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Mirrors > Home > ILE Home > Th. List > 0lt1sr | GIF version |
Description: 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) |
Ref | Expression |
---|---|
0lt1sr | ⊢ 0R <R 1R |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1pr 7386 | . . . . . 6 ⊢ 1P ∈ P | |
2 | addclpr 7369 | . . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P) | |
3 | 1, 1, 2 | mp2an 423 | . . . . 5 ⊢ (1P +P 1P) ∈ P |
4 | ltaddpr 7429 | . . . . 5 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P) → (1P +P 1P)<P ((1P +P 1P) +P 1P)) | |
5 | 3, 1, 4 | mp2an 423 | . . . 4 ⊢ (1P +P 1P)<P ((1P +P 1P) +P 1P) |
6 | addcomprg 7410 | . . . . 5 ⊢ ((1P ∈ P ∧ (1P +P 1P) ∈ P) → (1P +P (1P +P 1P)) = ((1P +P 1P) +P 1P)) | |
7 | 1, 3, 6 | mp2an 423 | . . . 4 ⊢ (1P +P (1P +P 1P)) = ((1P +P 1P) +P 1P) |
8 | 5, 7 | breqtrri 3963 | . . 3 ⊢ (1P +P 1P)<P (1P +P (1P +P 1P)) |
9 | ltsrprg 7579 | . . . 4 ⊢ (((1P ∈ P ∧ 1P ∈ P) ∧ ((1P +P 1P) ∈ P ∧ 1P ∈ P)) → ([〈1P, 1P〉] ~R <R [〈(1P +P 1P), 1P〉] ~R ↔ (1P +P 1P)<P (1P +P (1P +P 1P)))) | |
10 | 1, 1, 3, 1, 9 | mp4an 424 | . . 3 ⊢ ([〈1P, 1P〉] ~R <R [〈(1P +P 1P), 1P〉] ~R ↔ (1P +P 1P)<P (1P +P (1P +P 1P))) |
11 | 8, 10 | mpbir 145 | . 2 ⊢ [〈1P, 1P〉] ~R <R [〈(1P +P 1P), 1P〉] ~R |
12 | df-0r 7563 | . 2 ⊢ 0R = [〈1P, 1P〉] ~R | |
13 | df-1r 7564 | . 2 ⊢ 1R = [〈(1P +P 1P), 1P〉] ~R | |
14 | 11, 12, 13 | 3brtr4i 3966 | 1 ⊢ 0R <R 1R |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1332 ∈ wcel 1481 〈cop 3535 class class class wbr 3937 (class class class)co 5782 [cec 6435 Pcnp 7123 1Pc1p 7124 +P cpp 7125 <P cltp 7127 ~R cer 7128 0Rc0r 7130 1Rc1r 7131 <R cltr 7135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-2o 6322 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-lti 7139 df-plpq 7176 df-mpq 7177 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-mqqs 7182 df-1nqqs 7183 df-rq 7184 df-ltnqqs 7185 df-enq0 7256 df-nq0 7257 df-0nq0 7258 df-plq0 7259 df-mq0 7260 df-inp 7298 df-i1p 7299 df-iplp 7300 df-iltp 7302 df-enr 7558 df-nr 7559 df-ltr 7562 df-0r 7563 df-1r 7564 |
This theorem is referenced by: 1ne0sr 7598 ltadd1sr 7608 caucvgsrlemcl 7621 caucvgsrlemfv 7623 suplocsrlempr 7639 ax0lt1 7708 |
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