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Mirrors > Home > ILE Home > Th. List > mulcomprg | Unicode version |
Description: Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Ref | Expression |
---|---|
mulcomprg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prop 7437 | . . . . . . . . 9 | |
2 | elprnql 7443 | . . . . . . . . 9 | |
3 | 1, 2 | sylan 281 | . . . . . . . 8 |
4 | prop 7437 | . . . . . . . . . . . . 13 | |
5 | elprnql 7443 | . . . . . . . . . . . . 13 | |
6 | 4, 5 | sylan 281 | . . . . . . . . . . . 12 |
7 | mulcomnqg 7345 | . . . . . . . . . . . . 13 | |
8 | 7 | eqeq2d 2182 | . . . . . . . . . . . 12 |
9 | 6, 8 | sylan2 284 | . . . . . . . . . . 11 |
10 | 9 | anassrs 398 | . . . . . . . . . 10 |
11 | 10 | rexbidva 2467 | . . . . . . . . 9 |
12 | 11 | ancoms 266 | . . . . . . . 8 |
13 | 3, 12 | sylan2 284 | . . . . . . 7 |
14 | 13 | anassrs 398 | . . . . . 6 |
15 | 14 | rexbidva 2467 | . . . . 5 |
16 | rexcom 2634 | . . . . 5 | |
17 | 15, 16 | bitrdi 195 | . . . 4 |
18 | 17 | rabbidv 2719 | . . 3 |
19 | elprnqu 7444 | . . . . . . . . 9 | |
20 | 1, 19 | sylan 281 | . . . . . . . 8 |
21 | elprnqu 7444 | . . . . . . . . . . . . 13 | |
22 | 4, 21 | sylan 281 | . . . . . . . . . . . 12 |
23 | 22, 8 | sylan2 284 | . . . . . . . . . . 11 |
24 | 23 | anassrs 398 | . . . . . . . . . 10 |
25 | 24 | rexbidva 2467 | . . . . . . . . 9 |
26 | 25 | ancoms 266 | . . . . . . . 8 |
27 | 20, 26 | sylan2 284 | . . . . . . 7 |
28 | 27 | anassrs 398 | . . . . . 6 |
29 | 28 | rexbidva 2467 | . . . . 5 |
30 | rexcom 2634 | . . . . 5 | |
31 | 29, 30 | bitrdi 195 | . . . 4 |
32 | 31 | rabbidv 2719 | . . 3 |
33 | 18, 32 | opeq12d 3773 | . 2 |
34 | mpvlu 7501 | . . 3 | |
35 | 34 | ancoms 266 | . 2 |
36 | mpvlu 7501 | . 2 | |
37 | 33, 35, 36 | 3eqtr4rd 2214 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wrex 2449 crab 2452 cop 3586 cfv 5198 (class class class)co 5853 c1st 6117 c2nd 6118 cnq 7242 cmq 7245 cnp 7253 cmp 7256 |
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 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
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 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-oadd 6399 df-omul 6400 df-er 6513 df-ec 6515 df-qs 6519 df-ni 7266 df-mi 7268 df-mpq 7307 df-enq 7309 df-nqqs 7310 df-mqqs 7312 df-inp 7428 df-imp 7431 |
This theorem is referenced by: ltmprr 7604 mulcmpblnrlemg 7702 mulcomsrg 7719 mulasssrg 7720 m1m1sr 7723 recexgt0sr 7735 mulgt0sr 7740 mulextsr1lem 7742 recidpirqlemcalc 7819 |
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