Theorem addlocprlemgt 7895

Description: Lemma for addlocpr 7897. 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 7893	. . . . . 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 7836	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
15	1, 14	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
16		elprnql 7842	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝐷 ∈ (1st ‘𝐴)) → 𝐷 ∈ Q)
17	15, 6, 16	syl2anc 415	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ Q)
18		prop 7836	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
19	2, 18	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
20		elprnql 7842	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐸 ∈ (1st ‘𝐵)) → 𝐸 ∈ Q)
21	19, 9, 20	syl2anc 415	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Q)
22		addclnq 7736	. . . . . . . . . 10 ⊢ ((𝐷 ∈ Q ∧ 𝐸 ∈ Q) → (𝐷 +Q 𝐸) ∈ Q)
23	17, 21, 22	syl2anc 415	. . . . . . . . 9 ⊢ (𝜑 → (𝐷 +Q 𝐸) ∈ Q)
24		ltrelnq 7726	. . . . . . . . . . . 12 ⊢ <Q ⊆ (Q × Q)
25	24	brel 4825	. . . . . . . . . . 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 7736	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Q ∧ 𝑃 ∈ Q) → (𝑃 +Q 𝑃) ∈ Q)
29	4, 4, 28	syl2anc 415	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 +Q 𝑃) ∈ Q)
30		ltanqg 7761	. . . . . . . . 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 1278	. . . . . . . 8 ⊢ (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄 ↔ ((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) <Q ((𝑃 +Q 𝑃) +Q 𝑄)))
32		addcomnqg 7742	. . . . . . . . . 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 415	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
34		addcomnqg 7742	. . . . . . . . . 10 ⊢ (((𝑃 +Q 𝑃) ∈ Q ∧ 𝑄 ∈ Q) → ((𝑃 +Q 𝑃) +Q 𝑄) = (𝑄 +Q (𝑃 +Q 𝑃)))
35	29, 27, 34	syl2anc 415	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 +Q 𝑃) +Q 𝑄) = (𝑄 +Q (𝑃 +Q 𝑃)))
36	33, 35	breq12d 4141	. . . . . . . 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 4140	. . . . . . 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 7759	. . . . 5 ⊢ <Q Or Q
44	43, 24	sotri 5181	. . . 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 7891	. . . . 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 1277	. . . 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 1402   ∈ wcel 2209  ⟨cop 3711   class class class wbr 4128  ‘cfv 5375  (class class class)co 6079  1^st c1st 6366  2^nd c2nd 6367  Qcnq 7641   +_Q cplq 7643   <_Q cltq 7646  Pcnp 7652   +_P cpp 7654

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-eprel 4432  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-1o 6681  df-oadd 6685  df-omul 6686  df-er 6801  df-ec 6803  df-qs 6807  df-ni 7665  df-pli 7666  df-mi 7667  df-lti 7668  df-plpq 7705  df-mpq 7706  df-enq 7708  df-nqqs 7709  df-plqqs 7710  df-mqqs 7711  df-1nqqs 7712  df-rq 7713  df-ltnqqs 7714  df-inp 7827  df-iplp 7829

This theorem is referenced by:  addlocprlem  7896