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Theorem addlocprlem 7846

Description: Lemma for addlocpr 7847. 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 7676	. . . . . 6 ⊢ <_Q ⊆ (Q × Q)
3	2	brel 4801	. . . . 5 ⊢ (𝑄 <_Q 𝑅 → (𝑄 ∈ Q ∧ 𝑅 ∈ Q))
4	3	simpld 112	. . . 4 ⊢ (𝑄 <_Q 𝑅 → 𝑄 ∈ Q)
5	1, 4	syl 14	. . 3 ⊢ (𝜑 → 𝑄 ∈ Q)
6		addlocprlem.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ P)
7		prop 7786	. . . . . 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 7792	. . . . 5 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝐷 ∈ (1^st ‘𝐴)) → 𝐷 ∈ Q)
11	8, 9, 10	syl2anc 411	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Q)
12		addlocprlem.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ P)
13		prop 7786	. . . . . 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 7792	. . . . 5 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝐸 ∈ (1^st ‘𝐵)) → 𝐸 ∈ Q)
17	14, 15, 16	syl2anc 411	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Q)
18		addclnq 7686	. . . 4 ⊢ ((𝐷 ∈ Q ∧ 𝐸 ∈ Q) → (𝐷 +_Q 𝐸) ∈ Q)
19	11, 17, 18	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐷 +_Q 𝐸) ∈ Q)
20		nqtri3or 7707	. . 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 411	. 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 7842	. . . 4 ⊢ (𝜑 → (𝑄 <_Q (𝐷 +_Q 𝐸) → 𝑄 ∈ (1^st ‘(𝐴 +_P 𝐵))))
29		orc 720	. . . 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 7844	. . . 4 ⊢ (𝜑 → (𝑄 = (𝐷 +_Q 𝐸) → 𝑅 ∈ (2^nd ‘(𝐴 +_P 𝐵))))
32		olc 719	. . . 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 7845	. . . 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 1341	. 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 716 ∨ w3o 1004 = wceq 1398 ∈ wcel 2203 ⟨cop 3691 class class class wbr 4108 ‘cfv 5351 (class class class)co 6049 1^st c1st 6331 2^nd c2nd 6332 Qcnq 7591 +_Q cplq 7593 <_Q cltq 7596 Pcnp 7602 +_P cpp 7604

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-nul 4235 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-iinf 4709

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-ral 2525 df-rex 2526 df-reu 2527 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-nul 3508 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-br 4109 df-opab 4171 df-mpt 4172 df-tr 4208 df-eprel 4409 df-id 4413 df-po 4416 df-iso 4417 df-iord 4486 df-on 4488 df-suc 4491 df-iom 4712 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-ov 6052 df-oprab 6053 df-mpo 6054 df-1st 6333 df-2nd 6334 df-recs 6535 df-irdg 6600 df-1o 6646 df-oadd 6650 df-omul 6651 df-er 6766 df-ec 6768 df-qs 6772 df-ni 7615 df-pli 7616 df-mi 7617 df-lti 7618 df-plpq 7655 df-mpq 7656 df-enq 7658 df-nqqs 7659 df-plqqs 7660 df-mqqs 7661 df-1nqqs 7662 df-rq 7663 df-ltnqqs 7664 df-inp 7777 df-iplp 7779

This theorem is referenced by: addlocpr 7847

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