![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > m1p1sr | Structured version Visualization version GIF version |
Description: Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
m1p1sr | ⊢ (-1R +R 1R) = 0R |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-m1r 10276 | . . 3 ⊢ -1R = [〈1P, (1P +P 1P)〉] ~R | |
2 | df-1r 10275 | . . 3 ⊢ 1R = [〈(1P +P 1P), 1P〉] ~R | |
3 | 1, 2 | oveq12i 6982 | . 2 ⊢ (-1R +R 1R) = ([〈1P, (1P +P 1P)〉] ~R +R [〈(1P +P 1P), 1P〉] ~R ) |
4 | df-0r 10274 | . . 3 ⊢ 0R = [〈1P, 1P〉] ~R | |
5 | 1pr 10229 | . . . . 5 ⊢ 1P ∈ P | |
6 | addclpr 10232 | . . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P) | |
7 | 5, 5, 6 | mp2an 679 | . . . . 5 ⊢ (1P +P 1P) ∈ P |
8 | addsrpr 10289 | . . . . 5 ⊢ (((1P ∈ P ∧ (1P +P 1P) ∈ P) ∧ ((1P +P 1P) ∈ P ∧ 1P ∈ P)) → ([〈1P, (1P +P 1P)〉] ~R +R [〈(1P +P 1P), 1P〉] ~R ) = [〈(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)〉] ~R ) | |
9 | 5, 7, 7, 5, 8 | mp4an 680 | . . . 4 ⊢ ([〈1P, (1P +P 1P)〉] ~R +R [〈(1P +P 1P), 1P〉] ~R ) = [〈(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)〉] ~R |
10 | addasspr 10236 | . . . . . 6 ⊢ ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P)) | |
11 | 10 | oveq2i 6981 | . . . . 5 ⊢ (1P +P ((1P +P 1P) +P 1P)) = (1P +P (1P +P (1P +P 1P))) |
12 | addclpr 10232 | . . . . . . 7 ⊢ ((1P ∈ P ∧ (1P +P 1P) ∈ P) → (1P +P (1P +P 1P)) ∈ P) | |
13 | 5, 7, 12 | mp2an 679 | . . . . . 6 ⊢ (1P +P (1P +P 1P)) ∈ P |
14 | addclpr 10232 | . . . . . . 7 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P) → ((1P +P 1P) +P 1P) ∈ P) | |
15 | 7, 5, 14 | mp2an 679 | . . . . . 6 ⊢ ((1P +P 1P) +P 1P) ∈ P |
16 | enreceq 10280 | . . . . . 6 ⊢ (((1P ∈ P ∧ 1P ∈ P) ∧ ((1P +P (1P +P 1P)) ∈ P ∧ ((1P +P 1P) +P 1P) ∈ P)) → ([〈1P, 1P〉] ~R = [〈(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)〉] ~R ↔ (1P +P ((1P +P 1P) +P 1P)) = (1P +P (1P +P (1P +P 1P))))) | |
17 | 5, 5, 13, 15, 16 | mp4an 680 | . . . . 5 ⊢ ([〈1P, 1P〉] ~R = [〈(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)〉] ~R ↔ (1P +P ((1P +P 1P) +P 1P)) = (1P +P (1P +P (1P +P 1P)))) |
18 | 11, 17 | mpbir 223 | . . . 4 ⊢ [〈1P, 1P〉] ~R = [〈(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)〉] ~R |
19 | 9, 18 | eqtr4i 2799 | . . 3 ⊢ ([〈1P, (1P +P 1P)〉] ~R +R [〈(1P +P 1P), 1P〉] ~R ) = [〈1P, 1P〉] ~R |
20 | 4, 19 | eqtr4i 2799 | . 2 ⊢ 0R = ([〈1P, (1P +P 1P)〉] ~R +R [〈(1P +P 1P), 1P〉] ~R ) |
21 | 3, 20 | eqtr4i 2799 | 1 ⊢ (-1R +R 1R) = 0R |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1507 ∈ wcel 2050 〈cop 4441 (class class class)co 6970 [cec 8081 Pcnp 10073 1Pc1p 10074 +P cpp 10075 ~R cer 10078 0Rc0r 10080 1Rc1r 10081 -1Rcm1r 10082 +R cplr 10083 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8892 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-oadd 7903 df-omul 7904 df-er 8083 df-ec 8085 df-qs 8089 df-ni 10086 df-pli 10087 df-mi 10088 df-lti 10089 df-plpq 10122 df-mpq 10123 df-ltpq 10124 df-enq 10125 df-nq 10126 df-erq 10127 df-plq 10128 df-mq 10129 df-1nq 10130 df-rq 10131 df-ltnq 10132 df-np 10195 df-1p 10196 df-plp 10197 df-ltp 10199 df-enr 10269 df-nr 10270 df-plr 10271 df-0r 10274 df-1r 10275 df-m1r 10276 |
This theorem is referenced by: pn0sr 10315 supsrlem 10325 axi2m1 10373 |
Copyright terms: Public domain | W3C validator |