Theorem addlocprlemeqgt 6630

Description: Lemma for addlocpr 6634. 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	⊢ (𝜑 → 𝐴 ∈ P)
addlocprlem.b	⊢ (𝜑 → 𝐵 ∈ P)
addlocprlem.qr	⊢ (𝜑 → 𝑄 <Q 𝑅)
addlocprlem.p	⊢ (𝜑 → 𝑃 ∈ Q)
addlocprlem.qppr	⊢ (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)
addlocprlem.dlo	⊢ (𝜑 → 𝐷 ∈ (1st ‘𝐴))
addlocprlem.uup	⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴))
addlocprlem.du	⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))
addlocprlem.elo	⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵))
addlocprlem.tup	⊢ (𝜑 → 𝑇 ∈ (2nd ‘𝐵))
addlocprlem.et	⊢ (𝜑 → 𝑇 <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 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addlocprlem.du	. . 3 ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))
2		addlocprlem.et	. . 3 ⊢ (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))
3		addlocprlem.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ P)
4		prop 6573	. . . . . 6 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
5	3, 4	syl 14	. . . . 5 ⊢ (𝜑 → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
6		addlocprlem.uup	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴))
7		elprnqu 6580	. . . . 5 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑈 ∈ (2nd ‘𝐴)) → 𝑈 ∈ Q)
8	5, 6, 7	syl2anc 391	. . . 4 ⊢ (𝜑 → 𝑈 ∈ Q)
9		addlocprlem.dlo	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (1st ‘𝐴))
10		elprnql 6579	. . . . . 6 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝐷 ∈ (1st ‘𝐴)) → 𝐷 ∈ Q)
11	5, 9, 10	syl2anc 391	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Q)
12		addlocprlem.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ Q)
13		addclnq 6473	. . . . 5 ⊢ ((𝐷 ∈ Q ∧ 𝑃 ∈ Q) → (𝐷 +Q 𝑃) ∈ Q)
14	11, 12, 13	syl2anc 391	. . . 4 ⊢ (𝜑 → (𝐷 +Q 𝑃) ∈ Q)
15		addlocprlem.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ P)
16		prop 6573	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
17	15, 16	syl 14	. . . . 5 ⊢ (𝜑 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
18		addlocprlem.tup	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (2nd ‘𝐵))
19		elprnqu 6580	. . . . 5 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑇 ∈ (2nd ‘𝐵)) → 𝑇 ∈ Q)
20	17, 18, 19	syl2anc 391	. . . 4 ⊢ (𝜑 → 𝑇 ∈ Q)
21		addlocprlem.elo	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵))
22		elprnql 6579	. . . . . 6 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐸 ∈ (1st ‘𝐵)) → 𝐸 ∈ Q)
23	17, 21, 22	syl2anc 391	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Q)
24		addclnq 6473	. . . . 5 ⊢ ((𝐸 ∈ Q ∧ 𝑃 ∈ Q) → (𝐸 +Q 𝑃) ∈ Q)
25	23, 12, 24	syl2anc 391	. . . 4 ⊢ (𝜑 → (𝐸 +Q 𝑃) ∈ Q)
26		lt2addnq 6502	. . . 4 ⊢ (((𝑈 ∈ Q ∧ (𝐷 +Q 𝑃) ∈ Q) ∧ (𝑇 ∈ Q ∧ (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))))
27	8, 14, 20, 25, 26	syl22anc 1136	. . 3 ⊢ (𝜑 → ((𝑈 <Q (𝐷 +Q 𝑃) ∧ 𝑇 <Q (𝐸 +Q 𝑃)) → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃))))
28	1, 2, 27	mp2and 409	. 2 ⊢ (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃)))
29		addcomnqg 6479	. . . 4 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
30	29	adantl 262	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
31		addassnqg 6480	. . . 4 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
32	31	adantl 262	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
33		addclnq 6473	. . . 4 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) ∈ Q)
34	33	adantl 262	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) ∈ Q)
35	11, 12, 23, 30, 32, 12, 34	caov4d 5685	. 2 ⊢ (𝜑 → ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
36	28, 35	breqtrd 3788	1 ⊢ (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ⟨cop 3378   class class class wbr 3764  ‘cfv 4902  (class class class)co 5512  1^st c1st 5765  2^nd c2nd 5766  Qcnq 6378   +_Q cplq 6380   <_Q cltq 6383  Pcnp 6389

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-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-ltnqqs 6451  df-inp 6564

This theorem is referenced by:  addlocprlemeq  6631  addlocprlemgt  6632