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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 10770 | . . . 4 ⊢ 1P ∈ P | |
2 | addclpr 10773 | . . . . 5 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P) | |
3 | 1, 1, 2 | mp2an 689 | . . . 4 ⊢ (1P +P 1P) ∈ P |
4 | opelxpi 5626 | . . . 4 ⊢ ((1P ∈ P ∧ (1P +P 1P) ∈ P) → 〈1P, (1P +P 1P)〉 ∈ (P × P)) | |
5 | 1, 3, 4 | mp2an 689 | . . 3 ⊢ 〈1P, (1P +P 1P)〉 ∈ (P × P) |
6 | enrex 10822 | . . . 4 ⊢ ~R ∈ V | |
7 | 6 | ecelqsi 8543 | . . 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 10817 | . 2 ⊢ -1R = [〈1P, (1P +P 1P)〉] ~R | |
10 | df-nr 10811 | . 2 ⊢ R = ((P × P) / ~R ) | |
11 | 8, 9, 10 | 3eltr4i 2854 | 1 ⊢ -1R ∈ R |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 〈cop 4573 × cxp 5587 (class class class)co 7269 [cec 8477 / cqs 8478 Pcnp 10614 1Pc1p 10615 +P cpp 10616 ~R cer 10619 Rcnr 10620 -1Rcm1r 10623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-inf2 9375 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-oadd 8290 df-omul 8291 df-er 8479 df-ec 8481 df-qs 8485 df-ni 10627 df-pli 10628 df-mi 10629 df-lti 10630 df-plpq 10663 df-mpq 10664 df-ltpq 10665 df-enq 10666 df-nq 10667 df-erq 10668 df-plq 10669 df-mq 10670 df-1nq 10671 df-rq 10672 df-ltnq 10673 df-np 10736 df-1p 10737 df-plp 10738 df-enr 10810 df-nr 10811 df-m1r 10817 |
This theorem is referenced by: negexsr 10857 sqgt0sr 10861 map2psrpr 10865 supsrlem 10866 mulresr 10894 axmulf 10901 axmulass 10912 axdistr 10913 axi2m1 10914 axrnegex 10917 axcnre 10919 |
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