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Theorem addlocprlemeqgt 6687
Description: Lemma for addlocpr 6691. 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 6630 . . . . . 6 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
53, 4syl 14 . . . . 5 (𝜑 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
6 addlocprlem.uup . . . . 5 (𝜑𝑈 ∈ (2nd𝐴))
7 elprnqu 6637 . . . . 5 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑈 ∈ (2nd𝐴)) → 𝑈Q)
85, 6, 7syl2anc 397 . . . 4 (𝜑𝑈Q)
9 addlocprlem.dlo . . . . . 6 (𝜑𝐷 ∈ (1st𝐴))
10 elprnql 6636 . . . . . 6 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐷 ∈ (1st𝐴)) → 𝐷Q)
115, 9, 10syl2anc 397 . . . . 5 (𝜑𝐷Q)
12 addlocprlem.p . . . . 5 (𝜑𝑃Q)
13 addclnq 6530 . . . . 5 ((𝐷Q𝑃Q) → (𝐷 +Q 𝑃) ∈ Q)
1411, 12, 13syl2anc 397 . . . 4 (𝜑 → (𝐷 +Q 𝑃) ∈ Q)
15 addlocprlem.b . . . . . 6 (𝜑𝐵P)
16 prop 6630 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
1715, 16syl 14 . . . . 5 (𝜑 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
18 addlocprlem.tup . . . . 5 (𝜑𝑇 ∈ (2nd𝐵))
19 elprnqu 6637 . . . . 5 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑇 ∈ (2nd𝐵)) → 𝑇Q)
2017, 18, 19syl2anc 397 . . . 4 (𝜑𝑇Q)
21 addlocprlem.elo . . . . . 6 (𝜑𝐸 ∈ (1st𝐵))
22 elprnql 6636 . . . . . 6 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐸 ∈ (1st𝐵)) → 𝐸Q)
2317, 21, 22syl2anc 397 . . . . 5 (𝜑𝐸Q)
24 addclnq 6530 . . . . 5 ((𝐸Q𝑃Q) → (𝐸 +Q 𝑃) ∈ Q)
2523, 12, 24syl2anc 397 . . . 4 (𝜑 → (𝐸 +Q 𝑃) ∈ Q)
26 lt2addnq 6559 . . . 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 1147 . . 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 417 . 2 (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃)))
29 addcomnqg 6536 . . . 4 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3029adantl 266 . . 3 ((𝜑 ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
31 addassnqg 6537 . . . 4 ((𝑓Q𝑔QQ) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
3231adantl 266 . . 3 ((𝜑 ∧ (𝑓Q𝑔QQ)) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
33 addclnq 6530 . . . 4 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) ∈ Q)
3433adantl 266 . . 3 ((𝜑 ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) ∈ Q)
3511, 12, 23, 30, 32, 12, 34caov4d 5712 . 2 (𝜑 → ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
3628, 35breqtrd 3815 1 (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3a 896   = wceq 1259  wcel 1409  cop 3405   class class class wbr 3791  cfv 4929  (class class class)co 5539  1st c1st 5792  2nd c2nd 5793  Qcnq 6435   +Q cplq 6437   <Q cltq 6440  Pcnp 6446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-ltnqqs 6508  df-inp 6621
This theorem is referenced by:  addlocprlemeq  6688  addlocprlemgt  6689
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