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Theorem caucvgprlemcanl 7394
Description: Lemma for cauappcvgprlemladdrl 7407. 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 7369 . . . 4 ((𝑓P𝑔PP) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
21adantl 273 . . 3 ((𝜑 ∧ (𝑓P𝑔PP)) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
3 caucvgprlemcanl.r . . . 4 (𝜑𝑅Q)
4 nqprlu 7297 . . . 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 7297 . . . . 5 (𝑆Q → ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P)
97, 8syl 14 . . . 4 (𝜑 → ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P)
10 addclpr 7287 . . . 4 ((𝐿P ∧ ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩ ∈ P) → (𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) ∈ P)
116, 9, 10syl2anc 406 . . 3 (𝜑 → (𝐿 +P ⟨{𝑙𝑙 <Q 𝑆}, {𝑢𝑆 <Q 𝑢}⟩) ∈ P)
12 caucvgprlemcanl.q . . . 4 (𝜑𝑄Q)
13 nqprlu 7297 . . . 4 (𝑄Q → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P)
1412, 13syl 14 . . 3 (𝜑 → ⟨{𝑙𝑙 <Q 𝑄}, {𝑢𝑄 <Q 𝑢}⟩ ∈ P)
15 addcomprg 7328 . . . 4 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
1615adantl 273 . . 3 ((𝜑 ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
172, 5, 11, 14, 16caovord2d 5892 . 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 7301 . . 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 406 . 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 7311 . . . . 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 406 . . . 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 7311 . . . . . 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 406 . . . . 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 5742 . . . 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 3906 . . 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 7125 . . . . 5 ((𝑅Q𝑄Q) → (𝑅 +Q 𝑄) ∈ Q)
273, 12, 26syl2anc 406 . . . 4 (𝜑 → (𝑅 +Q 𝑄) ∈ Q)
28 addclnq 7125 . . . . . . 7 ((𝑆Q𝑄Q) → (𝑆 +Q 𝑄) ∈ Q)
297, 12, 28syl2anc 406 . . . . . 6 (𝜑 → (𝑆 +Q 𝑄) ∈ Q)
30 nqprlu 7297 . . . . . 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 7287 . . . . 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 406 . . . 4 (𝜑 → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩) ∈ P)
34 nqprl 7301 . . . 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 406 . . 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 7329 . . . . 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 1197 . . . 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 3905 . . 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 219 . 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 220 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 103  wb 104  w3a 943   = wceq 1312  wcel 1461  {cab 2099  cop 3494   class class class wbr 3893  cfv 5079  (class class class)co 5726  1st c1st 5988  Qcnq 7030   +Q cplq 7032   <Q cltq 7035  Pcnp 7041   +P cpp 7043  <P cltp 7045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-eprel 4169  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-irdg 6219  df-1o 6265  df-2o 6266  df-oadd 6269  df-omul 6270  df-er 6381  df-ec 6383  df-qs 6387  df-ni 7054  df-pli 7055  df-mi 7056  df-lti 7057  df-plpq 7094  df-mpq 7095  df-enq 7097  df-nqqs 7098  df-plqqs 7099  df-mqqs 7100  df-1nqqs 7101  df-rq 7102  df-ltnqqs 7103  df-enq0 7174  df-nq0 7175  df-0nq0 7176  df-plq0 7177  df-mq0 7178  df-inp 7216  df-iplp 7218  df-iltp 7220
This theorem is referenced by:  cauappcvgprlemladdrl  7407  caucvgprlemladdrl  7428
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