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Mirrors > Home > ILE Home > Th. List > 3dec | Unicode version |
Description: A "decimal constructor" which is used to build up "decimal integers" or "numeric terms" in base 10 with 3 "digits". (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
Ref | Expression |
---|---|
3dec.a |
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3dec.b |
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Ref | Expression |
---|---|
3dec |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdec10 9418 |
. 2
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2 | dfdec10 9418 |
. . . . . 6
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3 | 2 | oveq2i 5908 |
. . . . 5
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4 | 1nn 8961 |
. . . . . . . 8
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5 | 4 | decnncl2 9438 |
. . . . . . 7
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6 | 5 | nncni 8960 |
. . . . . 6
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7 | 3dec.a |
. . . . . . . 8
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8 | 7 | nn0cni 9219 |
. . . . . . 7
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9 | 6, 8 | mulcli 7993 |
. . . . . 6
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10 | 3dec.b |
. . . . . . 7
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11 | 10 | nn0cni 9219 |
. . . . . 6
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12 | 6, 9, 11 | adddii 7998 |
. . . . 5
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13 | 3, 12 | eqtri 2210 |
. . . 4
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14 | 6, 6, 8 | mulassi 7997 |
. . . . . . 7
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15 | 14 | eqcomi 2193 |
. . . . . 6
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16 | 6 | sqvali 10634 |
. . . . . . . 8
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17 | 16 | eqcomi 2193 |
. . . . . . 7
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18 | 17 | oveq1i 5907 |
. . . . . 6
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19 | 15, 18 | eqtri 2210 |
. . . . 5
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20 | 19 | oveq1i 5907 |
. . . 4
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21 | 13, 20 | eqtri 2210 |
. . 3
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22 | 21 | oveq1i 5907 |
. 2
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23 | 1, 22 | 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-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 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-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 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-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-frec 6417 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 df-inn 8951 df-2 9009 df-3 9010 df-4 9011 df-5 9012 df-6 9013 df-7 9014 df-8 9015 df-9 9016 df-n0 9208 df-z 9285 df-dec 9416 df-uz 9560 df-seqfrec 10479 df-exp 10554 |
This theorem is referenced by: 3dvds2dec 11906 |
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