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Mirrors > Home > MPE Home > Th. List > pcval | Structured version Visualization version Unicode version |
Description: The value of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 3-Oct-2014.) |
Ref | Expression |
---|---|
pcval.1 |
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pcval.2 |
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Ref | Expression |
---|---|
pcval |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 463 |
. . . . . 6
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2 | 1 | eqeq1d 2455 |
. . . . 5
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3 | eqeq1 2457 |
. . . . . . . 8
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4 | oveq1 6302 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 4 | breq1d 4415 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | 5 | rabbidv 3038 |
. . . . . . . . . . . 12
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7 | 6 | supeq1d 7965 |
. . . . . . . . . . 11
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8 | pcval.1 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 7, 8 | syl6eqr 2505 |
. . . . . . . . . 10
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10 | 4 | breq1d 4415 |
. . . . . . . . . . . . 13
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11 | 10 | rabbidv 3038 |
. . . . . . . . . . . 12
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12 | 11 | supeq1d 7965 |
. . . . . . . . . . 11
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13 | pcval.2 |
. . . . . . . . . . 11
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14 | 12, 13 | syl6eqr 2505 |
. . . . . . . . . 10
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15 | 9, 14 | oveq12d 6313 |
. . . . . . . . 9
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16 | 15 | eqeq2d 2463 |
. . . . . . . 8
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17 | 3, 16 | bi2anan9r 886 |
. . . . . . 7
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18 | 17 | 2rexbidv 2910 |
. . . . . 6
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19 | 18 | iotabidv 5570 |
. . . . 5
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20 | 2, 19 | ifbieq2d 3908 |
. . . 4
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21 | df-pc 14799 |
. . . 4
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22 | pnfex 11420 |
. . . . 5
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23 | iotaex 5566 |
. . . . 5
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24 | 22, 23 | ifex 3951 |
. . . 4
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25 | 20, 21, 24 | ovmpt2a 6432 |
. . 3
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26 | ifnefalse 3895 |
. . 3
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27 | 25, 26 | sylan9eq 2507 |
. 2
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28 | 27 | anasss 653 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 ax-cnex 9600 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-rab 2748 df-v 3049 df-sbc 3270 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-op 3977 df-uni 4202 df-br 4406 df-opab 4465 df-id 4752 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-iota 5549 df-fun 5587 df-fv 5593 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-sup 7961 df-pnf 9682 df-xr 9684 df-pc 14799 |
This theorem is referenced by: pczpre 14809 pcdiv 14814 |
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