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Mirrors > Home > ILE Home > Th. List > mulasspig | Unicode version |
Description: Multiplication of positive integers is associative. (Contributed by Jim Kingdon, 26-Aug-2019.) |
Ref | Expression |
---|---|
mulasspig |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 7141 |
. . 3
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2 | pinn 7141 |
. . 3
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3 | pinn 7141 |
. . 3
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4 | nnmass 6391 |
. . 3
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5 | 1, 2, 3, 4 | syl3an 1259 |
. 2
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6 | mulclpi 7160 |
. . . . 5
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7 | mulpiord 7149 |
. . . . 5
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8 | 6, 7 | sylan 281 |
. . . 4
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9 | mulpiord 7149 |
. . . . . 6
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10 | 9 | oveq1d 5797 |
. . . . 5
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11 | 10 | adantr 274 |
. . . 4
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12 | 8, 11 | eqtrd 2173 |
. . 3
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13 | 12 | 3impa 1177 |
. 2
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14 | mulclpi 7160 |
. . . . 5
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15 | mulpiord 7149 |
. . . . 5
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16 | 14, 15 | sylan2 284 |
. . . 4
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17 | mulpiord 7149 |
. . . . . 6
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18 | 17 | oveq2d 5798 |
. . . . 5
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19 | 18 | adantl 275 |
. . . 4
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20 | 16, 19 | eqtrd 2173 |
. . 3
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21 | 20 | 3impb 1178 |
. 2
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22 | 5, 13, 21 | 3eqtr4d 2183 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-oadd 6325 df-omul 6326 df-ni 7136 df-mi 7138 |
This theorem is referenced by: enqer 7190 addcmpblnq 7199 mulcmpblnq 7200 ordpipqqs 7206 addassnqg 7214 mulassnqg 7216 mulcanenq 7217 distrnqg 7219 ltsonq 7230 ltanqg 7232 ltmnqg 7233 ltexnqq 7240 |
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