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Mirrors > Home > MPE Home > Th. List > decmul2c | Structured version Unicode version |
Description: The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) |
Ref | Expression |
---|---|
decmul1c.1 |
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decmul1c.2 |
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decmul1c.3 |
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decmul1c.4 |
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decmul1c.5 |
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decmul1c.6 |
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decmul2c.7 |
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decmul2c.8 |
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Ref | Expression |
---|---|
decmul2c |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 10719 |
. . 3
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2 | decmul1c.1 |
. . 3
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3 | decmul1c.2 |
. . 3
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4 | decmul1c.3 |
. . 3
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5 | decmul1c.4 |
. . . 4
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6 | df-dec 10871 |
. . . 4
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7 | 5, 6 | eqtri 2483 |
. . 3
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8 | decmul1c.5 |
. . 3
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9 | decmul1c.6 |
. . 3
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10 | decmul2c.7 |
. . 3
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11 | decmul2c.8 |
. . . 4
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12 | df-dec 10871 |
. . . 4
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13 | 11, 12 | eqtri 2483 |
. . 3
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14 | 1, 2, 3, 4, 7, 8, 9, 10, 13 | nummul2c 10907 |
. 2
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15 | df-dec 10871 |
. 2
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16 | 14, 15 | eqtr4i 2486 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 ax-resscn 9454 ax-1cn 9455 ax-icn 9456 ax-addcl 9457 ax-addrcl 9458 ax-mulcl 9459 ax-mulrcl 9460 ax-mulcom 9461 ax-addass 9462 ax-mulass 9463 ax-distr 9464 ax-i2m1 9465 ax-1ne0 9466 ax-1rid 9467 ax-rnegex 9468 ax-rrecex 9469 ax-cnre 9470 ax-pre-lttri 9471 ax-pre-lttrn 9472 ax-pre-ltadd 9473 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-reu 2806 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-pss 3455 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-tp 3993 df-op 3995 df-uni 4203 df-iun 4284 df-br 4404 df-opab 4462 df-mpt 4463 df-tr 4497 df-eprel 4743 df-id 4747 df-po 4752 df-so 4753 df-fr 4790 df-we 4792 df-ord 4833 df-on 4834 df-lim 4835 df-suc 4836 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-riota 6164 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-om 6590 df-recs 6945 df-rdg 6979 df-er 7214 df-en 7424 df-dom 7425 df-sdom 7426 df-pnf 9535 df-mnf 9536 df-ltxr 9538 df-sub 9712 df-nn 10438 df-2 10495 df-3 10496 df-4 10497 df-5 10498 df-6 10499 df-7 10500 df-8 10501 df-9 10502 df-10 10503 df-n0 10695 df-dec 10871 |
This theorem is referenced by: 2exp8 14238 2exp16 14239 prmlem2 14269 37prm 14270 1259lem2 14278 1259lem3 14279 1259lem4 14280 1259prm 14282 2503lem1 14283 2503lem2 14284 2503prm 14286 4001lem1 14287 4001lem2 14288 4001lem3 14289 4001prm 14291 log2ublem3 22486 log2ub 22487 birthday 22491 |
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