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Theorem enqdc1 6215
Description: The equivalence relation for positive fractions is decidable. (Contributed by Jim Kingdon, 7-Sep-2019.)
Assertion
Ref Expression
enqdc1 (((A N B N) 𝐶 (N × N)) → DECIDA, B⟩ ~Q 𝐶)

Proof of Theorem enqdc1
StepHypRef Expression
1 xp1st 5711 . . . 4 (𝐶 (N × N) → (1st𝐶) N)
2 xp2nd 5712 . . . 4 (𝐶 (N × N) → (2nd𝐶) N)
31, 2jca 290 . . 3 (𝐶 (N × N) → ((1st𝐶) N (2nd𝐶) N))
4 enqdc 6214 . . 3 (((A N B N) ((1st𝐶) N (2nd𝐶) N)) → DECIDA, B⟩ ~Q ⟨(1st𝐶), (2nd𝐶)⟩)
53, 4sylan2 270 . 2 (((A N B N) 𝐶 (N × N)) → DECIDA, B⟩ ~Q ⟨(1st𝐶), (2nd𝐶)⟩)
6 1st2nd2 5720 . . . . 5 (𝐶 (N × N) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
76breq2d 3746 . . . 4 (𝐶 (N × N) → (⟨A, B⟩ ~Q 𝐶 ↔ ⟨A, B⟩ ~Q ⟨(1st𝐶), (2nd𝐶)⟩))
87dcbid 736 . . 3 (𝐶 (N × N) → (DECIDA, B⟩ ~Q 𝐶DECIDA, B⟩ ~Q ⟨(1st𝐶), (2nd𝐶)⟩))
98adantl 262 . 2 (((A N B N) 𝐶 (N × N)) → (DECIDA, B⟩ ~Q 𝐶DECIDA, B⟩ ~Q ⟨(1st𝐶), (2nd𝐶)⟩))
105, 9mpbird 156 1 (((A N B N) 𝐶 (N × N)) → DECIDA, B⟩ ~Q 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  DECID wdc 730   wcel 1370  cop 3349   class class class wbr 3734   × cxp 4266  cfv 4825  1st c1st 5684  2nd c2nd 5685  Ncnpi 6126   ~Q ceq 6133
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-nul 3853  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-setind 4200  ax-iinf 4234
This theorem depends on definitions:  df-bi 110  df-dc 731  df-3or 872  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-int 3586  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-tr 3825  df-id 4000  df-iord 4048  df-on 4050  df-suc 4053  df-iom 4237  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-1st 5686  df-2nd 5687  df-recs 5838  df-irdg 5874  df-oadd 5916  df-omul 5917  df-ni 6158  df-mi 6160  df-enq 6200
This theorem is referenced by: (None)
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