Theorem nnm0r 6058

Description: Multiplication with zero. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

Ref	Expression
nnm0r	⊢ (𝐴 ∈ ω → (∅ ·𝑜 𝐴) = ∅)

Proof of Theorem nnm0r

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 5520	. . 3 ⊢ (𝑥 = ∅ → (∅ ·𝑜 𝑥) = (∅ ·𝑜 ∅))
2	1	eqeq1d 2048	. 2 ⊢ (𝑥 = ∅ → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 ∅) = ∅))
3		oveq2 5520	. . 3 ⊢ (𝑥 = 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝑦))
4	3	eqeq1d 2048	. 2 ⊢ (𝑥 = 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 𝑦) = ∅))
5		oveq2 5520	. . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 suc 𝑦))
6	5	eqeq1d 2048	. 2 ⊢ (𝑥 = suc 𝑦 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 suc 𝑦) = ∅))
7		oveq2 5520	. . 3 ⊢ (𝑥 = 𝐴 → (∅ ·𝑜 𝑥) = (∅ ·𝑜 𝐴))
8	7	eqeq1d 2048	. 2 ⊢ (𝑥 = 𝐴 → ((∅ ·𝑜 𝑥) = ∅ ↔ (∅ ·𝑜 𝐴) = ∅))
9		0elon 4129	. . 3 ⊢ ∅ ∈ On
10		om0 6038	. . 3 ⊢ (∅ ∈ On → (∅ ·𝑜 ∅) = ∅)
11	9, 10	ax-mp 7	. 2 ⊢ (∅ ·𝑜 ∅) = ∅
12		oveq1 5519	. . . 4 ⊢ ((∅ ·𝑜 𝑦) = ∅ → ((∅ ·𝑜 𝑦) +𝑜 ∅) = (∅ +𝑜 ∅))
13		oa0 6037	. . . . 5 ⊢ (∅ ∈ On → (∅ +𝑜 ∅) = ∅)
14	9, 13	ax-mp 7	. . . 4 ⊢ (∅ +𝑜 ∅) = ∅
15	12, 14	syl6eq 2088	. . 3 ⊢ ((∅ ·𝑜 𝑦) = ∅ → ((∅ ·𝑜 𝑦) +𝑜 ∅) = ∅)
16		peano1 4317	. . . . 5 ⊢ ∅ ∈ ω
17		nnmsuc 6056	. . . . 5 ⊢ ((∅ ∈ ω ∧ 𝑦 ∈ ω) → (∅ ·𝑜 suc 𝑦) = ((∅ ·𝑜 𝑦) +𝑜 ∅))
18	16, 17	mpan 400	. . . 4 ⊢ (𝑦 ∈ ω → (∅ ·𝑜 suc 𝑦) = ((∅ ·𝑜 𝑦) +𝑜 ∅))
19	18	eqeq1d 2048	. . 3 ⊢ (𝑦 ∈ ω → ((∅ ·𝑜 suc 𝑦) = ∅ ↔ ((∅ ·𝑜 𝑦) +𝑜 ∅) = ∅))
20	15, 19	syl5ibr 145	. 2 ⊢ (𝑦 ∈ ω → ((∅ ·𝑜 𝑦) = ∅ → (∅ ·𝑜 suc 𝑦) = ∅))
21	2, 4, 6, 8, 11, 20	finds 4323	1 ⊢ (𝐴 ∈ ω → (∅ ·𝑜 𝐴) = ∅)

Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  ∅c0 3224  Oncon0 4100  suc csuc 4102  ωcom 4313  (class class class)co 5512   +_𝑜 coa 5998   ·_𝑜 comu 5999

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006

This theorem is referenced by:  nnmcom  6068  nnmord  6090  nnm00  6102  enq0tr  6532  nq0m0r  6554