![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 7322 | . . . 4 ⊢ (𝐴<P 𝐵 → ∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) | |
2 | 1 | adantr 272 | . . 3 ⊢ ((𝐴<P 𝐵 ∧ 𝐶 ∈ P) → ∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) |
3 | simplr 500 | . . . . . 6 ⊢ (((𝐴<P 𝐵 ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ (𝐴 +P 𝑥) = 𝐵)) → 𝐶 ∈ P) | |
4 | ltrelpr 7214 | . . . . . . . . . 10 ⊢ <P ⊆ (P × P) | |
5 | 4 | brel 4529 | . . . . . . . . 9 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P)) |
6 | 5 | simpld 111 | . . . . . . . 8 ⊢ (𝐴<P 𝐵 → 𝐴 ∈ P) |
7 | 6 | adantr 272 | . . . . . . 7 ⊢ ((𝐴<P 𝐵 ∧ 𝐶 ∈ P) → 𝐴 ∈ P) |
8 | 7 | adantr 272 | . . . . . 6 ⊢ (((𝐴<P 𝐵 ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ (𝐴 +P 𝑥) = 𝐵)) → 𝐴 ∈ P) |
9 | addclpr 7246 | . . . . . 6 ⊢ ((𝐶 ∈ P ∧ 𝐴 ∈ P) → (𝐶 +P 𝐴) ∈ P) | |
10 | 3, 8, 9 | syl2anc 406 | . . . . 5 ⊢ (((𝐴<P 𝐵 ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ (𝐴 +P 𝑥) = 𝐵)) → (𝐶 +P 𝐴) ∈ P) |
11 | simprl 501 | . . . . 5 ⊢ (((𝐴<P 𝐵 ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ (𝐴 +P 𝑥) = 𝐵)) → 𝑥 ∈ P) | |
12 | ltaddpr 7306 | . . . . 5 ⊢ (((𝐶 +P 𝐴) ∈ P ∧ 𝑥 ∈ P) → (𝐶 +P 𝐴)<P ((𝐶 +P 𝐴) +P 𝑥)) | |
13 | 10, 11, 12 | syl2anc 406 | . . . 4 ⊢ (((𝐴<P 𝐵 ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ (𝐴 +P 𝑥) = 𝐵)) → (𝐶 +P 𝐴)<P ((𝐶 +P 𝐴) +P 𝑥)) |
14 | addassprg 7288 | . . . . . 6 ⊢ ((𝐶 ∈ P ∧ 𝐴 ∈ P ∧ 𝑥 ∈ P) → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P (𝐴 +P 𝑥))) | |
15 | 3, 8, 11, 14 | syl3anc 1184 | . . . . 5 ⊢ (((𝐴<P 𝐵 ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ (𝐴 +P 𝑥) = 𝐵)) → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P (𝐴 +P 𝑥))) |
16 | oveq2 5714 | . . . . . 6 ⊢ ((𝐴 +P 𝑥) = 𝐵 → (𝐶 +P (𝐴 +P 𝑥)) = (𝐶 +P 𝐵)) | |
17 | 16 | ad2antll 478 | . . . . 5 ⊢ (((𝐴<P 𝐵 ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ (𝐴 +P 𝑥) = 𝐵)) → (𝐶 +P (𝐴 +P 𝑥)) = (𝐶 +P 𝐵)) |
18 | 15, 17 | eqtrd 2132 | . . . 4 ⊢ (((𝐴<P 𝐵 ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ (𝐴 +P 𝑥) = 𝐵)) → ((𝐶 +P 𝐴) +P 𝑥) = (𝐶 +P 𝐵)) |
19 | 13, 18 | breqtrd 3899 | . . 3 ⊢ (((𝐴<P 𝐵 ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ P ∧ (𝐴 +P 𝑥) = 𝐵)) → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)) |
20 | 2, 19 | rexlimddv 2513 | . 2 ⊢ ((𝐴<P 𝐵 ∧ 𝐶 ∈ P) → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)) |
21 | 20 | expcom 115 | 1 ⊢ (𝐶 ∈ P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1299 ∈ wcel 1448 ∃wrex 2376 class class class wbr 3875 (class class class)co 5706 Pcnp 7000 +P cpp 7002 <P cltp 7004 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 |
This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-eprel 4149 df-id 4153 df-po 4156 df-iso 4157 df-iord 4226 df-on 4228 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-irdg 6197 df-1o 6243 df-2o 6244 df-oadd 6247 df-omul 6248 df-er 6359 df-ec 6361 df-qs 6365 df-ni 7013 df-pli 7014 df-mi 7015 df-lti 7016 df-plpq 7053 df-mpq 7054 df-enq 7056 df-nqqs 7057 df-plqqs 7058 df-mqqs 7059 df-1nqqs 7060 df-rq 7061 df-ltnqqs 7062 df-enq0 7133 df-nq0 7134 df-0nq0 7135 df-plq0 7136 df-mq0 7137 df-inp 7175 df-iplp 7177 df-iltp 7179 |
This theorem is referenced by: ltaprg 7328 |
Copyright terms: Public domain | W3C validator |