Theorem nnm0r 6280

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 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) = ∅)

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