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Mirrors > Home > ILE Home > Th. List > addlocprlemeq | GIF version |
Description: Lemma for addlocpr 6726. 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 6722 | . . . . 5 ⊢ (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃))) |
13 | 12 | adantr 270 | . . . 4 ⊢ ((𝜑 ∧ 𝑄 = (𝐷 +Q 𝐸)) → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃))) |
14 | oveq1 5539 | . . . . 5 ⊢ (𝑄 = (𝐷 +Q 𝐸) → (𝑄 +Q (𝑃 +Q 𝑃)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃))) | |
15 | 5, 14 | sylan9req 2134 | . . . 4 ⊢ ((𝜑 ∧ 𝑄 = (𝐷 +Q 𝐸)) → 𝑅 = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃))) |
16 | 13, 15 | breqtrrd 3811 | . . 3 ⊢ ((𝜑 ∧ 𝑄 = (𝐷 +Q 𝐸)) → (𝑈 +Q 𝑇) <Q 𝑅) |
17 | 1, 7 | jca 300 | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ P ∧ 𝑈 ∈ (2nd ‘𝐴))) |
18 | 2, 10 | jca 300 | . . . . 5 ⊢ (𝜑 → (𝐵 ∈ P ∧ 𝑇 ∈ (2nd ‘𝐵))) |
19 | ltrelnq 6555 | . . . . . . . 8 ⊢ <Q ⊆ (Q × Q) | |
20 | 19 | brel 4410 | . . . . . . 7 ⊢ (𝑄 <Q 𝑅 → (𝑄 ∈ Q ∧ 𝑅 ∈ Q)) |
21 | 20 | simprd 112 | . . . . . 6 ⊢ (𝑄 <Q 𝑅 → 𝑅 ∈ Q) |
22 | 3, 21 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Q) |
23 | addnqpru 6720 | . . . . 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 1168 | . . . 4 ⊢ (𝜑 → ((𝑈 +Q 𝑇) <Q 𝑅 → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))) |
25 | 24 | adantr 270 | . . 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 113 | 1 ⊢ (𝜑 → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 class class class wbr 3785 ‘cfv 4922 (class class class)co 5532 1st c1st 5785 2nd c2nd 5786 Qcnq 6470 +Q cplq 6472 <Q cltq 6475 Pcnp 6481 +P cpp 6483 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-inp 6656 df-iplp 6658 |
This theorem is referenced by: addlocprlem 6725 |
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