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Mirrors > Home > MPE Home > Th. List > addclprlem1 | Structured version Visualization version GIF version |
Description: Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
addclprlem1 | ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elprnq 10402 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → 𝑔 ∈ Q) | |
2 | ltrnq 10390 | . . . . 5 ⊢ (𝑥 <Q (𝑔 +Q ℎ) ↔ (*Q‘(𝑔 +Q ℎ)) <Q (*Q‘𝑥)) | |
3 | ltmnq 10383 | . . . . . 6 ⊢ (𝑥 ∈ Q → ((*Q‘(𝑔 +Q ℎ)) <Q (*Q‘𝑥) ↔ (𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) <Q (𝑥 ·Q (*Q‘𝑥)))) | |
4 | ovex 7168 | . . . . . . 7 ⊢ (𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ∈ V | |
5 | ovex 7168 | . . . . . . 7 ⊢ (𝑥 ·Q (*Q‘𝑥)) ∈ V | |
6 | ltmnq 10383 | . . . . . . 7 ⊢ (𝑤 ∈ Q → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧))) | |
7 | vex 3444 | . . . . . . 7 ⊢ 𝑔 ∈ V | |
8 | mulcomnq 10364 | . . . . . . 7 ⊢ (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦) | |
9 | 4, 5, 6, 7, 8 | caovord2 7340 | . . . . . 6 ⊢ (𝑔 ∈ Q → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) <Q (𝑥 ·Q (*Q‘𝑥)) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q‘𝑥)) ·Q 𝑔))) |
10 | 3, 9 | sylan9bbr 514 | . . . . 5 ⊢ ((𝑔 ∈ Q ∧ 𝑥 ∈ Q) → ((*Q‘(𝑔 +Q ℎ)) <Q (*Q‘𝑥) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q‘𝑥)) ·Q 𝑔))) |
11 | 2, 10 | syl5bb 286 | . . . 4 ⊢ ((𝑔 ∈ Q ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q‘𝑥)) ·Q 𝑔))) |
12 | recidnq 10376 | . . . . . . 7 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q) | |
13 | 12 | oveq1d 7150 | . . . . . 6 ⊢ (𝑥 ∈ Q → ((𝑥 ·Q (*Q‘𝑥)) ·Q 𝑔) = (1Q ·Q 𝑔)) |
14 | mulcomnq 10364 | . . . . . . 7 ⊢ (1Q ·Q 𝑔) = (𝑔 ·Q 1Q) | |
15 | mulidnq 10374 | . . . . . . 7 ⊢ (𝑔 ∈ Q → (𝑔 ·Q 1Q) = 𝑔) | |
16 | 14, 15 | syl5eq 2845 | . . . . . 6 ⊢ (𝑔 ∈ Q → (1Q ·Q 𝑔) = 𝑔) |
17 | 13, 16 | sylan9eqr 2855 | . . . . 5 ⊢ ((𝑔 ∈ Q ∧ 𝑥 ∈ Q) → ((𝑥 ·Q (*Q‘𝑥)) ·Q 𝑔) = 𝑔) |
18 | 17 | breq2d 5042 | . . . 4 ⊢ ((𝑔 ∈ Q ∧ 𝑥 ∈ Q) → (((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q‘𝑥)) ·Q 𝑔) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q 𝑔)) |
19 | 11, 18 | bitrd 282 | . . 3 ⊢ ((𝑔 ∈ Q ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q 𝑔)) |
20 | 1, 19 | sylan 583 | . 2 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q 𝑔)) |
21 | prcdnq 10404 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → (((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q 𝑔 → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) ∈ 𝐴)) | |
22 | 21 | adantr 484 | . 2 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ 𝑥 ∈ Q) → (((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q 𝑔 → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) ∈ 𝐴)) |
23 | 20, 22 | sylbid 243 | 1 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Qcnq 10263 1Qc1q 10264 +Q cplq 10266 ·Q cmq 10267 *Qcrq 10268 <Q cltq 10269 Pcnp 10270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-omul 8090 df-er 8272 df-ni 10283 df-mi 10285 df-lti 10286 df-mpq 10320 df-ltpq 10321 df-enq 10322 df-nq 10323 df-erq 10324 df-mq 10326 df-1nq 10327 df-rq 10328 df-ltnq 10329 df-np 10392 |
This theorem is referenced by: addclprlem2 10428 |
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