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Mirrors > Home > ILE Home > Th. List > mulsrpr | Unicode version |
Description: Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
Ref | Expression |
---|---|
mulsrpr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 4630 | . . . 4 | |
2 | enrex 7669 | . . . . 5 | |
3 | 2 | ecelqsi 6546 | . . . 4 |
4 | 1, 3 | syl 14 | . . 3 |
5 | opelxpi 4630 | . . . 4 | |
6 | 2 | ecelqsi 6546 | . . . 4 |
7 | 5, 6 | syl 14 | . . 3 |
8 | 4, 7 | anim12i 336 | . 2 |
9 | eqid 2164 | . . . 4 | |
10 | eqid 2164 | . . . 4 | |
11 | 9, 10 | pm3.2i 270 | . . 3 |
12 | eqid 2164 | . . 3 | |
13 | opeq12 3754 | . . . . . . . . 9 | |
14 | 13 | eceq1d 6528 | . . . . . . . 8 |
15 | 14 | eqeq2d 2176 | . . . . . . 7 |
16 | 15 | anbi1d 461 | . . . . . 6 |
17 | simpl 108 | . . . . . . . . . . 11 | |
18 | 17 | oveq1d 5851 | . . . . . . . . . 10 |
19 | simpr 109 | . . . . . . . . . . 11 | |
20 | 19 | oveq1d 5851 | . . . . . . . . . 10 |
21 | 18, 20 | oveq12d 5854 | . . . . . . . . 9 |
22 | 17 | oveq1d 5851 | . . . . . . . . . 10 |
23 | 19 | oveq1d 5851 | . . . . . . . . . 10 |
24 | 22, 23 | oveq12d 5854 | . . . . . . . . 9 |
25 | 21, 24 | opeq12d 3760 | . . . . . . . 8 |
26 | 25 | eceq1d 6528 | . . . . . . 7 |
27 | 26 | eqeq2d 2176 | . . . . . 6 |
28 | 16, 27 | anbi12d 465 | . . . . 5 |
29 | 28 | spc2egv 2811 | . . . 4 |
30 | opeq12 3754 | . . . . . . . . . 10 | |
31 | 30 | eceq1d 6528 | . . . . . . . . 9 |
32 | 31 | eqeq2d 2176 | . . . . . . . 8 |
33 | 32 | anbi2d 460 | . . . . . . 7 |
34 | simpl 108 | . . . . . . . . . . . 12 | |
35 | 34 | oveq2d 5852 | . . . . . . . . . . 11 |
36 | simpr 109 | . . . . . . . . . . . 12 | |
37 | 36 | oveq2d 5852 | . . . . . . . . . . 11 |
38 | 35, 37 | oveq12d 5854 | . . . . . . . . . 10 |
39 | 36 | oveq2d 5852 | . . . . . . . . . . 11 |
40 | 34 | oveq2d 5852 | . . . . . . . . . . 11 |
41 | 39, 40 | oveq12d 5854 | . . . . . . . . . 10 |
42 | 38, 41 | opeq12d 3760 | . . . . . . . . 9 |
43 | 42 | eceq1d 6528 | . . . . . . . 8 |
44 | 43 | eqeq2d 2176 | . . . . . . 7 |
45 | 33, 44 | anbi12d 465 | . . . . . 6 |
46 | 45 | spc2egv 2811 | . . . . 5 |
47 | 46 | 2eximdv 1869 | . . . 4 |
48 | 29, 47 | sylan9 407 | . . 3 |
49 | 11, 12, 48 | mp2ani 429 | . 2 |
50 | ecexg 6496 | . . . 4 | |
51 | 2, 50 | ax-mp 5 | . . 3 |
52 | simp1 986 | . . . . . . . 8 | |
53 | 52 | eqeq1d 2173 | . . . . . . 7 |
54 | simp2 987 | . . . . . . . 8 | |
55 | 54 | eqeq1d 2173 | . . . . . . 7 |
56 | 53, 55 | anbi12d 465 | . . . . . 6 |
57 | simp3 988 | . . . . . . 7 | |
58 | 57 | eqeq1d 2173 | . . . . . 6 |
59 | 56, 58 | anbi12d 465 | . . . . 5 |
60 | 59 | 4exbidv 1857 | . . . 4 |
61 | mulsrmo 7676 | . . . 4 | |
62 | df-mr 7661 | . . . . 5 | |
63 | df-nr 7659 | . . . . . . . . 9 | |
64 | 63 | eleq2i 2231 | . . . . . . . 8 |
65 | 63 | eleq2i 2231 | . . . . . . . 8 |
66 | 64, 65 | anbi12i 456 | . . . . . . 7 |
67 | 66 | anbi1i 454 | . . . . . 6 |
68 | 67 | oprabbii 5888 | . . . . 5 |
69 | 62, 68 | eqtri 2185 | . . . 4 |
70 | 60, 61, 69 | ovig 5954 | . . 3 |
71 | 51, 70 | mp3an3 1315 | . 2 |
72 | 8, 49, 71 | sylc 62 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 967 wceq 1342 wex 1479 wcel 2135 cvv 2721 cop 3573 cxp 4596 (class class class)co 5836 coprab 5837 cec 6490 cqs 6491 cnp 7223 cpp 7225 cmp 7226 cer 7228 cnr 7229 cmr 7234 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-2o 6376 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 df-enq0 7356 df-nq0 7357 df-0nq0 7358 df-plq0 7359 df-mq0 7360 df-inp 7398 df-iplp 7400 df-imp 7401 df-enr 7658 df-nr 7659 df-mr 7661 |
This theorem is referenced by: mulclsr 7686 mulcomsrg 7689 mulasssrg 7690 distrsrg 7691 m1m1sr 7693 1idsr 7700 00sr 7701 recexgt0sr 7705 mulgt0sr 7710 mulextsr1 7713 recidpirq 7790 |
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