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Mirrors > Home > ILE Home > Th. List > addlocprlemlt | GIF version |
Description: Lemma for addlocpr 7312. The 𝑄 <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 |
---|---|
addlocprlemlt | ⊢ (𝜑 → (𝑄 <Q (𝐷 +Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 +P 𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addlocprlem.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ P) | |
2 | addlocprlem.dlo | . . 3 ⊢ (𝜑 → 𝐷 ∈ (1st ‘𝐴)) | |
3 | 1, 2 | jca 304 | . 2 ⊢ (𝜑 → (𝐴 ∈ P ∧ 𝐷 ∈ (1st ‘𝐴))) |
4 | addlocprlem.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ P) | |
5 | addlocprlem.elo | . . 3 ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵)) | |
6 | 4, 5 | jca 304 | . 2 ⊢ (𝜑 → (𝐵 ∈ P ∧ 𝐸 ∈ (1st ‘𝐵))) |
7 | addlocprlem.qr | . . 3 ⊢ (𝜑 → 𝑄 <Q 𝑅) | |
8 | ltrelnq 7141 | . . . . 5 ⊢ <Q ⊆ (Q × Q) | |
9 | 8 | brel 4561 | . . . 4 ⊢ (𝑄 <Q 𝑅 → (𝑄 ∈ Q ∧ 𝑅 ∈ Q)) |
10 | 9 | simpld 111 | . . 3 ⊢ (𝑄 <Q 𝑅 → 𝑄 ∈ Q) |
11 | 7, 10 | syl 14 | . 2 ⊢ (𝜑 → 𝑄 ∈ Q) |
12 | addnqprl 7305 | . 2 ⊢ ((((𝐴 ∈ P ∧ 𝐷 ∈ (1st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐸 ∈ (1st ‘𝐵))) ∧ 𝑄 ∈ Q) → (𝑄 <Q (𝐷 +Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 +P 𝐵)))) | |
13 | 3, 6, 11, 12 | syl21anc 1200 | 1 ⊢ (𝜑 → (𝑄 <Q (𝐷 +Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 +P 𝐵)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 class class class wbr 3899 ‘cfv 5093 (class class class)co 5742 1st c1st 6004 2nd c2nd 6005 Qcnq 7056 +Q cplq 7058 <Q cltq 7061 Pcnp 7067 +P cpp 7069 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-eprel 4181 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-1o 6281 df-oadd 6285 df-omul 6286 df-er 6397 df-ec 6399 df-qs 6403 df-ni 7080 df-pli 7081 df-mi 7082 df-lti 7083 df-plpq 7120 df-mpq 7121 df-enq 7123 df-nqqs 7124 df-plqqs 7125 df-mqqs 7126 df-1nqqs 7127 df-rq 7128 df-ltnqqs 7129 df-inp 7242 df-iplp 7244 |
This theorem is referenced by: addlocprlem 7311 |
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