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Theorem addlocprlemeqgt 7360
Description: Lemma for addlocpr 7364. This is a step used in both the 𝑄 = (𝐷 +Q 𝐸) and (𝐷 +Q 𝐸) <Q 𝑄 cases. (Contributed by Jim Kingdon, 7-Dec-2019.)
Hypotheses
Ref Expression
addlocprlem.a (𝜑𝐴P)
addlocprlem.b (𝜑𝐵P)
addlocprlem.qr (𝜑𝑄 <Q 𝑅)
addlocprlem.p (𝜑𝑃Q)
addlocprlem.qppr (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)
addlocprlem.dlo (𝜑𝐷 ∈ (1st𝐴))
addlocprlem.uup (𝜑𝑈 ∈ (2nd𝐴))
addlocprlem.du (𝜑𝑈 <Q (𝐷 +Q 𝑃))
addlocprlem.elo (𝜑𝐸 ∈ (1st𝐵))
addlocprlem.tup (𝜑𝑇 ∈ (2nd𝐵))
addlocprlem.et (𝜑𝑇 <Q (𝐸 +Q 𝑃))
Assertion
Ref Expression
addlocprlemeqgt (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))

Proof of Theorem addlocprlemeqgt
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addlocprlem.du . . 3 (𝜑𝑈 <Q (𝐷 +Q 𝑃))
2 addlocprlem.et . . 3 (𝜑𝑇 <Q (𝐸 +Q 𝑃))
3 addlocprlem.a . . . . . 6 (𝜑𝐴P)
4 prop 7303 . . . . . 6 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
53, 4syl 14 . . . . 5 (𝜑 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
6 addlocprlem.uup . . . . 5 (𝜑𝑈 ∈ (2nd𝐴))
7 elprnqu 7310 . . . . 5 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑈 ∈ (2nd𝐴)) → 𝑈Q)
85, 6, 7syl2anc 409 . . . 4 (𝜑𝑈Q)
9 addlocprlem.dlo . . . . . 6 (𝜑𝐷 ∈ (1st𝐴))
10 elprnql 7309 . . . . . 6 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐷 ∈ (1st𝐴)) → 𝐷Q)
115, 9, 10syl2anc 409 . . . . 5 (𝜑𝐷Q)
12 addlocprlem.p . . . . 5 (𝜑𝑃Q)
13 addclnq 7203 . . . . 5 ((𝐷Q𝑃Q) → (𝐷 +Q 𝑃) ∈ Q)
1411, 12, 13syl2anc 409 . . . 4 (𝜑 → (𝐷 +Q 𝑃) ∈ Q)
15 addlocprlem.b . . . . . 6 (𝜑𝐵P)
16 prop 7303 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
1715, 16syl 14 . . . . 5 (𝜑 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
18 addlocprlem.tup . . . . 5 (𝜑𝑇 ∈ (2nd𝐵))
19 elprnqu 7310 . . . . 5 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑇 ∈ (2nd𝐵)) → 𝑇Q)
2017, 18, 19syl2anc 409 . . . 4 (𝜑𝑇Q)
21 addlocprlem.elo . . . . . 6 (𝜑𝐸 ∈ (1st𝐵))
22 elprnql 7309 . . . . . 6 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐸 ∈ (1st𝐵)) → 𝐸Q)
2317, 21, 22syl2anc 409 . . . . 5 (𝜑𝐸Q)
24 addclnq 7203 . . . . 5 ((𝐸Q𝑃Q) → (𝐸 +Q 𝑃) ∈ Q)
2523, 12, 24syl2anc 409 . . . 4 (𝜑 → (𝐸 +Q 𝑃) ∈ Q)
26 lt2addnq 7232 . . . 4 (((𝑈Q ∧ (𝐷 +Q 𝑃) ∈ Q) ∧ (𝑇Q ∧ (𝐸 +Q 𝑃) ∈ Q)) → ((𝑈 <Q (𝐷 +Q 𝑃) ∧ 𝑇 <Q (𝐸 +Q 𝑃)) → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃))))
278, 14, 20, 25, 26syl22anc 1218 . . 3 (𝜑 → ((𝑈 <Q (𝐷 +Q 𝑃) ∧ 𝑇 <Q (𝐸 +Q 𝑃)) → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃))))
281, 2, 27mp2and 430 . 2 (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃)))
29 addcomnqg 7209 . . . 4 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3029adantl 275 . . 3 ((𝜑 ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
31 addassnqg 7210 . . . 4 ((𝑓Q𝑔QQ) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
3231adantl 275 . . 3 ((𝜑 ∧ (𝑓Q𝑔QQ)) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
33 addclnq 7203 . . . 4 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) ∈ Q)
3433adantl 275 . . 3 ((𝜑 ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) ∈ Q)
3511, 12, 23, 30, 32, 12, 34caov4d 5959 . 2 (𝜑 → ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
3628, 35breqtrd 3958 1 (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 963   = wceq 1332  wcel 1481  cop 3531   class class class wbr 3933  cfv 5127  (class class class)co 5778  1st c1st 6040  2nd c2nd 6041  Qcnq 7108   +Q cplq 7110   <Q cltq 7113  Pcnp 7119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-iinf 4506
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-eprel 4215  df-id 4219  df-po 4222  df-iso 4223  df-iord 4292  df-on 4294  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-recs 6206  df-irdg 6271  df-oadd 6321  df-omul 6322  df-er 6433  df-ec 6435  df-qs 6439  df-ni 7132  df-pli 7133  df-mi 7134  df-lti 7135  df-plpq 7172  df-enq 7175  df-nqqs 7176  df-plqqs 7177  df-ltnqqs 7181  df-inp 7294
This theorem is referenced by:  addlocprlemeq  7361  addlocprlemgt  7362
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