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Theorem addlocprlem 7244

Description: Lemma for addlocpr 7245. The result, in deduction form. (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 ∈ (1^st ‘𝐴))
addlocprlem.uup	⊢ (𝜑 → 𝑈 ∈ (2^nd ‘𝐴))
addlocprlem.du	⊢ (𝜑 → 𝑈 <_Q (𝐷 +_Q 𝑃))
addlocprlem.elo	⊢ (𝜑 → 𝐸 ∈ (1^st ‘𝐵))
addlocprlem.tup	⊢ (𝜑 → 𝑇 ∈ (2^nd ‘𝐵))
addlocprlem.et	⊢ (𝜑 → 𝑇 <_Q (𝐸 +_Q 𝑃))

**Assertion**
Ref	Expression
addlocprlem	⊢ (𝜑 → (𝑄 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∨ 𝑅 ∈ (2^nd ‘(𝐴 +_P 𝐵))))

**Proof of Theorem addlocprlem**
Step	Hyp	Ref	Expression
1		addlocprlem.qr	. . . 4 ⊢ (𝜑 → 𝑄 <_Q 𝑅)
2		ltrelnq 7074	. . . . . 6 ⊢ <_Q ⊆ (Q × Q)
3	2	brel 4529	. . . . 5 ⊢ (𝑄 <_Q 𝑅 → (𝑄 ∈ Q ∧ 𝑅 ∈ Q))
4	3	simpld 111	. . . 4 ⊢ (𝑄 <_Q 𝑅 → 𝑄 ∈ Q)
5	1, 4	syl 14	. . 3 ⊢ (𝜑 → 𝑄 ∈ Q)
6		addlocprlem.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ P)
7		prop 7184	. . . . . 6 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
8	6, 7	syl 14	. . . . 5 ⊢ (𝜑 → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
9		addlocprlem.dlo	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ (1^st ‘𝐴))
10		elprnql 7190	. . . . 5 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝐷 ∈ (1^st ‘𝐴)) → 𝐷 ∈ Q)
11	8, 9, 10	syl2anc 406	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Q)
12		addlocprlem.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ P)
13		prop 7184	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
14	12, 13	syl 14	. . . . 5 ⊢ (𝜑 → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
15		addlocprlem.elo	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (1^st ‘𝐵))
16		elprnql 7190	. . . . 5 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝐸 ∈ (1^st ‘𝐵)) → 𝐸 ∈ Q)
17	14, 15, 16	syl2anc 406	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Q)
18		addclnq 7084	. . . 4 ⊢ ((𝐷 ∈ Q ∧ 𝐸 ∈ Q) → (𝐷 +_Q 𝐸) ∈ Q)
19	11, 17, 18	syl2anc 406	. . 3 ⊢ (𝜑 → (𝐷 +_Q 𝐸) ∈ Q)
20		nqtri3or 7105	. . 3 ⊢ ((𝑄 ∈ Q ∧ (𝐷 +_Q 𝐸) ∈ Q) → (𝑄 <_Q (𝐷 +_Q 𝐸) ∨ 𝑄 = (𝐷 +_Q 𝐸) ∨ (𝐷 +_Q 𝐸) <_Q 𝑄))
21	5, 19, 20	syl2anc 406	. 2 ⊢ (𝜑 → (𝑄 <_Q (𝐷 +_Q 𝐸) ∨ 𝑄 = (𝐷 +_Q 𝐸) ∨ (𝐷 +_Q 𝐸) <_Q 𝑄))
22		addlocprlem.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ Q)
23		addlocprlem.qppr	. . . . 5 ⊢ (𝜑 → (𝑄 +_Q (𝑃 +_Q 𝑃)) = 𝑅)
24		addlocprlem.uup	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (2^nd ‘𝐴))
