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Mirrors > Home > ILE Home > Th. List > ex-fac | Unicode version |
Description: Example for df-fac 9967. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-fac |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-5 8376 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | 1 | fveq2i 5254 |
. . 3
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3 | 4nn0 8582 |
. . . 4
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4 | facp1 9971 |
. . . 4
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5 | 3, 4 | ax-mp 7 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | 2, 5 | eqtri 2103 |
. 2
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7 | fac4 9974 |
. . . 4
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8 | 4p1e5 8443 |
. . . 4
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9 | 7, 8 | oveq12i 5601 |
. . 3
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10 | 5nn0 8583 |
. . . 4
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11 | 2nn0 8580 |
. . . 4
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12 | eqid 2083 |
. . . 4
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13 | 0nn0 8578 |
. . . 4
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14 | 1nn0 8579 |
. . . . 5
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15 | 5cn 8394 |
. . . . . 6
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16 | 2cn 8385 |
. . . . . 6
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17 | 5t2e10 8869 |
. . . . . 6
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18 | 15, 16, 17 | mulcomli 7396 |
. . . . 5
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19 | 16 | addid2i 7526 |
. . . . 5
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20 | 14, 13, 11, 18, 19 | decaddi 8829 |
. . . 4
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21 | 4cn 8392 |
. . . . 5
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22 | 5t4e20 8871 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | 15, 21, 22 | mulcomli 7396 |
. . . 4
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24 | 10, 11, 3, 12, 13, 11, 20, 23 | decmul1c 8834 |
. . 3
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25 | 9, 24 | eqtri 2103 |
. 2
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26 | 6, 25 | eqtri 2103 |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-iinf 4365 ax-cnex 7337 ax-resscn 7338 ax-1cn 7339 ax-1re 7340 ax-icn 7341 ax-addcl 7342 ax-addrcl 7343 ax-mulcl 7344 ax-addcom 7346 ax-mulcom 7347 ax-addass 7348 ax-mulass 7349 ax-distr 7350 ax-i2m1 7351 ax-0lt1 7352 ax-1rid 7353 ax-0id 7354 ax-rnegex 7355 ax-cnre 7357 ax-pre-ltirr 7358 ax-pre-ltwlin 7359 ax-pre-lttrn 7360 ax-pre-ltadd 7362 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4083 df-iord 4156 df-on 4158 df-ilim 4159 df-suc 4161 df-iom 4368 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-rn 4410 df-res 4411 df-ima 4412 df-iota 4932 df-fun 4969 df-fn 4970 df-f 4971 df-f1 4972 df-fo 4973 df-f1o 4974 df-fv 4975 df-riota 5545 df-ov 5592 df-oprab 5593 df-mpt2 5594 df-1st 5844 df-2nd 5845 df-recs 6000 df-frec 6086 df-pnf 7425 df-mnf 7426 df-xr 7427 df-ltxr 7428 df-le 7429 df-sub 7556 df-neg 7557 df-inn 8315 df-2 8373 df-3 8374 df-4 8375 df-5 8376 df-6 8377 df-7 8378 df-8 8379 df-9 8380 df-n0 8564 df-z 8645 df-dec 8771 df-uz 8913 df-iseq 9739 df-fac 9967 |
This theorem is referenced by: (None) |
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