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Theorem caucvgprlemcanl 6740
Description: Lemma for cauappcvgprlemladdrl 6753. Cancelling a term from both sides. (Contributed by Jim Kingdon, 15-Aug-2020.)
Hypotheses
Ref Expression
caucvgprlemcanl.l (𝜑𝐿P)
caucvgprlemcanl.s (𝜑𝑆Q)
caucvgprlemcanl.r (𝜑𝑅Q)
caucvgprlemcanl.q (𝜑𝑄Q)
Assertion
Ref Expression
caucvgprlemcanl (𝜑 → ((𝑅 +Q 𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩)) ↔ 𝑅 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))))
Distinct variable groups:   𝑄,𝑙,𝑢   𝑅,𝑙,𝑢   𝑆,𝑙,𝑢
Allowed substitution hints:   𝜑(𝑢,𝑙)   𝐿(𝑢,𝑙)

Proof of Theorem caucvgprlemcanl
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltaprg 6715 . . . 4 ((𝑓P𝑔PP) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
21adantl 262 . . 3 ((𝜑 ∧ (𝑓P𝑔PP)) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
3 caucvgprlemcanl.r . . . 4 (𝜑𝑅Q)
4 nqprlu 6643 . . . 4 (𝑅Q → ⟨{𝑙𝑙 <Q 𝑅}, {𝑢𝑅 <Q 𝑢}⟩ ∈ P)
53, 4syl 14 . . 3 (𝜑 → ⟨{𝑙𝑙 <Q 𝑅}, {𝑢𝑅 <Q 𝑢}⟩ ∈ P)
6 caucvgprlemcanl.l . . . 4 (𝜑𝐿P)
7 caucvgprlemcanl.s . . . . 5 (𝜑𝑆Q)
8 nqprlu 6643 . . . . 5 (𝑆Q → ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P)
97, 8syl 14 . . . 4 (𝜑 → ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P)
10 addclpr 6633 . . . 4 ((𝐿P ∧ ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P) → (𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) ∈ P)
116, 9, 10syl2anc 391 . . 3 (𝜑 → (𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) ∈ P)
12 caucvgprlemcanl.q . . . 4 (𝜑𝑄Q)
13 nqprlu 6643 . . . 4 (𝑄Q → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P)
1412, 13syl 14 . . 3 (𝜑 → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P)
15 addcomprg 6674 . . . 4 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
1615adantl 262 . . 3 ((𝜑 ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
172, 5, 11, 14, 16caovord2d 5670 . 2 (𝜑 → (⟨{𝑙𝑙 <Q 𝑅}, {𝑢𝑅 <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) ↔ (⟨{𝑙𝑙 <Q 𝑅}, {𝑢𝑅 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)<P ((𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)))
18 nqprl 6647 . . 3 ((𝑅Q ∧ (𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) ∈ P) → (𝑅 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) ↔ ⟨{𝑙𝑙 <Q 𝑅}, {𝑢𝑅 <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)))
193, 11, 18syl2anc 391 . 2 (𝜑 → (𝑅 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)) ↔ ⟨{𝑙𝑙 <Q 𝑅}, {𝑢𝑅 <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩)))
20 addnqpr 6657 . . . . 5 ((𝑅Q𝑄Q) → ⟨{𝑙𝑙 <Q (𝑅 +Q 𝑄)}, {𝑢 ∣ (𝑅 +Q 𝑄) <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q 𝑅}, {𝑢𝑅 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))
213, 12, 20syl2anc 391 . . . 4 (𝜑 → ⟨{𝑙𝑙 <Q (𝑅 +Q 𝑄)}, {𝑢 ∣ (𝑅 +Q 𝑄) <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q 𝑅}, {𝑢𝑅 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))
22 addnqpr 6657 . . . . . 6 ((𝑆Q𝑄Q) → ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))
237, 12, 22syl2anc 391 . . . . 5 (𝜑 → ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))
2423oveq2d 5528 . . . 4 (𝜑 → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩) = (𝐿 +P (⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)))
2521, 24breq12d 3777 . . 3 (𝜑 → (⟨{𝑙𝑙 <Q (𝑅 +Q 𝑄)}, {𝑢 ∣ (𝑅 +Q 𝑄) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩) ↔ (⟨{𝑙𝑙 <Q 𝑅}, {𝑢𝑅 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)<P (𝐿 +P (⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))))
26 addclnq 6471 . . . . 5 ((𝑅Q𝑄Q) → (𝑅 +Q 𝑄) ∈ Q)
273, 12, 26syl2anc 391 . . . 4 (𝜑 → (𝑅 +Q 𝑄) ∈ Q)
28 addclnq 6471 . . . . . . 7 ((𝑆Q𝑄Q) → (𝑆 +Q 𝑄) ∈ Q)
297, 12, 28syl2anc 391 . . . . . 6 (𝜑 → (𝑆 +Q 𝑄) ∈ Q)
30 nqprlu 6643 . . . . . 6 ((𝑆 +Q 𝑄) ∈ Q → ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩ ∈ P)
3129, 30syl 14 . . . . 5 (𝜑 → ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩ ∈ P)
32 addclpr 6633 . . . . 5 ((𝐿P ∧ ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩ ∈ P) → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩) ∈ P)
336, 31, 32syl2anc 391 . . . 4 (𝜑 → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩) ∈ P)
34 nqprl 6647 . . . 4 (((𝑅 +Q 𝑄) ∈ Q ∧ (𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩) ∈ P) → ((𝑅 +Q 𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩)) ↔ ⟨{𝑙𝑙 <Q (𝑅 +Q 𝑄)}, {𝑢 ∣ (𝑅 +Q 𝑄) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩)))
3527, 33, 34syl2anc 391 . . 3 (𝜑 → ((𝑅 +Q 𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩)) ↔ ⟨{𝑙𝑙 <Q (𝑅 +Q 𝑄)}, {𝑢 ∣ (𝑅 +Q 𝑄) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩)))
36 addassprg 6675 . . . . 5 ((𝐿P ∧ ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P ∧ ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P) → ((𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) = (𝐿 +P (⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)))
376, 9, 14, 36syl3anc 1135 . . . 4 (𝜑 → ((𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) = (𝐿 +P (⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)))
3837breq2d 3776 . . 3 (𝜑 → ((⟨{𝑙𝑙 <Q 𝑅}, {𝑢𝑅 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)<P ((𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩) ↔ (⟨{𝑙𝑙 <Q 𝑅}, {𝑢𝑅 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)<P (𝐿 +P (⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩))))
3925, 35, 383bitr4d 209 . 2 (𝜑 → ((𝑅 +Q 𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩)) ↔ (⟨{𝑙𝑙 <Q 𝑅}, {𝑢𝑅 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)<P ((𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩)))
4017, 19, 393bitr4rd 210 1 (𝜑 → ((𝑅 +Q 𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩)) ↔ 𝑅 ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  w3a 885   = wceq 1243  wcel 1393  {cab 2026  cop 3378   class class class wbr 3764  cfv 4902  (class class class)co 5512  1st c1st 5765  Qcnq 6376   +Q cplq 6378   <Q cltq 6381  Pcnp 6387   +P cpp 6389  <P cltp 6391
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-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  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-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-iplp 6564  df-iltp 6566
This theorem is referenced by:  cauappcvgprlemladdrl  6753  caucvgprlemladdrl  6774
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