅)) |
19 | 18 | eqeq1d 2103 | . . 3 ⊢ (𝑦 ∈ ω → ((∅ ·o suc 𝑦) = ∅ ↔ ((∅ ·o 𝑦) +o ∅) = ∅)) |
20 | 15, 19 | syl5ibr 155 | . 2 ⊢ (𝑦 ∈ ω → ((∅ ·o 𝑦) = ∅ → (∅ ·o suc 𝑦) = ∅)) |
21 | 2, 4, 6, 8, 11, 20 | finds 4443 | 1 ⊢ (𝐴 ∈ ω → (∅ ·o 𝐴) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1296 ∈ wcel 1445 ∅c0 3302 Oncon0 4214 suc csuc 4216 ωcom 4433 (class class class)co 5690 +o coa 6216 ·o comu 6217 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-id 4144 df-iord 4217 df-on 4219 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-recs 6108 df-irdg 6173 df-oadd 6223 df-omul 6224 |
This theorem is referenced by: nnmcom 6290 nnmord 6316 nnm00 6328 enq0tr 7090 nq0m0r 7112 |
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