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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 6965 |
. . 3
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2 | pinn 6965 |
. . 3
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3 | pinn 6965 |
. . 3
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4 | nnmass 6288 |
. . 3
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5 | 1, 2, 3, 4 | syl3an 1223 |
. 2
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6 | mulclpi 6984 |
. . . . 5
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7 | mulpiord 6973 |
. . . . 5
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8 | 6, 7 | sylan 278 |
. . . 4
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9 | mulpiord 6973 |
. . . . . 6
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10 | 9 | oveq1d 5705 |
. . . . 5
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11 | 10 | adantr 271 |
. . . 4
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12 | 8, 11 | eqtrd 2127 |
. . 3
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13 | 12 | 3impa 1141 |
. 2
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14 | mulclpi 6984 |
. . . . 5
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15 | mulpiord 6973 |
. . . . 5
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16 | 14, 15 | sylan2 281 |
. . . 4
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17 | mulpiord 6973 |
. . . . . 6
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18 | 17 | oveq2d 5706 |
. . . . 5
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19 | 18 | adantl 272 |
. . . 4
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20 | 16, 19 | eqtrd 2127 |
. . 3
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21 | 20 | 3impb 1142 |
. 2
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22 | 5, 13, 21 | 3eqtr4d 2137 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-id 4144 df-iord 4217 df-on 4219 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-recs 6108 df-irdg 6173 df-oadd 6223 df-omul 6224 df-ni 6960 df-mi 6962 |
This theorem is referenced by: enqer 7014 addcmpblnq 7023 mulcmpblnq 7024 ordpipqqs 7030 addassnqg 7038 mulassnqg 7040 mulcanenq 7041 distrnqg 7043 ltsonq 7054 ltanqg 7056 ltmnqg 7057 ltexnqq 7064 |
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