Theorem addlocprlemgt 7475

Description: Lemma for addlocpr 7477. 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 7473	. . . . . 6 ⊢ (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
13	12	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
14		prop 7416	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
15	1, 14	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
16		elprnql 7422	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝐷 ∈ (1st ‘𝐴)) → 𝐷 ∈ Q)
17	15, 6, 16	syl2anc 409	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ Q)
18		prop 7416	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
19	2, 18	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
20		elprnql 7422	. . . . . . . . . . 11 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐸 ∈ (1st ‘𝐵)) → 𝐸 ∈ Q)
21	19, 9, 20	syl2anc 409	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ Q)
22		addclnq 7316	. . . . . . . . . 10 ⊢ ((𝐷 ∈ Q ∧ 𝐸 ∈ Q) → (𝐷 +Q 𝐸) ∈ Q)
23	17, 21, 22	syl2anc 409	. . . . . . . . 9 ⊢ (𝜑 → (𝐷 +Q 𝐸) ∈ Q)
24		ltrelnq 7306	. . . . . . . . . . . 12 ⊢ <Q ⊆ (Q × Q)
25	24	brel 4656	. . . . . . . . . . 11 ⊢ (𝑄 <Q 𝑅 → (𝑄 ∈ Q ∧ 𝑅 ∈ Q))
26	3, 25	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → (𝑄 ∈ Q ∧ 𝑅 ∈ Q))
27	26	simpld 111	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ Q)
28		addclnq 7316	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Q ∧ 𝑃 ∈ Q) → (𝑃 +Q 𝑃) ∈ Q)
29	4, 4, 28	syl2anc 409	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 +Q 𝑃) ∈ Q)
30		ltanqg 7341	. . . . . . . . 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 1228	. . . . . . . 8 ⊢ (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄 ↔ ((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) <Q ((𝑃 +Q 𝑃) +Q 𝑄)))
32		addcomnqg 7322	. . . . . . . . . 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 409	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
34		addcomnqg 7322	. . . . . . . . . 10 ⊢ (((𝑃 +Q 𝑃) ∈ Q ∧ 𝑄 ∈ Q) → ((𝑃 +Q 𝑃) +Q 𝑄) = (𝑄 +Q (𝑃 +Q 𝑃)))
35	29, 27, 34	syl2anc 409	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 +Q 𝑃) +Q 𝑄) = (𝑄 +Q (𝑃 +Q 𝑃)))
36	33, 35	breq12d 3995	. . . . . . . 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 187	. . . . . . 7 ⊢ (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄 ↔ ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q (𝑄 +Q (𝑃 +Q 𝑃))))
38	37	biimpa 294	. . . . . 6 ⊢ ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q (𝑄 +Q (𝑃 +Q 𝑃)))
39	5	breq2d 3994	. . . . . . 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 274	. . . . . 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 146	. . . . 5 ⊢ ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q 𝑅)
42	13, 41	jca 304	. . . 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 7339	. . . . 5 ⊢ <Q Or Q
44	43, 24	sotri 4999	. . . 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 304	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ P ∧ 𝑈 ∈ (2nd ‘𝐴)))
47	2, 10	jca 304	. . . . 5 ⊢ (𝜑 → (𝐵 ∈ P ∧ 𝑇 ∈ (2nd ‘𝐵)))
48	26	simprd 113	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Q)
49		addnqpru 7471	. . . . 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 1227	. . . 4 ⊢ (𝜑 → ((𝑈 +Q 𝑇) <Q 𝑅 → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
51	50	adantr 274	. . 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 114	1 ⊢ (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄 → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  ⟨cop 3579   class class class wbr 3982  ‘cfv 5188  (class class class)co 5842  1^st c1st 6106  2^nd c2nd 6107  Qcnq 7221   +_Q cplq 7223   <_Q cltq 7226  Pcnp 7232   +_P cpp 7234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-inp 7407  df-iplp 7409

This theorem is referenced by:  addlocprlem  7476