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Mirrors > Home > ILE Home > Th. List > ltaprlem | GIF version |
Description: Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) |
Ref | Expression |
---|---|
ltaprlem | ⊢ (𝐶 ∈ P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltexpri 6711 | . . . 4 ⊢ (𝐴<P 𝐵 → ∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) | |
2 | 1 | adantr 261 | . . 3 ⊢ ((𝐴<P 𝐵 ∧ 𝐶 ∈ P) → ∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) |
3 | simplr 482 | . . . . . 6 ⊢ (((𝐴<P 𝐵 ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ (𝐴 +P 𝑥) = 𝐵)) → 𝐶 ∈ P) | |
4 | ltrelpr 6603 | . . . . . . . . . 10 ⊢ <P ⊆ (P × P) | |
5 | 4 | brel 4392 | . . . . . . . . 9 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P)) |
6 | 5 | simpld 105 | . . . . . . . 8 ⊢ (𝐴<P 𝐵 → 𝐴 ∈ P) |
7 | 6 | adantr 261 | . . . . . . 7 ⊢ ((𝐴<P 𝐵 ∧ 𝐶 ∈ P) → 𝐴 ∈ P) |
8 | 7 | adantr 261 | . . . . . 6 ⊢ (((𝐴<P 𝐵 ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ (𝐴 +P 𝑥) = 𝐵)) → 𝐴 ∈ P) |
9 | addclpr 6635 | . . . . . 6 ⊢ ((𝐶 ∈ P ∧ 𝐴 ∈ P) → (𝐶 +P 𝐴) ∈ P) | |
10 | 3, 8, 9 | syl2anc 391 | . . . . 5 ⊢ (((𝐴<P 𝐵 ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ (𝐴 +P 𝑥) = 𝐵)) → (𝐶 +P 𝐴) ∈ P) |
11 | simprl 483 | . . . . 5 ⊢ (((𝐴<P 𝐵 ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ (𝐴 +P 𝑥) = 𝐵)) → 𝑥 ∈ P) | |
12 | ltaddpr 6695 | . . . . 5 ⊢ (((𝐶 +P 𝐴) ∈ P ∧ 𝑥 ∈ P) → (𝐶 +P 𝐴)<P ((𝐶 +P 𝐴) +P 𝑥)) | |
13 | 10, 11, 12 | syl2anc 391 | . . . 4 ⊢ (((𝐴<P 𝐵 ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ (𝐴 +P 𝑥) = 𝐵)) → (𝐶 +P 𝐴)<P ((𝐶 +P 𝐴) +P 𝑥)) |
14 | addassprg 6677 | . . . . . 6 ⊢ ((𝐶 ∈ P ∧ 𝐴 ∈ P ∧ 𝑥 ∈ P) → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P (𝐴 +P 𝑥))) | |
15 | 3, 8, 11, 14 | syl3anc 1135 | . . . . 5 ⊢ (((𝐴<P 𝐵 ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ (𝐴 +P 𝑥) = 𝐵)) → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P (𝐴 +P 𝑥))) |
16 | oveq2 5520 | . . . . . 6 ⊢ ((𝐴 +P 𝑥) = 𝐵 → (𝐶 +P (𝐴 +P 𝑥)) = (𝐶 +P 𝐵)) | |
17 | 16 | ad2antll 460 | . . . . 5 ⊢ (((𝐴<P 𝐵 ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ (𝐴 +P 𝑥) = 𝐵)) → (𝐶 +P (𝐴 +P 𝑥)) = (𝐶 +P 𝐵)) |
18 | 15, 17 | eqtrd 2072 | . . . 4 ⊢ (((𝐴<P 𝐵 ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ (𝐴 +P 𝑥) = 𝐵)) → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵)) |
19 | 13, 18 | breqtrd 3788 | . . 3 ⊢ (((𝐴<P 𝐵 ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ (𝐴 +P 𝑥) = 𝐵)) → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)) |
20 | 2, 19 | rexlimddv 2437 | . 2 ⊢ ((𝐴<P 𝐵 ∧ 𝐶 ∈ P) → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)) |
21 | 20 | expcom 109 | 1 ⊢ (𝐶 ∈ P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 ∃wrex 2307 class class class wbr 3764 (class class class)co 5512 Pcnp 6389 +P cpp 6391 <P cltp 6393 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-iplp 6566 df-iltp 6568 |
This theorem is referenced by: ltaprg 6717 |
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