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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 11087 | . . 3 ⊢ -1R = [〈1P, (1P +P 1P)〉] ~R | |
2 | df-1r 11086 | . . 3 ⊢ 1R = [〈(1P +P 1P), 1P〉] ~R | |
3 | 1, 2 | oveq12i 7431 | . 2 ⊢ (-1R +R 1R) = ([〈1P, (1P +P 1P)〉] ~R +R [〈(1P +P 1P), 1P〉] ~R ) |
4 | df-0r 11085 | . . 3 ⊢ 0R = [〈1P, 1P〉] ~R | |
5 | 1pr 11040 | . . . . 5 ⊢ 1P ∈ P | |
6 | addclpr 11043 | . . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P) | |
7 | 5, 5, 6 | mp2an 690 | . . . . 5 ⊢ (1P +P 1P) ∈ P |
8 | addsrpr 11100 | . . . . 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 11047 | . . . . . 6 ⊢ ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P)) | |
11 | 10 | oveq2i 7430 | . . . . 5 ⊢ (1P +P ((1P +P 1P) +P 1P)) = (1P +P (1P +P (1P +P 1P))) |
12 | addclpr 11043 | . . . . . . 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 11043 | . . . . . . 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 11091 | . . . . . 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 230 | . . . 4 ⊢ [〈1P, 1P〉] ~R = [〈(1P +P (1P +P 1P)), ((1P +P 1P) +P 1P)〉] ~R |
19 | 9, 18 | eqtr4i 2756 | . . 3 ⊢ ([〈1P, (1P +P 1P)〉] ~R +R [〈(1P +P 1P), 1P〉] ~R ) = [〈1P, 1P〉] ~R |
20 | 4, 19 | eqtr4i 2756 | . 2 ⊢ 0R = ([〈1P, (1P +P 1P)〉] ~R +R [〈(1P +P 1P), 1P〉] ~R ) |
21 | 3, 20 | eqtr4i 2756 | 1 ⊢ (-1R +R 1R) = 0R |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∈ wcel 2098 〈cop 4636 (class class class)co 7419 [cec 8723 Pcnp 10884 1Pc1p 10885 +P cpp 10886 ~R cer 10889 0Rc0r 10891 1Rc1r 10892 -1Rcm1r 10893 +R cplr 10894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-omul 8492 df-er 8725 df-ec 8727 df-qs 8731 df-ni 10897 df-pli 10898 df-mi 10899 df-lti 10900 df-plpq 10933 df-mpq 10934 df-ltpq 10935 df-enq 10936 df-nq 10937 df-erq 10938 df-plq 10939 df-mq 10940 df-1nq 10941 df-rq 10942 df-ltnq 10943 df-np 11006 df-1p 11007 df-plp 11008 df-ltp 11010 df-enr 11080 df-nr 11081 df-plr 11082 df-0r 11085 df-1r 11086 df-m1r 11087 |
This theorem is referenced by: pn0sr 11126 supsrlem 11136 axi2m1 11184 |
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