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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 10484 | . . 3 ⊢ -1R = [〈1P, (1P +P 1P)〉] ~R | |
2 | df-1r 10483 | . . 3 ⊢ 1R = [〈(1P +P 1P), 1P〉] ~R | |
3 | 1, 2 | oveq12i 7168 | . 2 ⊢ (-1R +R 1R) = ([〈1P, (1P +P 1P)〉] ~R +R [〈(1P +P 1P), 1P〉] ~R ) |
4 | df-0r 10482 | . . 3 ⊢ 0R = [〈1P, 1P〉] ~R | |
5 | 1pr 10437 | . . . . 5 ⊢ 1P ∈ P | |
6 | addclpr 10440 | . . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P) | |
7 | 5, 5, 6 | mp2an 690 | . . . . 5 ⊢ (1P +P 1P) ∈ P |
8 | addsrpr 10497 | . . . . 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 691 | . . . 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 10444 | . . . . . 6 ⊢ ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P)) | |
11 | 10 | oveq2i 7167 | . . . . 5 ⊢ (1P +P ((1P +P 1P) +P 1P)) = (1P +P (1P +P (1P +P 1P))) |
12 | addclpr 10440 | . . . . . . 7 ⊢ ((1P ∈ P ∧ (1P +P 1P) ∈ P) → (1P +P (1P +P 1P)) ∈ P) | |
13 | 5, 7, 12 | mp2an 690 | . . . . . 6 ⊢ (1P +P (1P +P 1P)) ∈ P |
14 | addclpr 10440 | . . . . . . 7 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P) → ((1P +P 1P) +P 1P) ∈ P) | |
15 | 7, 5, 14 | mp2an 690 | . . . . . 6 ⊢ ((1P +P 1P) +P 1P) ∈ P |
16 | enreceq 10488 | . . . . . 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 691 | . . . . 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 233 | . . . 4 ⊢ [〈1P, 1P〉] ~R = [〈(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)〉] ~R |
19 | 9, 18 | eqtr4i 2847 | . . 3 ⊢ ([〈1P, (1P +P 1P)〉] ~R +R [〈(1P +P 1P), 1P〉] ~R ) = [〈1P, 1P〉] ~R |
20 | 4, 19 | eqtr4i 2847 | . 2 ⊢ 0R = ([〈1P, (1P +P 1P)〉] ~R +R [〈(1P +P 1P), 1P〉] ~R ) |
21 | 3, 20 | eqtr4i 2847 | 1 ⊢ (-1R +R 1R) = 0R |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 〈cop 4573 (class class class)co 7156 [cec 8287 Pcnp 10281 1Pc1p 10282 +P cpp 10283 ~R cer 10286 0Rc0r 10288 1Rc1r 10289 -1Rcm1r 10290 +R cplr 10291 |
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-0r 10482 df-1r 10483 df-m1r 10484 |
This theorem is referenced by: pn0sr 10523 supsrlem 10533 axi2m1 10581 |
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