Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mappsrpr | Structured version Visualization version GIF version |
Description: Mapping 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 |
---|---|
mappsrpr.2 | ⊢ 𝐶 ∈ R |
Ref | Expression |
---|---|
mappsrpr | ⊢ ((𝐶 +R -1R) <R (𝐶 +R [〈𝐴, 1P〉] ~R ) ↔ 𝐴 ∈ P) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-m1r 9763 | . . . 4 ⊢ -1R = [〈1P, (1P +P 1P)〉] ~R | |
2 | 1 | breq1i 4590 | . . 3 ⊢ (-1R <R [〈𝐴, 1P〉] ~R ↔ [〈1P, (1P +P 1P)〉] ~R <R [〈𝐴, 1P〉] ~R ) |
3 | ltsrpr 9777 | . . 3 ⊢ ([〈1P, (1P +P 1P)〉] ~R <R [〈𝐴, 1P〉] ~R ↔ (1P +P 1P)<P ((1P +P 1P) +P 𝐴)) | |
4 | 2, 3 | bitri 263 | . 2 ⊢ (-1R <R [〈𝐴, 1P〉] ~R ↔ (1P +P 1P)<P ((1P +P 1P) +P 𝐴)) |
5 | mappsrpr.2 | . . 3 ⊢ 𝐶 ∈ R | |
6 | ltasr 9800 | . . 3 ⊢ (𝐶 ∈ R → (-1R <R [〈𝐴, 1P〉] ~R ↔ (𝐶 +R -1R) <R (𝐶 +R [〈𝐴, 1P〉] ~R ))) | |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ (-1R <R [〈𝐴, 1P〉] ~R ↔ (𝐶 +R -1R) <R (𝐶 +R [〈𝐴, 1P〉] ~R )) |
8 | ltrelpr 9699 | . . . . . 6 ⊢ <P ⊆ (P × P) | |
9 | 8 | brel 5090 | . . . . 5 ⊢ ((1P +P 1P)<P ((1P +P 1P) +P 𝐴) → ((1P +P 1P) ∈ P ∧ ((1P +P 1P) +P 𝐴) ∈ P)) |
10 | 9 | simprd 478 | . . . 4 ⊢ ((1P +P 1P)<P ((1P +P 1P) +P 𝐴) → ((1P +P 1P) +P 𝐴) ∈ P) |
11 | dmplp 9713 | . . . . . 6 ⊢ dom +P = (P × P) | |
12 | 0npr 9693 | . . . . . 6 ⊢ ¬ ∅ ∈ P | |
13 | 11, 12 | ndmovrcl 6718 | . . . . 5 ⊢ (((1P +P 1P) +P 𝐴) ∈ P → ((1P +P 1P) ∈ P ∧ 𝐴 ∈ P)) |
14 | 13 | simprd 478 | . . . 4 ⊢ (((1P +P 1P) +P 𝐴) ∈ P → 𝐴 ∈ P) |
15 | 10, 14 | syl 17 | . . 3 ⊢ ((1P +P 1P)<P ((1P +P 1P) +P 𝐴) → 𝐴 ∈ P) |
16 | 1pr 9716 | . . . . 5 ⊢ 1P ∈ P | |
17 | addclpr 9719 | . . . . 5 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P) | |
18 | 16, 16, 17 | mp2an 704 | . . . 4 ⊢ (1P +P 1P) ∈ P |
19 | ltaddpr 9735 | . . . 4 ⊢ (((1P +P 1P) ∈ P ∧ 𝐴 ∈ P) → (1P +P 1P)<P ((1P +P 1P) +P 𝐴)) | |
20 | 18, 19 | mpan 702 | . . 3 ⊢ (𝐴 ∈ P → (1P +P 1P)<P ((1P +P 1P) +P 𝐴)) |
21 | 15, 20 | impbii 198 | . 2 ⊢ ((1P +P 1P)<P ((1P +P 1P) +P 𝐴) ↔ 𝐴 ∈ P) |
22 | 4, 7, 21 | 3bitr3i 289 | 1 ⊢ ((𝐶 +R -1R) <R (𝐶 +R [〈𝐴, 1P〉] ~R ) ↔ 𝐴 ∈ P) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∈ wcel 1977 〈cop 4131 class class class wbr 4583 (class class class)co 6549 [cec 7627 Pcnp 9560 1Pc1p 9561 +P cpp 9562 <P cltp 9564 ~R cer 9565 Rcnr 9566 -1Rcm1r 9569 +R cplr 9570 <R cltr 9572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-ec 7631 df-qs 7635 df-ni 9573 df-pli 9574 df-mi 9575 df-lti 9576 df-plpq 9609 df-mpq 9610 df-ltpq 9611 df-enq 9612 df-nq 9613 df-erq 9614 df-plq 9615 df-mq 9616 df-1nq 9617 df-rq 9618 df-ltnq 9619 df-np 9682 df-1p 9683 df-plp 9684 df-ltp 9686 df-enr 9756 df-nr 9757 df-plr 9758 df-ltr 9760 df-m1r 9763 |
This theorem is referenced by: map2psrpr 9810 supsrlem 9811 |
Copyright terms: Public domain | W3C validator |