25		addlocprlem.du	. . . . 5 ⊢ (𝜑 → 𝑈 <_Q (𝐷 +_Q 𝑃))
26		addlocprlem.tup	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (2^nd ‘𝐵))
27		addlocprlem.et	. . . . 5 ⊢ (𝜑 → 𝑇 <_Q (𝐸 +_Q 𝑃))
28	6, 12, 1, 22, 23, 9, 24, 25, 15, 26, 27	addlocprlemlt 7240	. . . 4 ⊢ (𝜑 → (𝑄 <_Q (𝐷 +_Q 𝐸) → 𝑄 ∈ (1^st ‘(𝐴 +_P 𝐵))))
29		orc 674	. . . 4 ⊢ (𝑄 ∈ (1^st ‘(𝐴 +_P 𝐵)) → (𝑄 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∨ 𝑅 ∈ (2^nd ‘(𝐴 +_P 𝐵))))
30	28, 29	syl6 33	. . 3 ⊢ (𝜑 → (𝑄 <_Q (𝐷 +_Q 𝐸) → (𝑄 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∨ 𝑅 ∈ (2^nd ‘(𝐴 +_P 𝐵)))))
31	6, 12, 1, 22, 23, 9, 24, 25, 15, 26, 27	addlocprlemeq 7242	. . . 4 ⊢ (𝜑 → (𝑄 = (𝐷 +_Q 𝐸) → 𝑅 ∈ (2^nd ‘(𝐴 +_P 𝐵))))
32		olc 673	. . . 4 ⊢ (𝑅 ∈ (2^nd ‘(𝐴 +_P 𝐵)) → (𝑄 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∨ 𝑅 ∈ (2^nd ‘(𝐴 +_P 𝐵))))
33	31, 32	syl6 33	. . 3 ⊢ (𝜑 → (𝑄 = (𝐷 +_Q 𝐸) → (𝑄 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∨ 𝑅 ∈ (2^nd ‘(𝐴 +_P 𝐵)))))
34	6, 12, 1, 22, 23, 9, 24, 25, 15, 26, 27	addlocprlemgt 7243	. . . 4 ⊢ (𝜑 → ((𝐷 +_Q 𝐸) <_Q 𝑄 → 𝑅 ∈ (2^nd ‘(𝐴 +_P 𝐵))))
35	34, 32	syl6 33	. . 3 ⊢ (𝜑 → ((𝐷 +_Q 𝐸) <_Q 𝑄 → (𝑄 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∨ 𝑅 ∈ (2^nd ‘(𝐴 +_P 𝐵)))))
36	30, 33, 35	3jaod 1250	. 2 ⊢ (𝜑 → ((𝑄 <_Q (𝐷 +_Q 𝐸) ∨ 𝑄 = (𝐷 +_Q 𝐸) ∨ (𝐷 +_Q 𝐸) <_Q 𝑄) → (𝑄 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∨ 𝑅 ∈ (2^nd ‘(𝐴 +_P 𝐵)))))
37	21, 36	mpd 13	1 ⊢ (𝜑 → (𝑄 ∈ (1^st ‘(𝐴 +_P 𝐵)) ∨ 𝑅 ∈ (2^nd ‘(𝐴 +_P 𝐵))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∨ wo 670 ∨ w3o 929 = wceq 1299 ∈ wcel 1448 ⟨cop 3477 class class class wbr 3875 ‘cfv 5059 (class class class)co 5706 1^st c1st 5967 2^nd c2nd 5968 Qcnq 6989 +_Q cplq 6991 <_Q cltq 6994 Pcnp 7000 +_P cpp 7002

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440

This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-eprel 4149 df-id 4153 df-po 4156 df-iso 4157 df-iord 4226 df-on 4228 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-irdg 6197 df-1o 6243 df-oadd 6247 df-omul 6248 df-er 6359 df-ec 6361 df-qs 6365 df-ni 7013 df-pli 7014 df-mi 7015 df-lti 7016 df-plpq 7053 df-mpq 7054 df-enq 7056 df-nqqs 7057 df-plqqs 7058 df-mqqs 7059 df-1nqqs 7060 df-rq 7061 df-ltnqqs 7062 df-inp 7175 df-iplp 7177

This theorem is referenced by: addlocpr 7245

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