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Mirrors > Home > ILE Home > Th. List > prsrlt | GIF version |
Description: Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.) |
Ref | Expression |
---|---|
prsrlt | ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ [〈(𝐴 +P 1P), 1P〉] ~R <R [〈(𝐵 +P 1P), 1P〉] ~R )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1pr 6744 | . . . . 5 ⊢ 1P ∈ P | |
2 | 1 | a1i 9 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → 1P ∈ P) |
3 | simpr 108 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → 𝐵 ∈ P) | |
4 | addassprg 6769 | . . . 4 ⊢ ((1P ∈ P ∧ 𝐵 ∈ P ∧ 1P ∈ P) → ((1P +P 𝐵) +P 1P) = (1P +P (𝐵 +P 1P))) | |
5 | 2, 3, 2, 4 | syl3anc 1169 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((1P +P 𝐵) +P 1P) = (1P +P (𝐵 +P 1P))) |
6 | 5 | breq2d 3797 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝐴 +P 1P) +P 1P)<P ((1P +P 𝐵) +P 1P) ↔ ((𝐴 +P 1P) +P 1P)<P (1P +P (𝐵 +P 1P)))) |
7 | simpl 107 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → 𝐴 ∈ P) | |
8 | ltaprg 6809 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 1P ∈ P) → (𝐴<P 𝐵 ↔ (1P +P 𝐴)<P (1P +P 𝐵))) | |
9 | 7, 3, 2, 8 | syl3anc 1169 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ (1P +P 𝐴)<P (1P +P 𝐵))) |
10 | addcomprg 6768 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 1P ∈ P) → (𝐴 +P 1P) = (1P +P 𝐴)) | |
11 | 7, 2, 10 | syl2anc 403 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 1P) = (1P +P 𝐴)) |
12 | 11 | breq1d 3795 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 1P)<P (1P +P 𝐵) ↔ (1P +P 𝐴)<P (1P +P 𝐵))) |
13 | ltaprg 6809 | . . . . 5 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔))) | |
14 | 13 | adantl 271 | . . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P)) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔))) |
15 | addclpr 6727 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 1P ∈ P) → (𝐴 +P 1P) ∈ P) | |
16 | 7, 2, 15 | syl2anc 403 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 1P) ∈ P) |
17 | addclpr 6727 | . . . . 5 ⊢ ((1P ∈ P ∧ 𝐵 ∈ P) → (1P +P 𝐵) ∈ P) | |
18 | 2, 3, 17 | syl2anc 403 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (1P +P 𝐵) ∈ P) |
19 | addcomprg 6768 | . . . . 5 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓)) | |
20 | 19 | adantl 271 | . . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓)) |
21 | 14, 16, 18, 2, 20 | caovord2d 5690 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 1P)<P (1P +P 𝐵) ↔ ((𝐴 +P 1P) +P 1P)<P ((1P +P 𝐵) +P 1P))) |
22 | 9, 12, 21 | 3bitr2d 214 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ ((𝐴 +P 1P) +P 1P)<P ((1P +P 𝐵) +P 1P))) |
23 | addclpr 6727 | . . . 4 ⊢ ((𝐵 ∈ P ∧ 1P ∈ P) → (𝐵 +P 1P) ∈ P) | |
24 | 3, 2, 23 | syl2anc 403 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐵 +P 1P) ∈ P) |
25 | ltsrprg 6924 | . . 3 ⊢ ((((𝐴 +P 1P) ∈ P ∧ 1P ∈ P) ∧ ((𝐵 +P 1P) ∈ P ∧ 1P ∈ P)) → ([〈(𝐴 +P 1P), 1P〉] ~R <R [〈(𝐵 +P 1P), 1P〉] ~R ↔ ((𝐴 +P 1P) +P 1P)<P (1P +P (𝐵 +P 1P)))) | |
26 | 16, 2, 24, 2, 25 | syl22anc 1170 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ([〈(𝐴 +P 1P), 1P〉] ~R <R [〈(𝐵 +P 1P), 1P〉] ~R ↔ ((𝐴 +P 1P) +P 1P)<P (1P +P (𝐵 +P 1P)))) |
27 | 6, 22, 26 | 3bitr4d 218 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ [〈(𝐴 +P 1P), 1P〉] ~R <R [〈(𝐵 +P 1P), 1P〉] ~R )) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∧ w3a 919 = wceq 1284 ∈ wcel 1433 〈cop 3401 class class class wbr 3785 (class class class)co 5532 [cec 6127 Pcnp 6481 1Pc1p 6482 +P cpp 6483 <P cltp 6485 ~R cer 6486 <R cltr 6493 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-i1p 6657 df-iplp 6658 df-iltp 6660 df-enr 6903 df-nr 6904 df-ltr 6907 |
This theorem is referenced by: caucvgsrlemcau 6969 caucvgsrlembound 6970 caucvgsrlemgt1 6971 ltrennb 7022 |
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