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Theorem nqpnq0nq 6551
 Description: A positive fraction plus a non-negative fraction is a positive fraction. (Contributed by Jim Kingdon, 30-Nov-2019.)
Assertion
Ref Expression
nqpnq0nq ((𝐴Q𝐵Q0) → (𝐴 +Q0 𝐵) ∈ Q)

Proof of Theorem nqpnq0nq
Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nqpi 6476 . . . 4 (𝐴Q → ∃𝑥𝑦((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ))
2 nq0nn 6540 . . . 4 (𝐵Q0 → ∃𝑧𝑤((𝑧 ∈ ω ∧ 𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q0 ))
31, 2anim12i 321 . . 3 ((𝐴Q𝐵Q0) → (∃𝑥𝑦((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ∃𝑧𝑤((𝑧 ∈ ω ∧ 𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q0 )))
4 ee4anv 1809 . . 3 (∃𝑥𝑦𝑧𝑤(((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ((𝑧 ∈ ω ∧ 𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q0 )) ↔ (∃𝑥𝑦((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ∃𝑧𝑤((𝑧 ∈ ω ∧ 𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q0 )))
53, 4sylibr 137 . 2 ((𝐴Q𝐵Q0) → ∃𝑥𝑦𝑧𝑤(((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ((𝑧 ∈ ω ∧ 𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q0 )))
6 oveq12 5521 . . . . . . 7 ((𝐴 = [⟨𝑥, 𝑦⟩] ~Q𝐵 = [⟨𝑧, 𝑤⟩] ~Q0 ) → (𝐴 +Q0 𝐵) = ([⟨𝑥, 𝑦⟩] ~Q +Q0 [⟨𝑧, 𝑤⟩] ~Q0 ))
76ad2ant2l 477 . . . . . 6 ((((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ((𝑧 ∈ ω ∧ 𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q0 )) → (𝐴 +Q0 𝐵) = ([⟨𝑥, 𝑦⟩] ~Q +Q0 [⟨𝑧, 𝑤⟩] ~Q0 ))
8 nqnq0pi 6536 . . . . . . . . . 10 ((𝑥N𝑦N) → [⟨𝑥, 𝑦⟩] ~Q0 = [⟨𝑥, 𝑦⟩] ~Q )
98oveq1d 5527 . . . . . . . . 9 ((𝑥N𝑦N) → ([⟨𝑥, 𝑦⟩] ~Q0 +Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) = ([⟨𝑥, 𝑦⟩] ~Q +Q0 [⟨𝑧, 𝑤⟩] ~Q0 ))
109adantr 261 . . . . . . . 8 (((𝑥N𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → ([⟨𝑥, 𝑦⟩] ~Q0 +Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) = ([⟨𝑥, 𝑦⟩] ~Q +Q0 [⟨𝑧, 𝑤⟩] ~Q0 ))
11 pinn 6407 . . . . . . . . 9 (𝑥N𝑥 ∈ ω)
12 addnnnq0 6547 . . . . . . . . 9 (((𝑥 ∈ ω ∧ 𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → ([⟨𝑥, 𝑦⟩] ~Q0 +Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) = [⟨((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)), (𝑦 ·𝑜 𝑤)⟩] ~Q0 )
1311, 12sylanl1 382 . . . . . . . 8 (((𝑥N𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → ([⟨𝑥, 𝑦⟩] ~Q0 +Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) = [⟨((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)), (𝑦 ·𝑜 𝑤)⟩] ~Q0 )
1410, 13eqtr3d 2074 . . . . . . 7 (((𝑥N𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → ([⟨𝑥, 𝑦⟩] ~Q +Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) = [⟨((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)), (𝑦 ·𝑜 𝑤)⟩] ~Q0 )
