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Theorem mulcanenq0ec 6541
Description: Lemma for distributive law: cancellation of common factor. (Contributed by Jim Kingdon, 29-Nov-2019.)
Assertion
Ref Expression
mulcanenq0ec ((𝐴N𝐵 ∈ ω ∧ 𝐶N) → [⟨(𝐴 ·𝑜 𝐵), (𝐴 ·𝑜 𝐶)⟩] ~Q0 = [⟨𝐵, 𝐶⟩] ~Q0 )

Proof of Theorem mulcanenq0ec
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enq0er 6531 . . 3 ~Q0 Er (ω × N)
21a1i 9 . 2 ((𝐴N𝐵 ∈ ω ∧ 𝐶N) → ~Q0 Er (ω × N))
3 pinn 6405 . . . . 5 (𝐴N𝐴 ∈ ω)
433ad2ant1 925 . . . 4 ((𝐴N𝐵 ∈ ω ∧ 𝐶N) → 𝐴 ∈ ω)
5 simp2 905 . . . 4 ((𝐴N𝐵 ∈ ω ∧ 𝐶N) → 𝐵 ∈ ω)
6 pinn 6405 . . . . 5 (𝐶N𝐶 ∈ ω)
763ad2ant3 927 . . . 4 ((𝐴N𝐵 ∈ ω ∧ 𝐶N) → 𝐶 ∈ ω)
8 nnmcom 6068 . . . . 5 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ·𝑜 𝑦) = (𝑦 ·𝑜 𝑥))
98adantl 262 . . . 4 (((𝐴N𝐵 ∈ ω ∧ 𝐶N) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω)) → (𝑥 ·𝑜 𝑦) = (𝑦 ·𝑜 𝑥))
10 nnmass 6066 . . . . 5 ((𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑥 ·𝑜 𝑦) ·𝑜 𝑧) = (𝑥 ·𝑜 (𝑦 ·𝑜 𝑧)))
1110adantl 262 . . . 4 (((𝐴N𝐵 ∈ ω ∧ 𝐶N) ∧ (𝑥 ∈ ω ∧ 𝑦 ∈ ω ∧ 𝑧 ∈ ω)) → ((𝑥 ·𝑜 𝑦) ·𝑜 𝑧) = (𝑥 ·𝑜 (𝑦 ·𝑜 𝑧)))
124, 5, 7, 9, 11caov32d 5681 . . 3 ((𝐴N𝐵 ∈ ω ∧ 𝐶N) → ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = ((𝐴 ·𝑜 𝐶) ·𝑜 𝐵))
13 nnmcl 6060 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) ∈ ω)
143, 13sylan 267 . . . . . . 7 ((𝐴N𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) ∈ ω)
15 mulpiord 6413 . . . . . . . 8 ((𝐴N𝐶N) → (𝐴 ·N 𝐶) = (𝐴 ·𝑜 𝐶))
16 mulclpi 6424 . . . . . . . 8 ((𝐴N𝐶N) → (𝐴 ·N 𝐶) ∈ N)
1715, 16eqeltrrd 2115 . . . . . . 7 ((𝐴N𝐶N) → (𝐴 ·𝑜 𝐶) ∈ N)
1814, 17anim12i 321 . . . . . 6 (((𝐴N𝐵 ∈ ω) ∧ (𝐴N𝐶N)) → ((𝐴 ·𝑜 𝐵) ∈ ω ∧ (𝐴 ·𝑜 𝐶) ∈ N))
19 simpr 103 . . . . . . 7 (((𝐴N𝐴N) ∧ (𝐵 ∈ ω ∧ 𝐶N)) → (𝐵 ∈ ω ∧ 𝐶N))
2019an4s 522 . . . . . 6 (((𝐴N𝐵 ∈ ω) ∧ (𝐴N𝐶N)) → (𝐵 ∈ ω ∧ 𝐶N))
2118, 20jca 290 . . . . 5 (((𝐴N𝐵 ∈ ω) ∧ (𝐴N𝐶N)) → (((𝐴 ·𝑜 𝐵) ∈ ω ∧ (𝐴 ·𝑜 𝐶) ∈ N) ∧ (𝐵 ∈ ω ∧ 𝐶N)))
22213impdi 1190 . . . 4 ((𝐴N𝐵 ∈ ω ∧ 𝐶N) → (((𝐴 ·𝑜 𝐵) ∈ ω ∧ (𝐴 ·𝑜 𝐶) ∈ N) ∧ (𝐵 ∈ ω ∧ 𝐶N)))
23 enq0breq 6532 . . . 4 ((((𝐴 ·𝑜 𝐵) ∈ ω ∧ (𝐴 ·𝑜 𝐶) ∈ N) ∧ (𝐵 ∈ ω ∧ 𝐶N)) → (⟨(𝐴 ·𝑜 𝐵), (𝐴 ·𝑜 𝐶)⟩ ~Q0𝐵, 𝐶⟩ ↔ ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = ((𝐴 ·𝑜 𝐶) ·𝑜 𝐵)))
2422, 23syl 14 . . 3 ((𝐴N𝐵 ∈ ω ∧ 𝐶N) → (⟨(𝐴 ·𝑜 𝐵), (𝐴 ·𝑜 𝐶)⟩ ~Q0𝐵, 𝐶⟩ ↔ ((𝐴 ·𝑜 𝐵) ·𝑜 𝐶) = ((𝐴 ·𝑜 𝐶) ·𝑜 𝐵)))
2512, 24mpbird 156 . 2 ((𝐴N𝐵 ∈ ω ∧ 𝐶N) → ⟨(𝐴 ·𝑜 𝐵), (𝐴 ·𝑜 𝐶)⟩ ~Q0𝐵, 𝐶⟩)
262, 25erthi 6152 1 ((𝐴N𝐵 ∈ ω ∧ 𝐶N) → [⟨(𝐴 ·𝑜 𝐵), (𝐴 ·𝑜 𝐶)⟩] ~Q0 = [⟨𝐵, 𝐶⟩] ~Q0 )
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  w3a 885   = wceq 1243  wcel 1393  cop 3378   class class class wbr 3764  ωcom 4313   × cxp 4343  (class class class)co 5512   ·𝑜 comu 5999   Er wer 6103  [cec 6104  Ncnpi 6368   ·N cmi 6370   ~Q0 ceq0 6382
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-ni 6400  df-mi 6402  df-enq0 6520
This theorem is referenced by:  nnanq0  6554  distrnq0  6555
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