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Mirrors > Home > ILE Home > Th. List > nummul1c | Unicode version |
Description: The product of a decimal integer with a number. (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
nummul1c.1 |
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nummul1c.2 |
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nummul1c.3 |
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nummul1c.4 |
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nummul1c.5 |
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nummul1c.6 |
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nummul1c.7 |
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nummul1c.8 |
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nummul1c.9 |
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Ref | Expression |
---|---|
nummul1c |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nummul1c.5 |
. . . 4
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2 | nummul1c.1 |
. . . . 5
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3 | nummul1c.3 |
. . . . 5
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4 | nummul1c.4 |
. . . . 5
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5 | 2, 3, 4 | numcl 9427 |
. . . 4
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6 | 1, 5 | eqeltri 2262 |
. . 3
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7 | nummul1c.2 |
. . 3
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8 | 6, 7 | num0u 9425 |
. 2
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9 | 0nn0 9222 |
. . 3
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10 | 2, 9 | num0h 9426 |
. . 3
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11 | nummul1c.6 |
. . 3
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12 | nummul1c.7 |
. . 3
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13 | 12 | nn0cni 9219 |
. . . . . 6
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14 | 13 | addid2i 8131 |
. . . . 5
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15 | 14 | oveq2i 5908 |
. . . 4
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16 | nummul1c.8 |
. . . 4
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17 | 15, 16 | eqtri 2210 |
. . 3
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18 | 4, 7 | num0u 9425 |
. . . 4
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19 | nummul1c.9 |
. . . 4
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20 | 18, 19 | eqtr3i 2212 |
. . 3
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21 | 2, 3, 4, 9, 9, 1, 10, 7, 11, 12, 17, 20 | nummac 9459 |
. 2
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22 | 8, 21 | eqtri 2210 |
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 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-cnre 7953 |
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 5900 df-oprab 5901 df-mpo 5902 df-sub 8161 df-inn 8951 df-n0 9208 |
This theorem is referenced by: nummul2c 9464 decmul1 9478 decmul1c 9479 |
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