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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 10484 | . . . 4 ⊢ -1R = [〈1P, (1P +P 1P)〉] ~R | |
2 | 1 | breq1i 5073 | . . 3 ⊢ (-1R <R [〈𝐴, 1P〉] ~R ↔ [〈1P, (1P +P 1P)〉] ~R <R [〈𝐴, 1P〉] ~R ) |
3 | ltsrpr 10499 | . . 3 ⊢ ([〈1P, (1P +P 1P)〉] ~R <R [〈𝐴, 1P〉] ~R ↔ (1P +P 1P)<P ((1P +P 1P) +P 𝐴)) | |
4 | 2, 3 | bitri 277 | . 2 ⊢ (-1R <R [〈𝐴, 1P〉] ~R ↔ (1P +P 1P)<P ((1P +P 1P) +P 𝐴)) |
5 | mappsrpr.2 | . . 3 ⊢ 𝐶 ∈ R | |
6 | ltasr 10522 | . . 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 10420 | . . . . 5 ⊢ <P ⊆ (P × P) | |
9 | 8 | brel 5617 | . . . 4 ⊢ ((1P +P 1P)<P ((1P +P 1P) +P 𝐴) → ((1P +P 1P) ∈ P ∧ ((1P +P 1P) +P 𝐴) ∈ P)) |
10 | dmplp 10434 | . . . . . 6 ⊢ dom +P = (P × P) | |
11 | 0npr 10414 | . . . . . 6 ⊢ ¬ ∅ ∈ P | |
12 | 10, 11 | ndmovrcl 7334 | . . . . 5 ⊢ (((1P +P 1P) +P 𝐴) ∈ P → ((1P +P 1P) ∈ P ∧ 𝐴 ∈ P)) |
13 | 12 | simprd 498 | . . . 4 ⊢ (((1P +P 1P) +P 𝐴) ∈ P → 𝐴 ∈ P) |
14 | 9, 13 | simpl2im 506 | . . 3 ⊢ ((1P +P 1P)<P ((1P +P 1P) +P 𝐴) → 𝐴 ∈ P) |
15 | 1pr 10437 | . . . . 5 ⊢ 1P ∈ P | |
16 | addclpr 10440 | . . . . 5 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P) | |
17 | 15, 15, 16 | mp2an 690 | . . . 4 ⊢ (1P +P 1P) ∈ P |
18 | ltaddpr 10456 | . . . 4 ⊢ (((1P +P 1P) ∈ P ∧ 𝐴 ∈ P) → (1P +P 1P)<P ((1P +P 1P) +P 𝐴)) | |
19 | 17, 18 | mpan 688 | . . 3 ⊢ (𝐴 ∈ P → (1P +P 1P)<P ((1P +P 1P) +P 𝐴)) |
20 | 14, 19 | impbii 211 | . 2 ⊢ ((1P +P 1P)<P ((1P +P 1P) +P 𝐴) ↔ 𝐴 ∈ P) |
21 | 4, 7, 20 | 3bitr3i 303 | 1 ⊢ ((𝐶 +R -1R) <R (𝐶 +R [〈𝐴, 1P〉] ~R ) ↔ 𝐴 ∈ P) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2114 〈cop 4573 class class class wbr 5066 (class class class)co 7156 [cec 8287 Pcnp 10281 1Pc1p 10282 +P cpp 10283 <P cltp 10285 ~R cer 10286 Rcnr 10287 -1Rcm1r 10290 +R cplr 10291 <R cltr 10293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-omul 8107 df-er 8289 df-ec 8291 df-qs 8295 df-ni 10294 df-pli 10295 df-mi 10296 df-lti 10297 df-plpq 10330 df-mpq 10331 df-ltpq 10332 df-enq 10333 df-nq 10334 df-erq 10335 df-plq 10336 df-mq 10337 df-1nq 10338 df-rq 10339 df-ltnq 10340 df-np 10403 df-1p 10404 df-plp 10405 df-ltp 10407 df-enr 10477 df-nr 10478 df-plr 10479 df-ltr 10481 df-m1r 10484 |
This theorem is referenced by: map2psrpr 10532 supsrlem 10533 |
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