1514ad2ant2r 478 . . . . . 6 ((((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ((𝑧 ∈ ω ∧ 𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q0 )) → ([⟨𝑥, 𝑦⟩] ~Q +Q0 [⟨𝑧, 𝑤⟩] ~Q0 ) = [⟨((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)), (𝑦 ·𝑜 𝑤)⟩] ~Q0 )
167, 15eqtrd 2072 . . . . 5 ((((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ((𝑧 ∈ ω ∧ 𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q0 )) → (𝐴 +Q0 𝐵) = [⟨((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)), (𝑦 ·𝑜 𝑤)⟩] ~Q0 )
17 pinn 6407 . . . . . . . . . . . . . 14 (𝑦N𝑦 ∈ ω)
18 nnmcl 6060 . . . . . . . . . . . . . 14 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (𝑦 ·𝑜 𝑧) ∈ ω)
1917, 18sylan 267 . . . . . . . . . . . . 13 ((𝑦N𝑧 ∈ ω) → (𝑦 ·𝑜 𝑧) ∈ ω)
2019ad2ant2lr 479 . . . . . . . . . . . 12 (((𝑥N𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → (𝑦 ·𝑜 𝑧) ∈ ω)
21 mulpiord 6415 . . . . . . . . . . . . . 14 ((𝑥N𝑤N) → (𝑥 ·N 𝑤) = (𝑥 ·𝑜 𝑤))
22 mulclpi 6426 . . . . . . . . . . . . . 14 ((𝑥N𝑤N) → (𝑥 ·N 𝑤) ∈ N)
2321, 22eqeltrrd 2115 . . . . . . . . . . . . 13 ((𝑥N𝑤N) → (𝑥 ·𝑜 𝑤) ∈ N)
2423ad2ant2rl 480 . . . . . . . . . . . 12 (((𝑥N𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → (𝑥 ·𝑜 𝑤) ∈ N)
25 pinn 6407 . . . . . . . . . . . . 13 ((𝑥 ·𝑜 𝑤) ∈ N → (𝑥 ·𝑜 𝑤) ∈ ω)
26 nnacom 6063 . . . . . . . . . . . . 13 (((𝑦 ·𝑜 𝑧) ∈ ω ∧ (𝑥 ·𝑜 𝑤) ∈ ω) → ((𝑦 ·𝑜 𝑧) +𝑜 (𝑥 ·𝑜 𝑤)) = ((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)))
2725, 26sylan2 270 . . . . . . . . . . . 12 (((𝑦 ·𝑜 𝑧) ∈ ω ∧ (𝑥 ·𝑜 𝑤) ∈ N) → ((𝑦 ·𝑜 𝑧) +𝑜 (𝑥 ·𝑜 𝑤)) = ((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)))
2820, 24, 27syl2anc 391 . . . . . . . . . . 11 (((𝑥N𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → ((𝑦 ·𝑜 𝑧) +𝑜 (𝑥 ·𝑜 𝑤)) = ((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)))
29 nnppipi 6441 . . . . . . . . . . . 12 (((𝑦 ·𝑜 𝑧) ∈ ω ∧ (𝑥 ·𝑜 𝑤) ∈ N) → ((𝑦 ·𝑜 𝑧) +𝑜 (𝑥 ·𝑜 𝑤)) ∈ N)
3020, 24, 29syl2anc 391 . . . . . . . . . . 11 (((𝑥N𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → ((𝑦 ·𝑜 𝑧) +𝑜 (𝑥 ·𝑜 𝑤)) ∈ N)
3128, 30eqeltrrd 2115 . . . . . . . . . 10 (((𝑥N𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → ((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)) ∈ N)
32 mulpiord 6415 . . . . . . . . . . . 12 ((𝑦N𝑤N) → (𝑦 ·N 𝑤) = (𝑦 ·𝑜 𝑤))
33 mulclpi 6426 . . . . . . . . . . . 12 ((𝑦N𝑤N) → (𝑦 ·N 𝑤) ∈ N)
3432, 33eqeltrrd 2115 . . . . . . . . . . 11 ((𝑦N𝑤N) → (𝑦 ·𝑜 𝑤) ∈ N)
3534ad2ant2l 477 . . . . . . . . . 10 (((𝑥N𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → (𝑦 ·𝑜 𝑤) ∈ N)
36 opelxpi 4376 . . . . . . . . . 10 ((((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)) ∈ N ∧ (𝑦 ·𝑜 𝑤) ∈ N) → ⟨((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)), (𝑦 ·𝑜 𝑤)⟩ ∈ (N × N))
3731, 35, 36syl2anc 391 . . . . . . . . 9 (((𝑥N𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → ⟨((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)), (𝑦 ·𝑜 𝑤)⟩ ∈ (N × N))
38 enqex 6458 . . . . . . . . . 10 ~Q ∈ V
