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Theorem nnm0r 5990
 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 5460 . . 3 (x = ∅ → (∅ ·𝑜 x) = (∅ ·𝑜 ∅))
21eqeq1d 2045 . 2 (x = ∅ → ((∅ ·𝑜 x) = ∅ ↔ (∅ ·𝑜 ∅) = ∅))
3 oveq2 5460 . . 3 (x = y → (∅ ·𝑜 x) = (∅ ·𝑜 y))
43eqeq1d 2045 . 2 (x = y → ((∅ ·𝑜 x) = ∅ ↔ (∅ ·𝑜 y) = ∅))
5 oveq2 5460 . . 3 (x = suc y → (∅ ·𝑜 x) = (∅ ·𝑜 suc y))
65eqeq1d 2045 . 2 (x = suc y → ((∅ ·𝑜 x) = ∅ ↔ (∅ ·𝑜 suc y) = ∅))
7 oveq2 5460 . . 3 (x = A → (∅ ·𝑜 x) = (∅ ·𝑜 A))
87eqeq1d 2045 . 2 (x = A → ((∅ ·𝑜 x) = ∅ ↔ (∅ ·𝑜 A) = ∅))
9 0elon 4094 . . 3 On
10 om0 5970 . . 3 (∅ On → (∅ ·𝑜 ∅) = ∅)
119, 10ax-mp 7 . 2 (∅ ·𝑜 ∅) = ∅
12 oveq1 5459 . . . 4 ((∅ ·𝑜 y) = ∅ → ((∅ ·𝑜 y) +𝑜 ∅) = (∅ +𝑜 ∅))
13 oa0 5969 . . . . 5 (∅ On → (∅ +𝑜 ∅) = ∅)
149, 13ax-mp 7 . . . 4 (∅ +𝑜 ∅) = ∅
1512, 14syl6eq 2085 . . 3 ((∅ ·𝑜 y) = ∅ → ((∅ ·𝑜 y) +𝑜 ∅) = ∅)
16 peano1 4259 . . . . 5 𝜔
17 nnmsuc 5988 . . . . 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 4265 1 (A 𝜔 → (∅ ·𝑜 A) = ∅)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  ∅c0 3218  Oncon0 4065  suc csuc 4067  𝜔com 4255  (class class class)co 5452   +𝑜 coa 5930   ·𝑜 comu 5931 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 3862  ax-sep 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-setind 4219  ax-iinf 4253 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 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-int 3606  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-tr 3845  df-id 4020  df-iord 4068  df-on 4070  df-suc 4073  df-iom 4256  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852  df-ov 5455  df-oprab 5456  df-mpt2 5457  df-1st 5706  df-2nd 5707  df-recs 5858  df-irdg 5894  df-oadd 5937  df-omul 5938 This theorem is referenced by:  nnmcom  6000  nnmord  6019  nnm00  6031  enq0tr  6409  nq0m0r  6431
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