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Mirrors > Home > ILE Home > Th. List > addlocprlemeq | GIF version |
Description: Lemma for addlocpr 7344. The 𝑄 = (𝐷 +Q 𝐸) case. (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 |
---|---|
addlocprlemeq | ⊢ (𝜑 → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addlocprlem.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ P) | |
2 | addlocprlem.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ P) | |
3 | addlocprlem.qr | . . . . . 6 ⊢ (𝜑 → 𝑄 <Q 𝑅) | |
4 | addlocprlem.p | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Q) | |
5 | addlocprlem.qppr | . . . . . 6 ⊢ (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅) | |
6 | addlocprlem.dlo | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (1st ‘𝐴)) | |
7 | addlocprlem.uup | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴)) | |
8 | addlocprlem.du | . . . . . 6 ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃)) | |
9 | addlocprlem.elo | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵)) | |
10 | addlocprlem.tup | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ (2nd ‘𝐵)) | |
11 | addlocprlem.et | . . . . . 6 ⊢ (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃)) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | addlocprlemeqgt 7340 | . . . . 5 ⊢ (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃))) |
13 | 12 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝑄 = (𝐷 +Q 𝐸)) → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃))) |
14 | oveq1 5781 | . . . . 5 ⊢ (𝑄 = (𝐷 +Q 𝐸) → (𝑄 +Q (𝑃 +Q 𝑃)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃))) | |
15 | 5, 14 | sylan9req 2193 | . . . 4 ⊢ ((𝜑 ∧ 𝑄 = (𝐷 +Q 𝐸)) → 𝑅 = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃))) |
16 | 13, 15 | breqtrrd 3956 | . . 3 ⊢ ((𝜑 ∧ 𝑄 = (𝐷 +Q 𝐸)) → (𝑈 +Q 𝑇) <Q 𝑅) |
17 | 1, 7 | jca 304 | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ P ∧ 𝑈 ∈ (2nd ‘𝐴))) |
18 | 2, 10 | jca 304 | . . . . 5 ⊢ (𝜑 → (𝐵 ∈ P ∧ 𝑇 ∈ (2nd ‘𝐵))) |
19 | ltrelnq 7173 | . . . . . . . 8 ⊢ <Q ⊆ (Q × Q) | |
20 | 19 | brel 4591 | . . . . . . 7 ⊢ (𝑄 <Q 𝑅 → (𝑄 ∈ Q ∧ 𝑅 ∈ Q)) |
21 | 20 | simprd 113 | . . . . . 6 ⊢ (𝑄 <Q 𝑅 → 𝑅 ∈ Q) |
22 | 3, 21 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Q) |
23 | addnqpru 7338 | . . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝑈 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝑇 ∈ (2nd ‘𝐵))) ∧ 𝑅 ∈ Q) → ((𝑈 +Q 𝑇) <Q 𝑅 → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))) | |
24 | 17, 18, 22, 23 | syl21anc 1215 | . . . 4 ⊢ (𝜑 → ((𝑈 +Q 𝑇) <Q 𝑅 → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))) |
25 | 24 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝑄 = (𝐷 +Q 𝐸)) → ((𝑈 +Q 𝑇) <Q 𝑅 → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))) |
26 | 16, 25 | mpd 13 | . 2 ⊢ ((𝜑 ∧ 𝑄 = (𝐷 +Q 𝐸)) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))) |
27 | 26 | ex 114 | 1 ⊢ (𝜑 → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 1st c1st 6036 2nd c2nd 6037 Qcnq 7088 +Q cplq 7090 <Q cltq 7093 Pcnp 7099 +P cpp 7101 |
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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-1nqqs 7159 df-rq 7160 df-ltnqqs 7161 df-inp 7274 df-iplp 7276 |
This theorem is referenced by: addlocprlem 7343 |
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