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Mirrors > Home > MPE Home > Th. List > 0idsr | Structured version Visualization version Unicode version |
Description: The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0idsr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 9878 | . 2 | |
2 | oveq1 6657 | . . 3 | |
3 | id 22 | . . 3 | |
4 | 2, 3 | eqeq12d 2637 | . 2 |
5 | df-0r 9882 | . . . 4 | |
6 | 5 | oveq2i 6661 | . . 3 |
7 | 1pr 9837 | . . . . 5 | |
8 | addsrpr 9896 | . . . . 5 | |
9 | 7, 7, 8 | mpanr12 721 | . . . 4 |
10 | addclpr 9840 | . . . . . . 7 | |
11 | 7, 10 | mpan2 707 | . . . . . 6 |
12 | addclpr 9840 | . . . . . . 7 | |
13 | 7, 12 | mpan2 707 | . . . . . 6 |
14 | 11, 13 | anim12i 590 | . . . . 5 |
15 | vex 3203 | . . . . . . 7 | |
16 | vex 3203 | . . . . . . 7 | |
17 | 7 | elexi 3213 | . . . . . . 7 |
18 | addcompr 9843 | . . . . . . 7 | |
19 | addasspr 9844 | . . . . . . 7 | |
20 | 15, 16, 17, 18, 19 | caov12 6862 | . . . . . 6 |
21 | enreceq 9887 | . . . . . 6 | |
22 | 20, 21 | mpbiri 248 | . . . . 5 |
23 | 14, 22 | mpdan 702 | . . . 4 |
24 | 9, 23 | eqtr4d 2659 | . . 3 |
25 | 6, 24 | syl5eq 2668 | . 2 |
26 | 1, 4, 25 | ecoptocl 7837 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cop 4183 (class class class)co 6650 cec 7740 cnp 9681 c1p 9682 cpp 9683 cer 9686 cnr 9687 c0r 9688 cplr 9691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-ec 7744 df-qs 7748 df-ni 9694 df-pli 9695 df-mi 9696 df-lti 9697 df-plpq 9730 df-mpq 9731 df-ltpq 9732 df-enq 9733 df-nq 9734 df-erq 9735 df-plq 9736 df-mq 9737 df-1nq 9738 df-rq 9739 df-ltnq 9740 df-np 9803 df-1p 9804 df-plp 9805 df-ltp 9807 df-enr 9877 df-nr 9878 df-plr 9879 df-0r 9882 |
This theorem is referenced by: addgt0sr 9925 sqgt0sr 9927 map2psrpr 9931 supsrlem 9932 addresr 9959 mulresr 9960 axi2m1 9980 axcnre 9985 |
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