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Mirrors > Home > MPE Home > Th. List > ltexprlem2 | Structured version Visualization version GIF version |
Description: Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 3-Apr-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ltexprlem.1 | ⊢ 𝐶 = {𝑥 ∣ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)} |
Ref | Expression |
---|---|
ltexprlem2 | ⊢ (𝐵 ∈ P → 𝐶 ⊊ Q) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltexprlem.1 | . . . . 5 ⊢ 𝐶 = {𝑥 ∣ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)} | |
2 | 1 | abeq2i 2948 | . . . 4 ⊢ (𝑥 ∈ 𝐶 ↔ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)) |
3 | elprnq 10407 | . . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ (𝑦 +Q 𝑥) ∈ 𝐵) → (𝑦 +Q 𝑥) ∈ Q) | |
4 | addnqf 10364 | . . . . . . . . . . 11 ⊢ +Q :(Q × Q)⟶Q | |
5 | 4 | fdmi 6518 | . . . . . . . . . 10 ⊢ dom +Q = (Q × Q) |
6 | 0nnq 10340 | . . . . . . . . . 10 ⊢ ¬ ∅ ∈ Q | |
7 | 5, 6 | ndmovrcl 7328 | . . . . . . . . 9 ⊢ ((𝑦 +Q 𝑥) ∈ Q → (𝑦 ∈ Q ∧ 𝑥 ∈ Q)) |
8 | 3, 7 | syl 17 | . . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ (𝑦 +Q 𝑥) ∈ 𝐵) → (𝑦 ∈ Q ∧ 𝑥 ∈ Q)) |
9 | ltaddnq 10390 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → 𝑥 <Q (𝑥 +Q 𝑦)) | |
10 | 9 | ancoms 461 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ Q ∧ 𝑥 ∈ Q) → 𝑥 <Q (𝑥 +Q 𝑦)) |
11 | addcomnq 10367 | . . . . . . . . . 10 ⊢ (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥) | |
12 | 10, 11 | breqtrdi 5099 | . . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑥 ∈ Q) → 𝑥 <Q (𝑦 +Q 𝑥)) |
13 | prcdnq 10409 | . . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ (𝑦 +Q 𝑥) ∈ 𝐵) → (𝑥 <Q (𝑦 +Q 𝑥) → 𝑥 ∈ 𝐵)) | |
14 | 12, 13 | syl5 34 | . . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ (𝑦 +Q 𝑥) ∈ 𝐵) → ((𝑦 ∈ Q ∧ 𝑥 ∈ Q) → 𝑥 ∈ 𝐵)) |
15 | 8, 14 | mpd 15 | . . . . . . 7 ⊢ ((𝐵 ∈ P ∧ (𝑦 +Q 𝑥) ∈ 𝐵) → 𝑥 ∈ 𝐵) |
16 | 15 | ex 415 | . . . . . 6 ⊢ (𝐵 ∈ P → ((𝑦 +Q 𝑥) ∈ 𝐵 → 𝑥 ∈ 𝐵)) |
17 | 16 | adantld 493 | . . . . 5 ⊢ (𝐵 ∈ P → ((¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵) → 𝑥 ∈ 𝐵)) |
18 | 17 | exlimdv 1930 | . . . 4 ⊢ (𝐵 ∈ P → (∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵) → 𝑥 ∈ 𝐵)) |
19 | 2, 18 | syl5bi 244 | . . 3 ⊢ (𝐵 ∈ P → (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐵)) |
20 | 19 | ssrdv 3972 | . 2 ⊢ (𝐵 ∈ P → 𝐶 ⊆ 𝐵) |
21 | prpssnq 10406 | . 2 ⊢ (𝐵 ∈ P → 𝐵 ⊊ Q) | |
22 | 20, 21 | sspsstrd 4084 | 1 ⊢ (𝐵 ∈ P → 𝐶 ⊊ Q) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 {cab 2799 ⊊ wpss 3936 class class class wbr 5058 × cxp 5547 (class class class)co 7150 Qcnq 10268 +Q cplq 10271 <Q cltq 10274 Pcnp 10275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-omul 8101 df-er 8283 df-ni 10288 df-pli 10289 df-mi 10290 df-lti 10291 df-plpq 10324 df-mpq 10325 df-ltpq 10326 df-enq 10327 df-nq 10328 df-erq 10329 df-plq 10330 df-mq 10331 df-1nq 10332 df-ltnq 10334 df-np 10397 |
This theorem is referenced by: ltexprlem5 10456 |
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