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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 11102 | . . 3 ⊢ -1R = [〈1P, (1P +P 1P)〉] ~R | |
| 2 | df-1r 11101 | . . 3 ⊢ 1R = [〈(1P +P 1P), 1P〉] ~R | |
| 3 | 1, 2 | oveq12i 7443 | . 2 ⊢ (-1R +R 1R) = ([〈1P, (1P +P 1P)〉] ~R +R [〈(1P +P 1P), 1P〉] ~R ) |
| 4 | df-0r 11100 | . . 3 ⊢ 0R = [〈1P, 1P〉] ~R | |
| 5 | 1pr 11055 | . . . . 5 ⊢ 1P ∈ P | |
| 6 | addclpr 11058 | . . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P) | |
| 7 | 5, 5, 6 | mp2an 692 | . . . . 5 ⊢ (1P +P 1P) ∈ P |
| 8 | addsrpr 11115 | . . . . 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 693 | . . . 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 11062 | . . . . . 6 ⊢ ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P)) | |
| 11 | 10 | oveq2i 7442 | . . . . 5 ⊢ (1P +P ((1P +P 1P) +P 1P)) = (1P +P (1P +P (1P +P 1P))) |
| 12 | addclpr 11058 | . . . . . . 7 ⊢ ((1P ∈ P ∧ (1P +P 1P) ∈ P) → (1P +P (1P +P 1P)) ∈ P) | |
| 13 | 5, 7, 12 | mp2an 692 | . . . . . 6 ⊢ (1P +P (1P +P 1P)) ∈ P |
| 14 | addclpr 11058 | . . . . . . 7 ⊢ (((1P +P 1P) ∈ P ∧ 1P ∈ P) → ((1P +P 1P) +P 1P) ∈ P) | |
| 15 | 7, 5, 14 | mp2an 692 | . . . . . 6 ⊢ ((1P +P 1P) +P 1P) ∈ P |
| 16 | enreceq 11106 | . . . . . 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 693 | . . . . 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 231 | . . . 4 ⊢ [〈1P, 1P〉] ~R = [〈(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)〉] ~R |
| 19 | 9, 18 | eqtr4i 2768 | . . 3 ⊢ ([〈1P, (1P +P 1P)〉] ~R +R [〈(1P +P 1P), 1P〉] ~R ) = [〈1P, 1P〉] ~R |
| 20 | 4, 19 | eqtr4i 2768 | . 2 ⊢ 0R = ([〈1P, (1P +P 1P)〉] ~R +R [〈(1P +P 1P), 1P〉] ~R ) |
| 21 | 3, 20 | eqtr4i 2768 | 1 ⊢ (-1R +R 1R) = 0R |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 〈cop 4632 (class class class)co 7431 [cec 8743 Pcnp 10899 1Pc1p 10900 +P cpp 10901 ~R cer 10904 0Rc0r 10906 1Rc1r 10907 -1Rcm1r 10908 +R cplr 10909 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-omul 8511 df-er 8745 df-ec 8747 df-qs 8751 df-ni 10912 df-pli 10913 df-mi 10914 df-lti 10915 df-plpq 10948 df-mpq 10949 df-ltpq 10950 df-enq 10951 df-nq 10952 df-erq 10953 df-plq 10954 df-mq 10955 df-1nq 10956 df-rq 10957 df-ltnq 10958 df-np 11021 df-1p 11022 df-plp 11023 df-ltp 11025 df-enr 11095 df-nr 11096 df-plr 11097 df-0r 11100 df-1r 11101 df-m1r 11102 |
| This theorem is referenced by: pn0sr 11141 supsrlem 11151 axi2m1 11199 |
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