Theorem addlocprlemeqgt 6722

Description: Lemma for addlocpr 6726. 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 6665	. . . . . 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 6672	. . . . 5 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑈 ∈ (2nd ‘𝐴)) → 𝑈 ∈ Q)
8	5, 6, 7	syl2anc 403	. . . 4 ⊢ (𝜑 → 𝑈 ∈ Q)
9		addlocprlem.dlo	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (1st ‘𝐴))
10		elprnql 6671	. . . . . 6 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝐷 ∈ (1st ‘𝐴)) → 𝐷 ∈ Q)
11	5, 9, 10	syl2anc 403	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Q)
12		addlocprlem.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ Q)
13		addclnq 6565	. . . . 5 ⊢ ((𝐷 ∈ Q ∧ 𝑃 ∈ Q) → (𝐷 +Q 𝑃) ∈ Q)
14	11, 12, 13	syl2anc 403	. . . 4 ⊢ (𝜑 → (𝐷 +Q 𝑃) ∈ Q)
15		addlocprlem.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ P)
16		prop 6665	. . . . . 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 6672	. . . . 5 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑇 ∈ (2nd ‘𝐵)) → 𝑇 ∈ Q)
20	17, 18, 19	syl2anc 403	. . . 4 ⊢ (𝜑 → 𝑇 ∈ Q)
21		addlocprlem.elo	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵))
22		elprnql 6671	. . . . . 6 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐸 ∈ (1st ‘𝐵)) → 𝐸 ∈ Q)
23	17, 21, 22	syl2anc 403	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Q)
24		addclnq 6565	. . . . 5 ⊢ ((𝐸 ∈ Q ∧ 𝑃 ∈ Q) → (𝐸 +Q 𝑃) ∈ Q)
25	23, 12, 24	syl2anc 403	. . . 4 ⊢ (𝜑 → (𝐸 +Q 𝑃) ∈ Q)
26		lt2addnq 6594	. . . 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 1170	. . 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 423	. 2 ⊢ (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃)))
29		addcomnqg 6571	. . . 4 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
30	29	adantl 271	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
31		addassnqg 6572	. . . 4 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
32	31	adantl 271	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
33		addclnq 6565	. . . 4 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) ∈ Q)
34	33	adantl 271	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) ∈ Q)
35	11, 12, 23, 30, 32, 12, 34	caov4d 5705	. 2 ⊢ (𝜑 → ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
36	28, 35	breqtrd 3809	1 ⊢ (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))

Syntax hints:   → wi 4   ∧ wa 102   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  ⟨cop 3401   class class class wbr 3785  ‘cfv 4922  (class class class)co 5532  1^st c1st 5785  2^nd c2nd 5786  Qcnq 6470   +_Q cplq 6472   <_Q cltq 6475  Pcnp 6481

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-ltnqqs 6543  df-inp 6656

This theorem is referenced by:  addlocprlemeq  6723  addlocprlemgt  6724