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Mirrors > Home > ILE Home > Th. List > decmul10add | Unicode version |
Description: A multiplication of a number and a numeral expressed as addition with first summand as multiple of 10. (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
decmul10add.1 |
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decmul10add.2 |
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decmul10add.3 |
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decmul10add.4 |
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decmul10add.5 |
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Ref | Expression |
---|---|
decmul10add |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdec10 9422 |
. . 3
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2 | 1 | oveq2i 5911 |
. 2
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3 | decmul10add.3 |
. . . 4
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4 | 3 | nn0cni 9223 |
. . 3
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5 | 10nn0 9436 |
. . . . 5
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6 | decmul10add.1 |
. . . . 5
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7 | 5, 6 | nn0mulcli 9249 |
. . . 4
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8 | 7 | nn0cni 9223 |
. . 3
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9 | decmul10add.2 |
. . . 4
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10 | 9 | nn0cni 9223 |
. . 3
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11 | 4, 8, 10 | adddii 8002 |
. 2
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12 | 5 | nn0cni 9223 |
. . . . 5
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13 | 6 | nn0cni 9223 |
. . . . 5
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14 | 4, 12, 13 | mul12i 8138 |
. . . 4
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15 | 3, 6 | nn0mulcli 9249 |
. . . . 5
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16 | 15 | dec0u 9439 |
. . . 4
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17 | decmul10add.4 |
. . . . . 6
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18 | 17 | eqcomi 2193 |
. . . . 5
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19 | 18 | deceq1i 9425 |
. . . 4
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20 | 14, 16, 19 | 3eqtri 2214 |
. . 3
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21 | decmul10add.5 |
. . . 4
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22 | 21 | eqcomi 2193 |
. . 3
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23 | 20, 22 | oveq12i 5912 |
. 2
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24 | 2, 11, 23 | 3eqtri 2214 |
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 4139 ax-pow 4195 ax-pr 4230 ax-setind 4557 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-addcom 7946 ax-mulcom 7947 ax-addass 7948 ax-mulass 7949 ax-distr 7950 ax-i2m1 7951 ax-1rid 7953 ax-0id 7954 ax-rnegex 7955 ax-cnre 7957 |
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 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-br 4022 df-opab 4083 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-iota 5199 df-fun 5240 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-sub 8165 df-inn 8955 df-2 9013 df-3 9014 df-4 9015 df-5 9016 df-6 9017 df-7 9018 df-8 9019 df-9 9020 df-n0 9212 df-dec 9420 |
This theorem is referenced by: (None) |
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