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Theorem addlocprlem 7336

Description: Lemma for addlocpr 7337. 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 7166	. . . . . 6 ⊢ <_Q ⊆ (Q × Q)
3	2	brel 4586	. . . . 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 7276	. . . . . 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 7282	. . . . 5 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝐷 ∈ (1^st ‘𝐴)) → 𝐷 ∈ Q)
11	8, 9, 10	syl2anc 408	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Q)
12		addlocprlem.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ P)
13		prop 7276	. . . . . 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 7282	. . . . 5 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝐸 ∈ (1^st ‘𝐵)) → 𝐸 ∈ Q)
17	14, 15, 16	syl2anc 408	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Q)
18		addclnq 7176	. . . 4 ⊢ ((𝐷 ∈ Q ∧ 𝐸 ∈ Q) → (𝐷 +_Q 𝐸) ∈ Q)
19	11, 17, 18	syl2anc 408	. . 3 ⊢ (𝜑 → (𝐷 +_Q 𝐸) ∈ Q)
20		nqtri3or 7197	. . 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 408	. 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 7332	. . . 4 ⊢ (𝜑 → (𝑄 <_Q (𝐷 +_Q 𝐸) → 𝑄 ∈ (1^st ‘(𝐴 +_P 𝐵))))
29		orc 701	. . . 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 7334	. . . 4 ⊢ (𝜑 → (𝑄 = (𝐷 +_Q 𝐸) → 𝑅 ∈ (2^nd ‘(𝐴 +_P 𝐵))))
32		olc 700	. . . 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 7335	. . . 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 1282	. 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 697 ∨ w3o 961 = wceq 1331 ∈ wcel 1480 ⟨cop 3525 class class class wbr 3924 ‘cfv 5118 (class class class)co 5767 1^st c1st 6029 2^nd c2nd 6030 Qcnq 7081 +_Q cplq 7083 <_Q cltq 7086 Pcnp 7092 +_P cpp 7094

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-eprel 4206 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-1o 6306 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-pli 7106 df-mi 7107 df-lti 7108 df-plpq 7145 df-mpq 7146 df-enq 7148 df-nqqs 7149 df-plqqs 7150 df-mqqs 7151 df-1nqqs 7152 df-rq 7153 df-ltnqqs 7154 df-inp 7267 df-iplp 7269

This theorem is referenced by: addlocpr 7337

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