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Theorem addlocprlem 7476

Description: Lemma for addlocpr 7477. 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 7306	. . . . . 6 ⊢ <_Q ⊆ (Q × Q)
3	2	brel 4656	. . . . 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 7416	. . . . . 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 7422	. . . . 5 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝐷 ∈ (1^st ‘𝐴)) → 𝐷 ∈ Q)
11	8, 9, 10	syl2anc 409	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Q)
12		addlocprlem.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ P)
13		prop 7416	. . . . . 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 7422	. . . . 5 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝐸 ∈ (1^st ‘𝐵)) → 𝐸 ∈ Q)
17	14, 15, 16	syl2anc 409	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Q)
18		addclnq 7316	. . . 4 ⊢ ((𝐷 ∈ Q ∧ 𝐸 ∈ Q) → (𝐷 +_Q 𝐸) ∈ Q)
19	11, 17, 18	syl2anc 409	. . 3 ⊢ (𝜑 → (𝐷 +_Q 𝐸) ∈ Q)
20		nqtri3or 7337	. . 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 409	. 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 7472	. . . 4 ⊢ (𝜑 → (𝑄 <_Q (𝐷 +_Q 𝐸) → 𝑄 ∈ (1^st ‘(𝐴 +_P 𝐵))))
29		orc 702	. . . 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 7474	. . . 4 ⊢ (𝜑 → (𝑄 = (𝐷 +_Q 𝐸) → 𝑅 ∈ (2^nd ‘(𝐴 +_P 𝐵))))
32		olc 701	. . . 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 7475	. . . 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 1294	. 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 698 ∨ w3o 967 = 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: addlocpr 7477

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