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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 11107 | . 2 ⊢ ([〈1P, 1P〉] ~R <R [〈𝐴, 𝐵〉] ~R ↔ (1P +P 𝐵)<P (1P +P 𝐴)) | |
2 | df-0r 11090 | . . 3 ⊢ 0R = [〈1P, 1P〉] ~R | |
3 | 2 | breq1i 5156 | . 2 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R ↔ [〈1P, 1P〉] ~R <R [〈𝐴, 𝐵〉] ~R ) |
4 | 1pr 11045 | . . 3 ⊢ 1P ∈ P | |
5 | ltapr 11075 | . . 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 302 | 1 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R ↔ 𝐵<P 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2098 〈cop 4636 class class class wbr 5149 (class class class)co 7419 [cec 8723 Pcnp 10889 1Pc1p 10890 +P cpp 10891 <P cltp 10893 ~R cer 10894 0Rc0r 10896 <R cltr 10901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9671 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-omul 8492 df-er 8725 df-ec 8727 df-qs 8731 df-ni 10902 df-pli 10903 df-mi 10904 df-lti 10905 df-plpq 10938 df-mpq 10939 df-ltpq 10940 df-enq 10941 df-nq 10942 df-erq 10943 df-plq 10944 df-mq 10945 df-1nq 10946 df-rq 10947 df-ltnq 10948 df-np 11011 df-1p 11012 df-plp 11013 df-ltp 11015 df-enr 11085 df-nr 11086 df-ltr 11089 df-0r 11090 |
This theorem is referenced by: recexsrlem 11133 mulgt0sr 11135 |
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