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Mirrors > Home > MPE Home > Th. List > gt0srpr | Structured version Visualization version GIF version |
Description: Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
gt0srpr | ⊢ (0R <R [〈𝐴, 𝐵〉] ~R ↔ 𝐵<P 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltsrpr 10502 | . 2 ⊢ ([〈1P, 1P〉] ~R <R [〈𝐴, 𝐵〉] ~R ↔ (1P +P 𝐵)<P (1P +P 𝐴)) | |
2 | df-0r 10485 | . . 3 ⊢ 0R = [〈1P, 1P〉] ~R | |
3 | 2 | breq1i 5076 | . 2 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R ↔ [〈1P, 1P〉] ~R <R [〈𝐴, 𝐵〉] ~R ) |
4 | 1pr 10440 | . . 3 ⊢ 1P ∈ P | |
5 | ltapr 10470 | . . 3 ⊢ (1P ∈ P → (𝐵<P 𝐴 ↔ (1P +P 𝐵)<P (1P +P 𝐴))) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ (𝐵<P 𝐴 ↔ (1P +P 𝐵)<P (1P +P 𝐴)) |
7 | 1, 3, 6 | 3bitr4i 305 | 1 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R ↔ 𝐵<P 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2113 〈cop 4576 class class class wbr 5069 (class class class)co 7159 [cec 8290 Pcnp 10284 1Pc1p 10285 +P cpp 10286 <P cltp 10288 ~R cer 10289 0Rc0r 10291 <R cltr 10296 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-omul 8110 df-er 8292 df-ec 8294 df-qs 8298 df-ni 10297 df-pli 10298 df-mi 10299 df-lti 10300 df-plpq 10333 df-mpq 10334 df-ltpq 10335 df-enq 10336 df-nq 10337 df-erq 10338 df-plq 10339 df-mq 10340 df-1nq 10341 df-rq 10342 df-ltnq 10343 df-np 10406 df-1p 10407 df-plp 10408 df-ltp 10410 df-enr 10480 df-nr 10481 df-ltr 10484 df-0r 10485 |
This theorem is referenced by: recexsrlem 10528 mulgt0sr 10530 |
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