Theorem nnm0r 8622

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 7413	. . 3 ⊢ (𝑥 = ∅ → (∅ ·o 𝑥) = (∅ ·o ∅))
2	1	eqeq1d 2737	. 2 ⊢ (𝑥 = ∅ → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o ∅) = ∅))
3		oveq2 7413	. . 3 ⊢ (𝑥 = 𝑦 → (∅ ·o 𝑥) = (∅ ·o 𝑦))
4	3	eqeq1d 2737	. 2 ⊢ (𝑥 = 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝑦) = ∅))
5		oveq2 7413	. . 3 ⊢ (𝑥 = suc 𝑦 → (∅ ·o 𝑥) = (∅ ·o suc 𝑦))
6	5	eqeq1d 2737	. 2 ⊢ (𝑥 = suc 𝑦 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o suc 𝑦) = ∅))
7		oveq2 7413	. . 3 ⊢ (𝑥 = 𝐴 → (∅ ·o 𝑥) = (∅ ·o 𝐴))
8	7	eqeq1d 2737	. 2 ⊢ (𝑥 = 𝐴 → ((∅ ·o 𝑥) = ∅ ↔ (∅ ·o 𝐴) = ∅))
9		0elon 6407	. . 3 ⊢ ∅ ∈ On
10		om0 8529	. . 3 ⊢ (∅ ∈ On → (∅ ·o ∅) = ∅)
11	9, 10	ax-mp 5	. 2 ⊢ (∅ ·o ∅) = ∅
12		oveq1 7412	. . . 4 ⊢ ((∅ ·o 𝑦) = ∅ → ((∅ ·o 𝑦) +o ∅) = (∅ +o ∅))
13		oa0 8528	. . . . 5 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅)
14	9, 13	ax-mp 5	. . . 4 ⊢ (∅ +o ∅) = ∅
15	12, 14	eqtrdi 2786	. . 3 ⊢ ((∅ ·o 𝑦) = ∅ → ((∅ ·o 𝑦) +o ∅) = ∅)
16		peano1 7884	. . . . 5 ⊢ ∅ ∈ ω
17		nnmsuc 8619	. . . . 5 ⊢ ((∅ ∈ ω ∧ 𝑦 ∈ ω) → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅))
18	16, 17	mpan 690	. . . 4 ⊢ (𝑦 ∈ ω → (∅ ·o suc 𝑦) = ((∅ ·o 𝑦) +o ∅))
19	18	eqeq1d 2737	. . 3 ⊢ (𝑦 ∈ ω → ((∅ ·o suc 𝑦) = ∅ ↔ ((∅ ·o 𝑦) +o ∅) = ∅))
20	15, 19	imbitrrid 246	. 2 ⊢ (𝑦 ∈ ω → ((∅ ·o 𝑦) = ∅ → (∅ ·o suc 𝑦) = ∅))
21	2, 4, 6, 8, 11, 20	finds 7892	1 ⊢ (𝐴 ∈ ω → (∅ ·o 𝐴) = ∅)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ∅c0 4308  Oncon0 6352  suc csuc 6354  (class class class)co 7405  ωcom 7861   +_o coa 8477   ·_o comu 8478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-oadd 8484  df-omul 8485

This theorem is referenced by:  nnmcom  8638  nnmord  8644  nnmwordi  8647