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Theorem addlocprlem 7595

Description: Lemma for addlocpr 7596. 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 7425	. . . . . 6 ⊢ <_Q ⊆ (Q × Q)
3	2	brel 4711	. . . . 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 7535	. . . . . 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 7541	. . . . 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 7535	. . . . . 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 7541	. . . . 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 7435	. . . 4 ⊢ ((𝐷 ∈ Q ∧ 𝐸 ∈ Q) → (𝐷 +_Q 𝐸) ∈ Q)
19	11, 17, 18	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐷 +_Q 𝐸) ∈ Q)
20		nqtri3or 7456	. . 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 7591	. . . 4 ⊢ (𝜑 → (𝑄 <_Q (𝐷 +_Q 𝐸) → 𝑄 ∈ (1^st ‘(𝐴 +_P 𝐵))))
29		orc 713	. . . 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 7593	. . . 4 ⊢ (𝜑 → (𝑄 = (𝐷 +_Q 𝐸) → 𝑅 ∈ (2^nd ‘(𝐴 +_P 𝐵))))
32		olc 712	. . . 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 7594	. . . 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 1315	. 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 709 ∨ w3o 979 = wceq 1364 ∈ wcel 2164 ⟨cop 3621 class class class wbr 4029 ‘cfv 5254 (class class class)co 5918 1^st c1st 6191 2^nd c2nd 6192 Qcnq 7340 +_Q cplq 7342 <_Q cltq 7345 Pcnp 7351 +_P cpp 7353

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-eprel 4320 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-1o 6469 df-oadd 6473 df-omul 6474 df-er 6587 df-ec 6589 df-qs 6593 df-ni 7364 df-pli 7365 df-mi 7366 df-lti 7367 df-plpq 7404 df-mpq 7405 df-enq 7407 df-nqqs 7408 df-plqqs 7409 df-mqqs 7410 df-1nqqs 7411 df-rq 7412 df-ltnqqs 7413 df-inp 7526 df-iplp 7528

This theorem is referenced by: addlocpr 7596

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