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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 10186 | . 2 ⊢ ([〈1P, 1P〉] ~R <R [〈𝐴, 𝐵〉] ~R ↔ (1P +P 𝐵)<P (1P +P 𝐴)) | |
2 | df-0r 10170 | . . 3 ⊢ 0R = [〈1P, 1P〉] ~R | |
3 | 2 | breq1i 4850 | . 2 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R ↔ [〈1P, 1P〉] ~R <R [〈𝐴, 𝐵〉] ~R ) |
4 | 1pr 10125 | . . 3 ⊢ 1P ∈ P | |
5 | ltapr 10155 | . . 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 295 | 1 ⊢ (0R <R [〈𝐴, 𝐵〉] ~R ↔ 𝐵<P 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∈ wcel 2157 〈cop 4374 class class class wbr 4843 (class class class)co 6878 [cec 7980 Pcnp 9969 1Pc1p 9970 +P cpp 9971 <P cltp 9973 ~R cer 9974 0Rc0r 9976 <R cltr 9981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-omul 7804 df-er 7982 df-ec 7984 df-qs 7988 df-ni 9982 df-pli 9983 df-mi 9984 df-lti 9985 df-plpq 10018 df-mpq 10019 df-ltpq 10020 df-enq 10021 df-nq 10022 df-erq 10023 df-plq 10024 df-mq 10025 df-1nq 10026 df-rq 10027 df-ltnq 10028 df-np 10091 df-1p 10092 df-plp 10093 df-ltp 10095 df-enr 10165 df-nr 10166 df-ltr 10169 df-0r 10170 |
This theorem is referenced by: recexsrlem 10212 mulgt0sr 10214 |
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