Theorem addlocprlemeqgt 7742

Description: Lemma for addlocpr 7746. 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 7685	. . . . . 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 7692	. . . . 5 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑈 ∈ (2nd ‘𝐴)) → 𝑈 ∈ Q)
8	5, 6, 7	syl2anc 411	. . . 4 ⊢ (𝜑 → 𝑈 ∈ Q)
9		addlocprlem.dlo	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (1st ‘𝐴))
10		elprnql 7691	. . . . . 6 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝐷 ∈ (1st ‘𝐴)) → 𝐷 ∈ Q)
11	5, 9, 10	syl2anc 411	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Q)
12		addlocprlem.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ Q)
13		addclnq 7585	. . . . 5 ⊢ ((𝐷 ∈ Q ∧ 𝑃 ∈ Q) → (𝐷 +Q 𝑃) ∈ Q)
14	11, 12, 13	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝐷 +Q 𝑃) ∈ Q)
15		addlocprlem.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ P)
16		prop 7685	. . . . . 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 7692	. . . . 5 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑇 ∈ (2nd ‘𝐵)) → 𝑇 ∈ Q)
20	17, 18, 19	syl2anc 411	. . . 4 ⊢ (𝜑 → 𝑇 ∈ Q)
21		addlocprlem.elo	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵))
22		elprnql 7691	. . . . . 6 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐸 ∈ (1st ‘𝐵)) → 𝐸 ∈ Q)
23	17, 21, 22	syl2anc 411	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Q)
24		addclnq 7585	. . . . 5 ⊢ ((𝐸 ∈ Q ∧ 𝑃 ∈ Q) → (𝐸 +Q 𝑃) ∈ Q)
25	23, 12, 24	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝐸 +Q 𝑃) ∈ Q)
26		lt2addnq 7614	. . . 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 1272	. . 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 433	. 2 ⊢ (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃)))
29		addcomnqg 7591	. . . 4 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
30	29	adantl 277	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
31		addassnqg 7592	. . . 4 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
32	31	adantl 277	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
33		addclnq 7585	. . . 4 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) ∈ Q)
34	33	adantl 277	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) ∈ Q)
35	11, 12, 23, 30, 32, 12, 34	caov4d 6202	. 2 ⊢ (𝜑 → ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
36	28, 35	breqtrd 4112	1 ⊢ (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ⟨cop 3670   class class class wbr 4086  ‘cfv 5324  (class class class)co 6013  1^st c1st 6296  2^nd c2nd 6297  Qcnq 7490   +_Q cplq 7492   <_Q cltq 7495  Pcnp 7501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-ltnqqs 7563  df-inp 7676

This theorem is referenced by:  addlocprlemeq  7743  addlocprlemgt  7744