Theorem addlocprlemgt 7797

Description: Lemma for addlocpr 7799. 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 7795	. . . . . 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 7738	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
15	1, 14	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
16		elprnql 7744	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝐷 ∈ (1st ‘𝐴)) → 𝐷 ∈ Q)
17	15, 6, 16	syl2anc 411	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ Q)
18		prop 7738	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
19	2, 18	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
20		elprnql 7744	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐸 ∈ (1st ‘𝐵)) → 𝐸 ∈ Q)
21	19, 9, 20	syl2anc 411	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Q)
22		addclnq 7638	. . . . . . . . . 10 ⊢ ((𝐷 ∈ Q ∧ 𝐸 ∈ Q) → (𝐷 +Q 𝐸) ∈ Q)
23	17, 21, 22	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → (𝐷 +Q 𝐸) ∈ Q)
24		ltrelnq 7628	. . . . . . . . . . . 12 ⊢ <Q ⊆ (Q × Q)
25	24	brel 4784	. . . . . . . . . . 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 7638	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Q ∧ 𝑃 ∈ Q) → (𝑃 +Q 𝑃) ∈ Q)
29	4, 4, 28	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 +Q 𝑃) ∈ Q)
30		ltanqg 7663	. . . . . . . . 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 1274	. . . . . . . 8 ⊢ (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄 ↔ ((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) <Q ((𝑃 +Q 𝑃) +Q 𝑄)))
32		addcomnqg 7644	. . . . . . . . . 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 7644	. . . . . . . . . 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 4106	. . . . . . . 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 4105	. . . . . . 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 7661	. . . . 5 ⊢ <Q Or Q
44	43, 24	sotri 5139	. . . 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 7793	. . . . 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 1273	. . . 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 1398   ∈ wcel 2202  ⟨cop 3676   class class class wbr 4093  ‘cfv 5333  (class class class)co 6028  1^st c1st 6310  2^nd c2nd 6311  Qcnq 7543   +_Q cplq 7545   <_Q cltq 7548  Pcnp 7554   +_P cpp 7556

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-pli 7568  df-mi 7569  df-lti 7570  df-plpq 7607  df-mpq 7608  df-enq 7610  df-nqqs 7611  df-plqqs 7612  df-mqqs 7613  df-1nqqs 7614  df-rq 7615  df-ltnqqs 7616  df-inp 7729  df-iplp 7731

This theorem is referenced by:  addlocprlem  7798