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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 10148 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → 𝑔 ∈ Q) | |
2 | ltrnq 10136 | . . . . 5 ⊢ (𝑥 <Q (𝑔 +Q ℎ) ↔ (*Q‘(𝑔 +Q ℎ)) <Q (*Q‘𝑥)) | |
3 | ltmnq 10129 | . . . . . 6 ⊢ (𝑥 ∈ Q → ((*Q‘(𝑔 +Q ℎ)) <Q (*Q‘𝑥) ↔ (𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) <Q (𝑥 ·Q (*Q‘𝑥)))) | |
4 | ovex 6954 | . . . . . . 7 ⊢ (𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ∈ V | |
5 | ovex 6954 | . . . . . . 7 ⊢ (𝑥 ·Q (*Q‘𝑥)) ∈ V | |
6 | ltmnq 10129 | . . . . . . 7 ⊢ (𝑤 ∈ Q → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧))) | |
7 | vex 3400 | . . . . . . 7 ⊢ 𝑔 ∈ V | |
8 | mulcomnq 10110 | . . . . . . 7 ⊢ (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦) | |
9 | 4, 5, 6, 7, 8 | caovord2 7123 | . . . . . 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 506 | . . . . 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 275 | . . . 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 10122 | . . . . . . 7 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q) | |
13 | 12 | oveq1d 6937 | . . . . . 6 ⊢ (𝑥 ∈ Q → ((𝑥 ·Q (*Q‘𝑥)) ·Q 𝑔) = (1Q ·Q 𝑔)) |
14 | mulcomnq 10110 | . . . . . . 7 ⊢ (1Q ·Q 𝑔) = (𝑔 ·Q 1Q) | |
15 | mulidnq 10120 | . . . . . . 7 ⊢ (𝑔 ∈ Q → (𝑔 ·Q 1Q) = 𝑔) | |
16 | 14, 15 | syl5eq 2825 | . . . . . 6 ⊢ (𝑔 ∈ Q → (1Q ·Q 𝑔) = 𝑔) |
17 | 13, 16 | sylan9eqr 2835 | . . . . 5 ⊢ ((𝑔 ∈ Q ∧ 𝑥 ∈ Q) → ((𝑥 ·Q (*Q‘𝑥)) ·Q 𝑔) = 𝑔) |
18 | 17 | breq2d 4898 | . . . 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 271 | . . 3 ⊢ ((𝑔 ∈ Q ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q 𝑔)) |
20 | 1, 19 | sylan 575 | . 2 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q 𝑔)) |
21 | prcdnq 10150 | . . 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 474 | . 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 232 | 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 198 ∧ wa 386 ∈ wcel 2106 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 Qcnq 10009 1Qc1q 10010 +Q cplq 10012 ·Q cmq 10013 *Qcrq 10014 <Q cltq 10015 Pcnp 10016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-omul 7848 df-er 8026 df-ni 10029 df-mi 10031 df-lti 10032 df-mpq 10066 df-ltpq 10067 df-enq 10068 df-nq 10069 df-erq 10070 df-mq 10072 df-1nq 10073 df-rq 10074 df-ltnq 10075 df-np 10138 |
This theorem is referenced by: addclprlem2 10174 |
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