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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 8950 |
. . . 4
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6 | 1, 5 | eqeltri 2161 |
. . 3
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7 | nummul1c.2 |
. . 3
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8 | 6, 7 | num0u 8948 |
. 2
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9 | 0nn0 8749 |
. . 3
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10 | 2, 9 | num0h 8949 |
. . 3
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11 | nummul1c.6 |
. . 3
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12 | nummul1c.7 |
. . 3
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13 | 12 | nn0cni 8746 |
. . . . . 6
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14 | 13 | addid2i 7686 |
. . . . 5
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15 | 14 | oveq2i 5677 |
. . . 4
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16 | nummul1c.8 |
. . . 4
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17 | 15, 16 | eqtri 2109 |
. . 3
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18 | 4, 7 | num0u 8948 |
. . . 4
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19 | nummul1c.9 |
. . . 4
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20 | 18, 19 | eqtr3i 2111 |
. . 3
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21 | 2, 3, 4, 9, 9, 1, 10, 7, 11, 12, 17, 20 | nummac 8982 |
. 2
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22 | 8, 21 | eqtri 2109 |
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 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-addcom 7506 ax-mulcom 7507 ax-addass 7508 ax-mulass 7509 ax-distr 7510 ax-i2m1 7511 ax-1rid 7513 ax-0id 7514 ax-rnegex 7515 ax-cnre 7517 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-opab 3906 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-sub 7716 df-inn 8484 df-n0 8735 |
This theorem is referenced by: nummul2c 8987 decmul1 9001 decmul1c 9002 |
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