![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 5903 | . . 3 ⊢ (𝑥 = ∅ → (∅ ·o 𝑥) = (∅ ·o ∅)) | |
2 | 1 | eqeq1d 2198 | . 2 ⊢ (𝑥 = ∅ → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o ∅) = ∅)) |
3 | oveq2 5903 | . . 3 ⊢ (𝑥 = 𝑦 → (∅ ·o 𝑥) = (∅ ·o 𝑦)) | |
4 | 3 | eqeq1d 2198 | . 2 ⊢ (𝑥 = 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝑦) = ∅)) |
5 | oveq2 5903 | . . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ·o 𝑥) = (∅ ·o suc 𝑦)) | |
6 | 5 | eqeq1d 2198 | . 2 ⊢ (𝑥 = suc 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o suc 𝑦) = ∅)) |
7 | oveq2 5903 | . . 3 ⊢ (𝑥 = 𝐴 → (∅ ·o 𝑥) = (∅ ·o 𝐴)) | |
8 | 7 | eqeq1d 2198 | . 2 ⊢ (𝑥 = 𝐴 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝐴) = ∅)) |
9 | 0elon 4410 | . . 3 ⊢ ∅ ∈ On | |
10 | om0 6482 | . . 3 ⊢ (∅ ∈ On → (∅ ·o ∅) = ∅) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ (∅ ·o ∅) = ∅ |
12 | oveq1 5902 | . . . 4 ⊢ ((∅ ·o 𝑦) = ∅ → ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅)) | |
13 | oa0 6481 | . . . . 5 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅) | |
14 | 9, 13 | ax-mp 5 | . . . 4 ⊢ (∅ +o ∅) = ∅ |
15 | 12, 14 | eqtrdi 2238 | . . 3 ⊢ ((∅ ·o 𝑦) = ∅ → ((∅ ·o 𝑦) +o ∅) = ∅) |
16 | peano1 4611 | . . . . 5 ⊢ ∅ ∈ ω | |
17 | nnmsuc 6501 | . . . . 5 ⊢ ((∅ ∈ ω ∧ 𝑦 ∈ ω) → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅)) | |
18 | 16, 17 | mpan 424 | . . . 4 ⊢ (𝑦 ∈ ω → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅)) |
19 | 18 | eqeq1d 2198 | . . 3 ⊢ (𝑦 ∈ ω → ((∅ ·o suc 𝑦) = ∅ ↔ ((∅ ·o 𝑦) +o ∅) = ∅)) |
20 | 15, 19 | imbitrrid 156 | . 2 ⊢ (𝑦 ∈ ω → ((∅ ·o 𝑦) = ∅ → (∅ ·o suc 𝑦) = ∅)) |
21 | 2, 4, 6, 8, 11, 20 | finds 4617 | 1 ⊢ (𝐴 ∈ ω → (∅ ·o 𝐴) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 ∅c0 3437 Oncon0 4381 suc csuc 4383 ωcom 4607 (class class class)co 5895 +o coa 6437 ·o comu 6438 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-recs 6329 df-irdg 6394 df-oadd 6444 df-omul 6445 |
This theorem is referenced by: nnmcom 6513 nnmord 6541 nnm00 6554 enq0tr 7462 nq0m0r 7484 |
Copyright terms: Public domain | W3C validator |