Theorem nnm0r 5997

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 5463	. . 3 ⊢ (x = ∅ → (∅ ·𝑜 x) = (∅ ·𝑜 ∅))
2	1	eqeq1d 2045	. 2 ⊢ (x = ∅ → ((∅ ·𝑜 x) = ∅ ↔ (∅ ·𝑜 ∅) = ∅))
3		oveq2 5463	. . 3 ⊢ (x = y → (∅ ·𝑜 x) = (∅ ·𝑜 y))
4	3	eqeq1d 2045	. 2 ⊢ (x = y → ((∅ ·𝑜 x) = ∅ ↔ (∅ ·𝑜 y) = ∅))
5		oveq2 5463	. . 3 ⊢ (x = suc y → (∅ ·𝑜 x) = (∅ ·𝑜 suc y))
6	5	eqeq1d 2045	. 2 ⊢ (x = suc y → ((∅ ·𝑜 x) = ∅ ↔ (∅ ·𝑜 suc y) = ∅))
7		oveq2 5463	. . 3 ⊢ (x = A → (∅ ·𝑜 x) = (∅ ·𝑜 A))
8	7	eqeq1d 2045	. 2 ⊢ (x = A → ((∅ ·𝑜 x) = ∅ ↔ (∅ ·𝑜 A) = ∅))
9		0elon 4095	. . 3 ⊢ ∅ ∈ On
10		om0 5977	. . 3 ⊢ (∅ ∈ On → (∅ ·𝑜 ∅) = ∅)
11	9, 10	ax-mp 7	. 2 ⊢ (∅ ·𝑜 ∅) = ∅
12		oveq1 5462	. . . 4 ⊢ ((∅ ·𝑜 y) = ∅ → ((∅ ·𝑜 y) +𝑜 ∅) = (∅ +𝑜 ∅))
13		oa0 5976	. . . . 5 ⊢ (∅ ∈ On → (∅ +𝑜 ∅) = ∅)
14	9, 13	ax-mp 7	. . . 4 ⊢ (∅ +𝑜 ∅) = ∅
15	12, 14	syl6eq 2085	. . 3 ⊢ ((∅ ·𝑜 y) = ∅ → ((∅ ·𝑜 y) +𝑜 ∅) = ∅)
16		peano1 4260	. . . . 5 ⊢ ∅ ∈ 𝜔
17		nnmsuc 5995	. . . . 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 2045	. . 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 4266	1 ⊢ (A ∈ 𝜔 → (∅ ·𝑜 A) = ∅)

Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  ∅c0 3218  Oncon0 4066  suc csuc 4068  𝜔com 4256  (class class class)co 5455   +_𝑜 coa 5937   ·_𝑜 comu 5938

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945

This theorem is referenced by:  nnmcom  6007  nnmord  6026  nnm00  6038  enq0tr  6417  nq0m0r  6439