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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.
StepHypRef Expression
1 oveq2 5463 . . 3 (x = ∅ → (∅ ·𝑜 x) = (∅ ·𝑜 ∅))
21eqeq1d 2045 . 2 (x = ∅ → ((∅ ·𝑜 x) = ∅ ↔ (∅ ·𝑜 ∅) = ∅))
3 oveq2 5463 . . 3 (x = y → (∅ ·𝑜 x) = (∅ ·𝑜 y))
43eqeq1d 2045 . 2 (x = y → ((∅ ·𝑜 x) = ∅ ↔ (∅ ·𝑜 y) = ∅))
5 oveq2 5463 . . 3 (x = suc y → (∅ ·𝑜 x) = (∅ ·𝑜 suc y))
65eqeq1d 2045 . 2 (x = suc y → ((∅ ·𝑜 x) = ∅ ↔ (∅ ·𝑜 suc y) = ∅))
7 oveq2 5463 . . 3 (x = A → (∅ ·𝑜 x) = (∅ ·𝑜 A))
87eqeq1d 2045 . 2 (x = A → ((∅ ·𝑜 x) = ∅ ↔ (∅ ·𝑜 A) = ∅))
9 0elon 4095 . . 3 On
10 om0 5977 . . 3 (∅ On → (∅ ·𝑜 ∅) = ∅)
119, 10ax-mp 7 . 2 (∅ ·𝑜 ∅) = ∅
12 oveq1 5462 . . . 4 ((∅ ·𝑜 y) = ∅ → ((∅ ·𝑜 y) +𝑜 ∅) = (∅ +𝑜 ∅))
13 oa0 5976 . . . . 5 (∅ On → (∅ +𝑜 ∅) = ∅)
149, 13ax-mp 7 . . . 4 (∅ +𝑜 ∅) = ∅
1512, 14syl6eq 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 ∅))
1816, 17mpan 400 . . . 4 (y 𝜔 → (∅ ·𝑜 suc y) = ((∅ ·𝑜 y) +𝑜 ∅))
1918eqeq1d 2045 . . 3 (y 𝜔 → ((∅ ·𝑜 suc y) = ∅ ↔ ((∅ ·𝑜 y) +𝑜 ∅) = ∅))
2015, 19syl5ibr 145 . 2 (y 𝜔 → ((∅ ·𝑜 y) = ∅ → (∅ ·𝑜 suc y) = ∅))
212, 4, 6, 8, 11, 20finds 4266 1 (A 𝜔 → (∅ ·𝑜 A) = ∅)
Colors of variables: wff set class
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-bnd 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
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