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Mirrors > Home > ILE Home > Th. List > m1p1sr | GIF version |
Description: Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) |
Ref | Expression |
---|---|
m1p1sr | ⊢ (-1R +R 1R) = 0R |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-m1r 7682 | . . 3 ⊢ -1R = [〈1P, (1P +P 1P)〉] ~R | |
2 | df-1r 7681 | . . 3 ⊢ 1R = [〈(1P +P 1P), 1P〉] ~R | |
3 | 1, 2 | oveq12i 5862 | . 2 ⊢ (-1R +R 1R) = ([〈1P, (1P +P 1P)〉] ~R +R [〈(1P +P 1P), 1P〉] ~R ) |
4 | df-0r 7680 | . . 3 ⊢ 0R = [〈1P, 1P〉] ~R | |
5 | 1pr 7503 | . . . . 5 ⊢ 1P ∈ P | |
6 | addclpr 7486 | . . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P) | |
7 | 5, 5, 6 | mp2an 424 | . . . . 5 ⊢ (1P +P 1P) ∈ P |
8 | addsrpr 7694 | . . . . 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 425 | . . . 4 ⊢ ([〈1P, (1P +P 1P)〉] ~R +R [〈(1P +P 1P), 1P〉] ~R ) = [〈(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)〉] ~R |
10 | addassprg 7528 | . . . . . . 7 ⊢ ((1P ∈ P ∧ 1P ∈ P ∧ 1P ∈ P) → ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P))) | |
11 | 5, 5, 5, 10 | mp3an 1332 | . . . . . 6 ⊢ ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P)) |
12 | 11 | oveq2i 5861 | . . . . 5 ⊢ (1P +P ((1P +P 1P) +P 1P)) = (1P +P (1P +P (1P +P 1P))) |
13 | addclpr 7486 | . . . . . . 7 ⊢ ((1P ∈ P ∧ (1P +P 1P) ∈ P) → (1P +P (1P +P 1P)) ∈ P) | |
14 | 5, 7, 13 | mp2an 424 | . . . . . 6 ⊢ (1P +P (1P +P 1P)) ∈ P |
15 | addclpr 7486 | . . . . . . 7 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P) → ((1P +P 1P) +P 1P) ∈ P) | |
16 | 7, 5, 15 | mp2an 424 | . . . . . 6 ⊢ ((1P +P 1P) +P 1P) ∈ P |
17 | enreceq 7685 | . . . . . 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))))) | |
18 | 5, 5, 14, 16, 17 | mp4an 425 | . . . . 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)))) |
19 | 12, 18 | mpbir 145 | . . . 4 ⊢ [〈1P, 1P〉] ~R = [〈(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)〉] ~R |
20 | 9, 19 | eqtr4i 2194 | . . 3 ⊢ ([〈1P, (1P +P 1P)〉] ~R +R [〈(1P +P 1P), 1P〉] ~R ) = [〈1P, 1P〉] ~R |
21 | 4, 20 | eqtr4i 2194 | . 2 ⊢ 0R = ([〈1P, (1P +P 1P)〉] ~R +R [〈(1P +P 1P), 1P〉] ~R ) |
22 | 3, 21 | eqtr4i 2194 | 1 ⊢ (-1R +R 1R) = 0R |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1348 ∈ wcel 2141 〈cop 3584 (class class class)co 5850 [cec 6507 Pcnp 7240 1Pc1p 7241 +P cpp 7242 ~R cer 7245 0Rc0r 7247 1Rc1r 7248 -1Rcm1r 7249 +R cplr 7250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-eprel 4272 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-recs 6281 df-irdg 6346 df-1o 6392 df-2o 6393 df-oadd 6396 df-omul 6397 df-er 6509 df-ec 6511 df-qs 6515 df-ni 7253 df-pli 7254 df-mi 7255 df-lti 7256 df-plpq 7293 df-mpq 7294 df-enq 7296 df-nqqs 7297 df-plqqs 7298 df-mqqs 7299 df-1nqqs 7300 df-rq 7301 df-ltnqqs 7302 df-enq0 7373 df-nq0 7374 df-0nq0 7375 df-plq0 7376 df-mq0 7377 df-inp 7415 df-i1p 7416 df-iplp 7417 df-enr 7675 df-nr 7676 df-plr 7677 df-0r 7680 df-1r 7681 df-m1r 7682 |
This theorem is referenced by: pn0sr 7720 ltm1sr 7726 caucvgsrlemoffres 7749 caucvgsr 7751 suplocsrlempr 7756 axi2m1 7824 |
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