3938ecelqsi 6160 . . . . . . . . 9 (⟨((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)), (𝑦 ·𝑜 𝑤)⟩ ∈ (N × N) → [⟨((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)), (𝑦 ·𝑜 𝑤)⟩] ~Q ∈ ((N × N) / ~Q ))
4037, 39syl 14 . . . . . . . 8 (((𝑥N𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → [⟨((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)), (𝑦 ·𝑜 𝑤)⟩] ~Q ∈ ((N × N) / ~Q ))
41 df-nqqs 6446 . . . . . . . 8 Q = ((N × N) / ~Q )
4240, 41syl6eleqr 2131 . . . . . . 7 (((𝑥N𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → [⟨((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)), (𝑦 ·𝑜 𝑤)⟩] ~QQ)
43 nqnq0pi 6536 . . . . . . . . 9 ((((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)) ∈ N ∧ (𝑦 ·𝑜 𝑤) ∈ N) → [⟨((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)), (𝑦 ·𝑜 𝑤)⟩] ~Q0 = [⟨((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)), (𝑦 ·𝑜 𝑤)⟩] ~Q )
4443eleq1d 2106 . . . . . . . 8 ((((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)) ∈ N ∧ (𝑦 ·𝑜 𝑤) ∈ N) → ([⟨((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)), (𝑦 ·𝑜 𝑤)⟩] ~Q0Q ↔ [⟨((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)), (𝑦 ·𝑜 𝑤)⟩] ~QQ))
4531, 35, 44syl2anc 391 . . . . . . 7 (((𝑥N𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → ([⟨((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)), (𝑦 ·𝑜 𝑤)⟩] ~Q0Q ↔ [⟨((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)), (𝑦 ·𝑜 𝑤)⟩] ~QQ))
4642, 45mpbird 156 . . . . . 6 (((𝑥N𝑦N) ∧ (𝑧 ∈ ω ∧ 𝑤N)) → [⟨((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)), (𝑦 ·𝑜 𝑤)⟩] ~Q0Q)
4746ad2ant2r 478 . . . . 5 ((((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ((𝑧 ∈ ω ∧ 𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q0 )) → [⟨((𝑥 ·𝑜 𝑤) +𝑜 (𝑦 ·𝑜 𝑧)), (𝑦 ·𝑜 𝑤)⟩] ~Q0Q)
4816, 47eqeltrd 2114 . . . 4 ((((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ((𝑧 ∈ ω ∧ 𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q0 )) → (𝐴 +Q0 𝐵) ∈ Q)
4948exlimivv 1776 . . 3 (∃𝑧𝑤(((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ((𝑧 ∈ ω ∧ 𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q0 )) → (𝐴 +Q0 𝐵) ∈ Q)
5049exlimivv 1776 . 2 (∃𝑥𝑦𝑧𝑤(((𝑥N𝑦N) ∧ 𝐴 = [⟨𝑥, 𝑦⟩] ~Q ) ∧ ((𝑧 ∈ ω ∧ 𝑤N) ∧ 𝐵 = [⟨𝑧, 𝑤⟩] ~Q0 )) → (𝐴 +Q0 𝐵) ∈ Q)
515, 50syl 14 1 ((𝐴Q𝐵Q0) → (𝐴 +Q0 𝐵) ∈ Q)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ⟨cop 3378  ωcom 4313   × cxp 4343  (class class class)co 5512   +𝑜 coa 5998   ·𝑜 comu 5999  [cec 6104   / cqs 6105  Ncnpi 6370   ·N cmi 6372   ~Q ceq 6377  Qcnq 6378   ~Q0 ceq0 6384  Q0cnq0 6385   +Q0 cplq0 6387 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-qs 6112  df-ni 6402  df-mi 6404  df-enq 6445  df-nqqs 6446  df-enq0 6522  df-nq0 6523  df-plq0 6525 This theorem is referenced by:  prarloclemcalc  6600
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