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Mirrors > Home > MPE Home > Th. List > m1r | Structured version Visualization version GIF version |
Description: The constant -1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
m1r | ⊢ -1R ∈ R |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1pr 10439 | . . . 4 ⊢ 1P ∈ P | |
2 | addclpr 10442 | . . . . 5 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P) | |
3 | 1, 1, 2 | mp2an 690 | . . . 4 ⊢ (1P +P 1P) ∈ P |
4 | opelxpi 5594 | . . . 4 ⊢ ((1P ∈ P ∧ (1P +P 1P) ∈ P) → 〈1P, (1P +P 1P)〉 ∈ (P × P)) | |
5 | 1, 3, 4 | mp2an 690 | . . 3 ⊢ 〈1P, (1P +P 1P)〉 ∈ (P × P) |
6 | enrex 10491 | . . . 4 ⊢ ~R ∈ V | |
7 | 6 | ecelqsi 8355 | . . 3 ⊢ (〈1P, (1P +P 1P)〉 ∈ (P × P) → [〈1P, (1P +P 1P)〉] ~R ∈ ((P × P) / ~R )) |
8 | 5, 7 | ax-mp 5 | . 2 ⊢ [〈1P, (1P +P 1P)〉] ~R ∈ ((P × P) / ~R ) |
9 | df-m1r 10486 | . 2 ⊢ -1R = [〈1P, (1P +P 1P)〉] ~R | |
10 | df-nr 10480 | . 2 ⊢ R = ((P × P) / ~R ) | |
11 | 8, 9, 10 | 3eltr4i 2928 | 1 ⊢ -1R ∈ R |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 〈cop 4575 × cxp 5555 (class class class)co 7158 [cec 8289 / cqs 8290 Pcnp 10283 1Pc1p 10284 +P cpp 10285 ~R cer 10288 Rcnr 10289 -1Rcm1r 10292 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-omul 8109 df-er 8291 df-ec 8293 df-qs 8297 df-ni 10296 df-pli 10297 df-mi 10298 df-lti 10299 df-plpq 10332 df-mpq 10333 df-ltpq 10334 df-enq 10335 df-nq 10336 df-erq 10337 df-plq 10338 df-mq 10339 df-1nq 10340 df-rq 10341 df-ltnq 10342 df-np 10405 df-1p 10406 df-plp 10407 df-enr 10479 df-nr 10480 df-m1r 10486 |
This theorem is referenced by: negexsr 10526 sqgt0sr 10530 map2psrpr 10534 supsrlem 10535 mulresr 10563 axmulf 10570 axmulass 10581 axdistr 10582 axi2m1 10583 axrnegex 10586 axcnre 10588 |
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