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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 5551 | . . 3 ⊢ (𝑥 = ∅ → (∅ ·𝑜 𝑥) = (∅ ·𝑜 ∅)) | |
2 | 1 | eqeq1d 2090 | . 2 ⊢ (𝑥 = ∅ → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 ∅) = ∅)) |
3 | oveq2 5551 | . . 3 ⊢ (𝑥 = 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝑦)) | |
4 | 3 | eqeq1d 2090 | . 2 ⊢ (𝑥 = 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 𝑦) = ∅)) |
5 | oveq2 5551 | . . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 suc 𝑦)) | |
6 | 5 | eqeq1d 2090 | . 2 ⊢ (𝑥 = suc 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 suc 𝑦) = ∅)) |
7 | oveq2 5551 | . . 3 ⊢ (𝑥 = 𝐴 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝐴)) | |
8 | 7 | eqeq1d 2090 | . 2 ⊢ (𝑥 = 𝐴 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 𝐴) = ∅)) |
9 | 0elon 4155 | . . 3 ⊢ ∅ ∈ On | |
10 | om0 6102 | . . 3 ⊢ (∅ ∈ On → (∅ ·𝑜 ∅) = ∅) | |
11 | 9, 10 | ax-mp 7 | . 2 ⊢ (∅ ·𝑜 ∅) = ∅ |
12 | oveq1 5550 | . . . 4 ⊢ ((∅ ·𝑜 𝑦) = ∅ → ((∅ ·𝑜 𝑦) +𝑜 ∅) = (∅ +𝑜 ∅)) | |
13 | oa0 6101 | . . . . 5 ⊢ (∅ ∈ On → (∅ +𝑜 ∅) = ∅) | |
14 | 9, 13 | ax-mp 7 | . . . 4 ⊢ (∅ +𝑜 ∅) = ∅ |
15 | 12, 14 | syl6eq 2130 | . . 3 ⊢ ((∅ ·𝑜 𝑦) = ∅ → ((∅ ·𝑜 𝑦) +𝑜 ∅) = ∅) |
16 | peano1 4343 | . . . . 5 ⊢ ∅ ∈ ω | |
17 | nnmsuc 6121 | . . . . 5 ⊢ ((∅ ∈ ω ∧ 𝑦 ∈ ω) → (∅ ·𝑜 suc 𝑦) = ((∅ ·𝑜 𝑦) +𝑜 ∅)) | |
18 | 16, 17 | mpan 415 | . . . 4 ⊢ (𝑦 ∈ ω → (∅ ·𝑜 suc 𝑦) = ((∅ ·𝑜 𝑦) +𝑜 ∅)) |
19 | 18 | eqeq1d 2090 | . . 3 ⊢ (𝑦 ∈ ω → ((∅ ·𝑜 suc 𝑦) = ∅ ↔ ((∅ ·𝑜 𝑦) +𝑜 ∅) = ∅)) |
20 | 15, 19 | syl5ibr 154 | . 2 ⊢ (𝑦 ∈ ω → ((∅ ·𝑜 𝑦) = ∅ → (∅ ·𝑜 suc 𝑦) = ∅)) |
21 | 2, 4, 6, 8, 11, 20 | finds 4349 | 1 ⊢ (𝐴 ∈ ω → (∅ ·𝑜 𝐴) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∈ wcel 1434 ∅c0 3258 Oncon0 4126 suc csuc 4128 ωcom 4339 (class class class)co 5543 +𝑜 coa 6062 ·𝑜 comu 6063 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-coll 3901 ax-sep 3904 ax-nul 3912 ax-pow 3956 ax-pr 3972 ax-un 4196 ax-setind 4288 ax-iinf 4337 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-ral 2354 df-rex 2355 df-reu 2356 df-rab 2358 df-v 2604 df-sbc 2817 df-csb 2910 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-nul 3259 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-int 3645 df-iun 3688 df-br 3794 df-opab 3848 df-mpt 3849 df-tr 3884 df-id 4056 df-iord 4129 df-on 4131 df-suc 4134 df-iom 4340 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-rn 4382 df-res 4383 df-ima 4384 df-iota 4897 df-fun 4934 df-fn 4935 df-f 4936 df-f1 4937 df-fo 4938 df-f1o 4939 df-fv 4940 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-1st 5798 df-2nd 5799 df-recs 5954 df-irdg 6019 df-oadd 6069 df-omul 6070 |
This theorem is referenced by: nnmcom 6133 nnmord 6156 nnm00 6168 enq0tr 6686 nq0m0r 6708 |
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