Theorem addlocprlemgt 7709

Description: Lemma for addlocpr 7711. The (𝐷 +_Q 𝐸) <_Q 𝑄 case. (Contributed by Jim Kingdon, 6-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
addlocprlemgt	⊢ (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄 → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))

Proof of Theorem addlocprlemgt

Step	Hyp	Ref	Expression
1		addlocprlem.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ P)
2		addlocprlem.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ P)
3		addlocprlem.qr	. . . . . . 7 ⊢ (𝜑 → 𝑄 <Q 𝑅)
4		addlocprlem.p	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ Q)
5		addlocprlem.qppr	. . . . . . 7 ⊢ (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)
6		addlocprlem.dlo	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (1st ‘𝐴))
7		addlocprlem.uup	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴))
8		addlocprlem.du	. . . . . . 7 ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))
9		addlocprlem.elo	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵))
10		addlocprlem.tup	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (2nd ‘𝐵))
11		addlocprlem.et	. . . . . . 7 ⊢ (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	addlocprlemeqgt 7707	. . . . . 6 ⊢ (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
13	12	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
14		prop 7650	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
15	1, 14	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
16		elprnql 7656	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝐷 ∈ (1st ‘𝐴)) → 𝐷 ∈ Q)
17	15, 6, 16	syl2anc 411	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ Q)
18		prop 7650	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
19	2, 18	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
20		elprnql 7656	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐸 ∈ (1st ‘𝐵)) → 𝐸 ∈ Q)
21	19, 9, 20	syl2anc 411	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Q)
22		addclnq 7550	. . . . . . . . . 10 ⊢ ((𝐷 ∈ Q ∧ 𝐸 ∈ Q) → (𝐷 +Q 𝐸) ∈ Q)
23	17, 21, 22	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → (𝐷 +Q 𝐸) ∈ Q)
24		ltrelnq 7540	. . . . . . . . . . . 12 ⊢ <Q ⊆ (Q × Q)
25	24	brel 4768	. . . . . . . . . . 11 ⊢ (𝑄 <Q 𝑅 → (𝑄 ∈ Q ∧ 𝑅 ∈ Q))
26	3, 25	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄 ∈ Q ∧ 𝑅 ∈ Q))
27	26	simpld 112	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ Q)
28		addclnq 7550	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Q ∧ 𝑃 ∈ Q) → (𝑃 +Q 𝑃) ∈ Q)
29	4, 4, 28	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 +Q 𝑃) ∈ Q)
30		ltanqg 7575	. . . . . . . . 9 ⊢ (((𝐷 +Q 𝐸) ∈ Q ∧ 𝑄 ∈ Q ∧ (𝑃 +Q 𝑃) ∈ Q) → ((𝐷 +Q 𝐸) <Q 𝑄 ↔ ((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) <Q ((𝑃 +Q 𝑃) +Q 𝑄)))
31	23, 27, 29, 30	syl3anc 1271	. . . . . . . 8 ⊢ (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄 ↔ ((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) <Q ((𝑃 +Q 𝑃) +Q 𝑄)))
32		addcomnqg 7556	. . . . . . . . . 10 ⊢ (((𝑃 +Q 𝑃) ∈ Q ∧ (𝐷 +Q 𝐸) ∈ Q) → ((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
33	29, 23, 32	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
34		addcomnqg 7556	. . . . . . . . . 10 ⊢ (((𝑃 +Q 𝑃) ∈ Q ∧ 𝑄 ∈ Q) → ((𝑃 +Q 𝑃) +Q 𝑄) = (𝑄 +Q (𝑃 +Q 𝑃)))
35	29, 27, 34	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 +Q 𝑃) +Q 𝑄) = (𝑄 +Q (𝑃 +Q 𝑃)))
36	33, 35	breq12d 4095	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) <Q ((𝑃 +Q 𝑃) +Q 𝑄) ↔ ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q (𝑄 +Q (𝑃 +Q 𝑃))))
37	31, 36	bitrd 188	. . . . . . 7 ⊢ (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄 ↔ ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q (𝑄 +Q (𝑃 +Q 𝑃))))
38	37	biimpa 296	. . . . . 6 ⊢ ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q (𝑄 +Q (𝑃 +Q 𝑃)))
39	5	breq2d 4094	. . . . . . 7 ⊢ (𝜑 → (((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q (𝑄 +Q (𝑃 +Q 𝑃)) ↔ ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q 𝑅))
40	39	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → (((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q (𝑄 +Q (𝑃 +Q 𝑃)) ↔ ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q 𝑅))
41	38, 40	mpbid 147	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q 𝑅)
42	13, 41	jca 306	. . . 4 ⊢ ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → ((𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) ∧ ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q 𝑅))
43		ltsonq 7573	. . . . 5 ⊢ <Q Or Q
44	43, 24	sotri 5120	. . . 4 ⊢ (((𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) ∧ ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q 𝑅) → (𝑈 +Q 𝑇) <Q 𝑅)
45	42, 44	syl 14	. . 3 ⊢ ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → (𝑈 +Q 𝑇) <Q 𝑅)
46	1, 7	jca 306	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ P ∧ 𝑈 ∈ (2nd ‘𝐴)))
47	2, 10	jca 306	. . . . 5 ⊢ (𝜑 → (𝐵 ∈ P ∧ 𝑇 ∈ (2nd ‘𝐵)))
48	26	simprd 114	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Q)
49		addnqpru 7705	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝑈 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝑇 ∈ (2nd ‘𝐵))) ∧ 𝑅 ∈ Q) → ((𝑈 +Q 𝑇) <Q 𝑅 → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
50	46, 47, 48, 49	syl21anc 1270	. . . 4 ⊢ (𝜑 → ((𝑈 +Q 𝑇) <Q 𝑅 → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
51	50	adantr 276	. . 3 ⊢ ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → ((𝑈 +Q 𝑇) <Q 𝑅 → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
52	45, 51	mpd 13	. 2 ⊢ ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))
53	52	ex 115	1 ⊢ (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄 → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ⟨cop 3669   class class class wbr 4082  ‘cfv 5314  (class class class)co 5994  1^st c1st 6274  2^nd c2nd 6275  Qcnq 7455   +_Q cplq 7457   <_Q cltq 7460  Pcnp 7466   +_P cpp 7468

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4377  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-irdg 6506  df-1o 6552  df-oadd 6556  df-omul 6557  df-er 6670  df-ec 6672  df-qs 6676  df-ni 7479  df-pli 7480  df-mi 7481  df-lti 7482  df-plpq 7519  df-mpq 7520  df-enq 7522  df-nqqs 7523  df-plqqs 7524  df-mqqs 7525  df-1nqqs 7526  df-rq 7527  df-ltnqqs 7528  df-inp 7641  df-iplp 7643

This theorem is referenced by:  addlocprlem  7710