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Mirrors > Home > ILE Home > Th. List > addlocprlemeq | GIF version |
Description: Lemma for addlocpr 7596. 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 7592 | . . . . 5 ⊢ (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃))) |
13 | 12 | adantr 276 | . . . 4 ⊢ ((𝜑 ∧ 𝑄 = (𝐷 +Q 𝐸)) → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃))) |
14 | oveq1 5925 | . . . . 5 ⊢ (𝑄 = (𝐷 +Q 𝐸) → (𝑄 +Q (𝑃 +Q 𝑃)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃))) | |
15 | 5, 14 | sylan9req 2247 | . . . 4 ⊢ ((𝜑 ∧ 𝑄 = (𝐷 +Q 𝐸)) → 𝑅 = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃))) |
16 | 13, 15 | breqtrrd 4057 | . . 3 ⊢ ((𝜑 ∧ 𝑄 = (𝐷 +Q 𝐸)) → (𝑈 +Q 𝑇) <Q 𝑅) |
17 | 1, 7 | jca 306 | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ P ∧ 𝑈 ∈ (2nd ‘𝐴))) |
18 | 2, 10 | jca 306 | . . . . 5 ⊢ (𝜑 → (𝐵 ∈ P ∧ 𝑇 ∈ (2nd ‘𝐵))) |
19 | ltrelnq 7425 | . . . . . . . 8 ⊢ <Q ⊆ (Q × Q) | |
20 | 19 | brel 4711 | . . . . . . 7 ⊢ (𝑄 <Q 𝑅 → (𝑄 ∈ Q ∧ 𝑅 ∈ Q)) |
21 | 20 | simprd 114 | . . . . . 6 ⊢ (𝑄 <Q 𝑅 → 𝑅 ∈ Q) |
22 | 3, 21 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Q) |
23 | addnqpru 7590 | . . . . 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 1248 | . . . 4 ⊢ (𝜑 → ((𝑈 +Q 𝑇) <Q 𝑅 → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))) |
25 | 24 | adantr 276 | . . 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 115 | 1 ⊢ (𝜑 → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 class class class wbr 4029 ‘cfv 5254 (class class class)co 5918 1st c1st 6191 2nd 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: addlocprlem 7595 |
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