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Mirrors > Home > ILE Home > Th. List > addlocprlemeq | GIF version |
Description: Lemma for addlocpr 7368. 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 7364 | . . . . 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 5789 | . . . . 5 ⊢ (𝑄 = (𝐷 +Q 𝐸) → (𝑄 +Q (𝑃 +Q 𝑃)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃))) | |
15 | 5, 14 | sylan9req 2194 | . . . 4 ⊢ ((𝜑 ∧ 𝑄 = (𝐷 +Q 𝐸)) → 𝑅 = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃))) |
16 | 13, 15 | breqtrrd 3964 | . . 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 7197 | . . . . . . . 8 ⊢ <Q ⊆ (Q × Q) | |
20 | 19 | brel 4599 | . . . . . . 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 7362 | . . . . 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 1216 | . . . 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 1332 ∈ wcel 1481 class class class wbr 3937 ‘cfv 5131 (class class class)co 5782 1st c1st 6044 2nd c2nd 6045 Qcnq 7112 +Q cplq 7114 <Q cltq 7117 Pcnp 7123 +P cpp 7125 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-lti 7139 df-plpq 7176 df-mpq 7177 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-mqqs 7182 df-1nqqs 7183 df-rq 7184 df-ltnqqs 7185 df-inp 7298 df-iplp 7300 |
This theorem is referenced by: addlocprlem 7367 |
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