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Mirrors > Home > ILE Home > Th. List > decmul1 | Unicode version |
Description: The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
decmul1.p |
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decmul1.a |
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decmul1.b |
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decmul1.n |
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decmul1.0 |
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decmul1.c |
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decmul1.d |
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Ref | Expression |
---|---|
decmul1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 9431 |
. . 3
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2 | decmul1.p |
. . 3
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3 | decmul1.a |
. . 3
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4 | decmul1.b |
. . 3
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5 | decmul1.n |
. . . 4
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6 | dfdec10 9417 |
. . . 4
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7 | 5, 6 | eqtri 2210 |
. . 3
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8 | decmul1.0 |
. . 3
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9 | 0nn0 9221 |
. . 3
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10 | 3, 2 | nn0mulcli 9244 |
. . . . . 6
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11 | 10 | nn0cni 9218 |
. . . . 5
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12 | 11 | addid1i 8129 |
. . . 4
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13 | decmul1.c |
. . . 4
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14 | 12, 13 | eqtri 2210 |
. . 3
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15 | decmul1.d |
. . . . 5
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16 | 15 | oveq2i 5907 |
. . . 4
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17 | 4, 2 | nn0mulcli 9244 |
. . . . . 6
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18 | 17 | nn0cni 9218 |
. . . . 5
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19 | 18 | addid2i 8130 |
. . . 4
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20 | 1 | nn0cni 9218 |
. . . . . . 7
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21 | 20 | mul01i 8378 |
. . . . . 6
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22 | 21 | eqcomi 2193 |
. . . . 5
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23 | 22 | oveq1i 5906 |
. . . 4
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24 | 16, 19, 23 | 3eqtr3i 2218 |
. . 3
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25 | 1, 2, 3, 4, 7, 8, 9, 14, 24 | nummul1c 9462 |
. 2
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26 | dfdec10 9417 |
. 2
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27 | 25, 26 | eqtr4i 2213 |
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 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-cnre 7952 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-sub 8160 df-inn 8950 df-2 9008 df-3 9009 df-4 9010 df-5 9011 df-6 9012 df-7 9013 df-8 9014 df-9 9015 df-n0 9207 df-dec 9415 |
This theorem is referenced by: sq10 10724 |
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