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Mirrors > Home > MPE Home > Th. List > ltpsrpr | Structured version Visualization version GIF version |
Description: Mapping of order from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ltpsrpr.3 | ⊢ 𝐶 ∈ R |
Ref | Expression |
---|---|
ltpsrpr | ⊢ ((𝐶 +R [〈𝐴, 1P〉] ~R ) <R (𝐶 +R [〈𝐵, 1P〉] ~R ) ↔ 𝐴<P 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltpsrpr.3 | . . 3 ⊢ 𝐶 ∈ R | |
2 | ltasr 11132 | . . 3 ⊢ (𝐶 ∈ R → ([〈𝐴, 1P〉] ~R <R [〈𝐵, 1P〉] ~R ↔ (𝐶 +R [〈𝐴, 1P〉] ~R ) <R (𝐶 +R [〈𝐵, 1P〉] ~R ))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ([〈𝐴, 1P〉] ~R <R [〈𝐵, 1P〉] ~R ↔ (𝐶 +R [〈𝐴, 1P〉] ~R ) <R (𝐶 +R [〈𝐵, 1P〉] ~R )) |
4 | addcompr 11053 | . . . 4 ⊢ (𝐴 +P 1P) = (1P +P 𝐴) | |
5 | 4 | breq1i 5151 | . . 3 ⊢ ((𝐴 +P 1P)<P (1P +P 𝐵) ↔ (1P +P 𝐴)<P (1P +P 𝐵)) |
6 | ltsrpr 11109 | . . 3 ⊢ ([〈𝐴, 1P〉] ~R <R [〈𝐵, 1P〉] ~R ↔ (𝐴 +P 1P)<P (1P +P 𝐵)) | |
7 | 1pr 11047 | . . . 4 ⊢ 1P ∈ P | |
8 | ltapr 11077 | . . . 4 ⊢ (1P ∈ P → (𝐴<P 𝐵 ↔ (1P +P 𝐴)<P (1P +P 𝐵))) | |
9 | 7, 8 | ax-mp 5 | . . 3 ⊢ (𝐴<P 𝐵 ↔ (1P +P 𝐴)<P (1P +P 𝐵)) |
10 | 5, 6, 9 | 3bitr4i 302 | . 2 ⊢ ([〈𝐴, 1P〉] ~R <R [〈𝐵, 1P〉] ~R ↔ 𝐴<P 𝐵) |
11 | 3, 10 | bitr3i 276 | 1 ⊢ ((𝐶 +R [〈𝐴, 1P〉] ~R ) <R (𝐶 +R [〈𝐵, 1P〉] ~R ) ↔ 𝐴<P 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2099 〈cop 4630 class class class wbr 5144 (class class class)co 7414 [cec 8722 Pcnp 10891 1Pc1p 10892 +P cpp 10893 <P cltp 10895 ~R cer 10896 Rcnr 10897 +R cplr 10901 <R cltr 10903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-inf2 9675 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-int 4948 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-oadd 8490 df-omul 8491 df-er 8724 df-ec 8726 df-qs 8730 df-ni 10904 df-pli 10905 df-mi 10906 df-lti 10907 df-plpq 10940 df-mpq 10941 df-ltpq 10942 df-enq 10943 df-nq 10944 df-erq 10945 df-plq 10946 df-mq 10947 df-1nq 10948 df-rq 10949 df-ltnq 10950 df-np 11013 df-1p 11014 df-plp 11015 df-ltp 11017 df-enr 11087 df-nr 11088 df-plr 11089 df-ltr 11091 |
This theorem is referenced by: supsrlem 11143 |
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