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Mirrors > Home > ILE Home > Th. List > addlocprlem | GIF version |
Description: Lemma for addlocpr 6634. The result, in deduction form. (Contributed by Jim Kingdon, 6-Dec-2019.) |
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 ∈ (1st ‘𝐴)) |
addlocprlem.uup | ⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴)) |
addlocprlem.du | ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃)) |
addlocprlem.elo | ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵)) |
addlocprlem.tup | ⊢ (𝜑 → 𝑇 ∈ (2nd ‘𝐵)) |
addlocprlem.et | ⊢ (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃)) |
Ref | Expression |
---|---|
addlocprlem | ⊢ (𝜑 → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addlocprlem.qr | . . . 4 ⊢ (𝜑 → 𝑄 <Q 𝑅) | |
2 | ltrelnq 6463 | . . . . . 6 ⊢ <Q ⊆ (Q × Q) | |
3 | 2 | brel 4392 | . . . . 5 ⊢ (𝑄 <Q 𝑅 → (𝑄 ∈ Q ∧ 𝑅 ∈ Q)) |
4 | 3 | simpld 105 | . . . 4 ⊢ (𝑄 <Q 𝑅 → 𝑄 ∈ Q) |
5 | 1, 4 | syl 14 | . . 3 ⊢ (𝜑 → 𝑄 ∈ Q) |
6 | addlocprlem.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ P) | |
7 | prop 6573 | . . . . . 6 ⊢ (𝐴 ∈ P → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ P) | |
8 | 6, 7 | syl 14 | . . . . 5 ⊢ (𝜑 → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ P) |
9 | addlocprlem.dlo | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ (1st ‘𝐴)) | |
10 | elprnql 6579 | . . . . 5 ⊢ ((〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ P ∧ 𝐷 ∈ (1st ‘𝐴)) → 𝐷 ∈ Q) | |
11 | 8, 9, 10 | syl2anc 391 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ Q) |
12 | addlocprlem.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ P) | |
13 | prop 6573 | . . . . . 6 ⊢ (𝐵 ∈ P → 〈(1st ‘𝐵), (2nd ‘𝐵)〉 ∈ P) | |
14 | 12, 13 | syl 14 | . . . . 5 ⊢ (𝜑 → 〈(1st ‘𝐵), (2nd ‘𝐵)〉 ∈ P) |
15 | addlocprlem.elo | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵)) | |
16 | elprnql 6579 | . . . . 5 ⊢ ((〈(1st ‘𝐵), (2nd ‘𝐵)〉 ∈ P ∧ 𝐸 ∈ (1st ‘𝐵)) → 𝐸 ∈ Q) | |
17 | 14, 15, 16 | syl2anc 391 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ Q) |
18 | addclnq 6473 | . . . 4 ⊢ ((𝐷 ∈ Q ∧ 𝐸 ∈ Q) → (𝐷 +Q 𝐸) ∈ Q) | |
19 | 11, 17, 18 | syl2anc 391 | . . 3 ⊢ (𝜑 → (𝐷 +Q 𝐸) ∈ Q) |
20 | nqtri3or 6494 | . . 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 391 | . 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 ∈ (2nd ‘𝐴)) | |
25 | addlocprlem.du | . . . . 5 ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃)) | |
26 | addlocprlem.tup | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (2nd ‘𝐵)) | |
27 | addlocprlem.et | . . . . 5 ⊢ (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃)) | |
28 | 6, 12, 1, 22, 23, 9, 24, 25, 15, 26, 27 | addlocprlemlt 6629 | . . . 4 ⊢ (𝜑 → (𝑄 <Q (𝐷 +Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 +P 𝐵)))) |
29 | orc 633 | . . . 4 ⊢ (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))) | |
30 | 28, 29 | syl6 29 | . . 3 ⊢ (𝜑 → (𝑄 <Q (𝐷 +Q 𝐸) → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))) |
31 | 6, 12, 1, 22, 23, 9, 24, 25, 15, 26, 27 | addlocprlemeq 6631 | . . . 4 ⊢ (𝜑 → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))) |
32 | olc 632 | . . . 4 ⊢ (𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)) → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))) | |
33 | 31, 32 | syl6 29 | . . 3 ⊢ (𝜑 → (𝑄 = (𝐷 +Q 𝐸) → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))) |
34 | 6, 12, 1, 22, 23, 9, 24, 25, 15, 26, 27 | addlocprlemgt 6632 | . . . 4 ⊢ (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄 → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))) |
35 | 34, 32 | syl6 29 | . . 3 ⊢ (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄 → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))) |
36 | 30, 33, 35 | 3jaod 1199 | . 2 ⊢ (𝜑 → ((𝑄 <Q (𝐷 +Q 𝐸) ∨ 𝑄 = (𝐷 +Q 𝐸) ∨ (𝐷 +Q 𝐸) <Q 𝑄) → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))) |
37 | 21, 36 | mpd 13 | 1 ⊢ (𝜑 → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 629 ∨ w3o 884 = wceq 1243 ∈ wcel 1393 〈cop 3378 class class class wbr 3764 ‘cfv 4902 (class class class)co 5512 1st c1st 5765 2nd c2nd 5766 Qcnq 6378 +Q cplq 6380 <Q cltq 6383 Pcnp 6389 +P cpp 6391 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-inp 6564 df-iplp 6566 |
This theorem is referenced by: addlocpr 6634 |
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