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Mirrors > Home > ILE Home > Th. List > mulasssrg | Unicode version |
Description: Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.) |
Ref | Expression |
---|---|
mulasssrg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 7676 | . 2 | |
2 | mulsrpr 7695 | . 2 | |
3 | mulsrpr 7695 | . 2 | |
4 | mulsrpr 7695 | . 2 | |
5 | mulsrpr 7695 | . 2 | |
6 | mulclpr 7521 | . . . . 5 | |
7 | 6 | ad2ant2r 506 | . . . 4 |
8 | mulclpr 7521 | . . . . 5 | |
9 | 8 | ad2ant2l 505 | . . . 4 |
10 | addclpr 7486 | . . . 4 | |
11 | 7, 9, 10 | syl2anc 409 | . . 3 |
12 | mulclpr 7521 | . . . . 5 | |
13 | 12 | ad2ant2rl 508 | . . . 4 |
14 | mulclpr 7521 | . . . . 5 | |
15 | 14 | ad2ant2lr 507 | . . . 4 |
16 | addclpr 7486 | . . . 4 | |
17 | 13, 15, 16 | syl2anc 409 | . . 3 |
18 | 11, 17 | jca 304 | . 2 |
19 | mulclpr 7521 | . . . . 5 | |
20 | 19 | ad2ant2r 506 | . . . 4 |
21 | mulclpr 7521 | . . . . 5 | |
22 | 21 | ad2ant2l 505 | . . . 4 |
23 | addclpr 7486 | . . . 4 | |
24 | 20, 22, 23 | syl2anc 409 | . . 3 |
25 | mulclpr 7521 | . . . . 5 | |
26 | 25 | ad2ant2rl 508 | . . . 4 |
27 | mulclpr 7521 | . . . . 5 | |
28 | 27 | ad2ant2lr 507 | . . . 4 |
29 | addclpr 7486 | . . . 4 | |
30 | 26, 28, 29 | syl2anc 409 | . . 3 |
31 | 24, 30 | jca 304 | . 2 |
32 | mulcomprg 7529 | . . . 4 | |
33 | 32 | adantl 275 | . . 3 |
34 | distrprg 7537 | . . . . . 6 | |
35 | 34 | adantl 275 | . . . . 5 |
36 | simp1 992 | . . . . 5 | |
37 | simp2 993 | . . . . 5 | |
38 | simp3 994 | . . . . 5 | |
39 | addclpr 7486 | . . . . . 6 | |
40 | 39 | adantl 275 | . . . . 5 |
41 | mulcomprg 7529 | . . . . . 6 | |
42 | 41 | adantl 275 | . . . . 5 |
43 | 35, 36, 37, 38, 40, 42 | caovdir2d 6026 | . . . 4 |
44 | 43 | adantl 275 | . . 3 |
45 | mulassprg 7530 | . . . 4 | |
46 | 45 | adantl 275 | . . 3 |
47 | mulclpr 7521 | . . . 4 | |
48 | 47 | adantl 275 | . . 3 |
49 | simp1l 1016 | . . 3 | |
50 | simp1r 1017 | . . 3 | |
51 | simp2l 1018 | . . 3 | |
52 | simp2r 1019 | . . 3 | |
53 | simp3l 1020 | . . 3 | |
54 | simp3r 1021 | . . 3 | |
55 | addcomprg 7527 | . . . 4 | |
56 | 55 | adantl 275 | . . 3 |
57 | addassprg 7528 | . . . 4 | |
58 | 57 | adantl 275 | . . 3 |
59 | addclpr 7486 | . . . 4 | |
60 | 59 | adantl 275 | . . 3 |
61 | 33, 44, 46, 48, 49, 50, 51, 52, 53, 54, 56, 58, 60 | caovlem2d 6042 | . 2 |
62 | 33, 44, 46, 48, 49, 50, 51, 52, 54, 53, 56, 58, 60 | caovlem2d 6042 | . 2 |
63 | 1, 2, 3, 4, 5, 18, 31, 61, 62 | ecoviass 6619 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 973 wceq 1348 wcel 2141 (class class class)co 5850 cnp 7240 cpp 7242 cmp 7243 cer 7245 cnr 7246 cmr 7251 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-eprel 4272 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-recs 6281 df-irdg 6346 df-1o 6392 df-2o 6393 df-oadd 6396 df-omul 6397 df-er 6509 df-ec 6511 df-qs 6515 df-ni 7253 df-pli 7254 df-mi 7255 df-lti 7256 df-plpq 7293 df-mpq 7294 df-enq 7296 df-nqqs 7297 df-plqqs 7298 df-mqqs 7299 df-1nqqs 7300 df-rq 7301 df-ltnqqs 7302 df-enq0 7373 df-nq0 7374 df-0nq0 7375 df-plq0 7376 df-mq0 7377 df-inp 7415 df-iplp 7417 df-imp 7418 df-enr 7675 df-nr 7676 df-mr 7678 |
This theorem is referenced by: axmulass 7822 |
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