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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 11091 | . . 3 ⊢ -1R = [⟨1P, (1P +P 1P)⟩] ~R | |
2 | df-1r 11090 | . . 3 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R | |
3 | 1, 2 | oveq12i 7436 | . 2 ⊢ (-1R +R 1R) = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) |
4 | df-0r 11089 | . . 3 ⊢ 0R = [⟨1P, 1P⟩] ~R | |
5 | 1pr 11044 | . . . . 5 ⊢ 1P ∈ P | |
6 | addclpr 11047 | . . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P) | |
7 | 5, 5, 6 | mp2an 690 | . . . . 5 ⊢ (1P +P 1P) ∈ P |
8 | addsrpr 11104 | . . . . 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 11051 | . . . . . 6 ⊢ ((1P +P 1P) +P 1P) = (1P +P (1P +P 1P)) | |
11 | 10 | oveq2i 7435 | . . . . 5 ⊢ (1P +P ((1P +P 1P) +P 1P)) = (1P +P (1P +P (1P +P 1P))) |
12 | addclpr 11047 | . . . . . . 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 11047 | . . . . . . 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 11095 | . . . . . 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 2758 | . . 3 ⊢ ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨1P, 1P⟩] ~R |
20 | 4, 19 | eqtr4i 2758 | . 2 ⊢ 0R = ([⟨1P, (1P +P 1P)⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) |
21 | 3, 20 | eqtr4i 2758 | 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 7424 [cec 8727 Pcnp 10888 1Pc1p 10889 +P cpp 10890 ~R cer 10893 0Rc0r 10895 1Rc1r 10896 -1Rcm1r 10897 +R cplr 10898 |
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 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-inf2 9670 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-oadd 8495 df-omul 8496 df-er 8729 df-ec 8731 df-qs 8735 df-ni 10901 df-pli 10902 df-mi 10903 df-lti 10904 df-plpq 10937 df-mpq 10938 df-ltpq 10939 df-enq 10940 df-nq 10941 df-erq 10942 df-plq 10943 df-mq 10944 df-1nq 10945 df-rq 10946 df-ltnq 10947 df-np 11010 df-1p 11011 df-plp 11012 df-ltp 11014 df-enr 11084 df-nr 11085 df-plr 11086 df-0r 11089 df-1r 11090 df-m1r 11091 |
This theorem is referenced by: pn0sr 11130 supsrlem 11140 axi2m1 11188 |
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