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Mirrors > Home > MPE Home > Th. List > mulclsr | Structured version Visualization version Unicode version |
Description: Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulclsr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 9878 | . . 3 | |
2 | oveq1 6657 | . . . 4 | |
3 | 2 | eleq1d 2686 | . . 3 |
4 | oveq2 6658 | . . . 4 | |
5 | 4 | eleq1d 2686 | . . 3 |
6 | mulsrpr 9897 | . . . 4 | |
7 | mulclpr 9842 | . . . . . . . 8 | |
8 | mulclpr 9842 | . . . . . . . 8 | |
9 | addclpr 9840 | . . . . . . . 8 | |
10 | 7, 8, 9 | syl2an 494 | . . . . . . 7 |
11 | 10 | an4s 869 | . . . . . 6 |
12 | mulclpr 9842 | . . . . . . . 8 | |
13 | mulclpr 9842 | . . . . . . . 8 | |
14 | addclpr 9840 | . . . . . . . 8 | |
15 | 12, 13, 14 | syl2an 494 | . . . . . . 7 |
16 | 15 | an42s 870 | . . . . . 6 |
17 | 11, 16 | jca 554 | . . . . 5 |
18 | opelxpi 5148 | . . . . 5 | |
19 | enrex 9888 | . . . . . 6 | |
20 | 19 | ecelqsi 7803 | . . . . 5 |
21 | 17, 18, 20 | 3syl 18 | . . . 4 |
22 | 6, 21 | eqeltrd 2701 | . . 3 |
23 | 1, 3, 5, 22 | 2ecoptocl 7838 | . 2 |
24 | 23, 1 | syl6eleqr 2712 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cop 4183 cxp 5112 (class class class)co 6650 cec 7740 cqs 7741 cnp 9681 cpp 9683 cmp 9684 cer 9686 cnr 9687 cmr 9692 |
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-plp 9805 df-mp 9806 df-ltp 9807 df-enr 9877 df-nr 9878 df-mr 9880 |
This theorem is referenced by: dmmulsr 9907 negexsr 9923 sqgt0sr 9927 recexsr 9928 map2psrpr 9931 mulresr 9960 axmulf 9967 axmulrcl 9975 axmulass 9978 axdistr 9979 axrnegex 9983 |
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