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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > omlmod1i2N | Structured version Unicode version |
Description: Analog of modular law atmod1i2 33809 that holds in any OML. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
omlmod.b |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
omlmod.l |
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omlmod.j |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
omlmod.m |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
omlmod.c |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
omlmod1i2N |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 988 |
. . 3
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2 | simp23 1023 |
. . 3
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3 | simp21 1021 |
. . 3
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4 | simp22 1022 |
. . 3
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5 | simp3l 1016 |
. . . . 5
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6 | omlmod.b |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | omlmod.l |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | omlmod.c |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 6, 7, 8 | lecmtN 33207 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | 1, 3, 2, 9 | syl3anc 1219 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | 5, 10 | mpd 15 |
. . . 4
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12 | 6, 8 | cmtcomN 33200 |
. . . . 5
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13 | 1, 3, 2, 12 | syl3anc 1219 |
. . . 4
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14 | 11, 13 | mpbid 210 |
. . 3
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15 | simp3r 1017 |
. . . 4
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16 | 6, 8 | cmtcomN 33200 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 1, 4, 2, 16 | syl3anc 1219 |
. . . 4
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18 | 15, 17 | mpbid 210 |
. . 3
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19 | omlmod.j |
. . . 4
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20 | omlmod.m |
. . . 4
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21 | 6, 19, 20, 8 | omlfh1N 33209 |
. . 3
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22 | 1, 2, 3, 4, 14, 18, 21 | syl132anc 1237 |
. 2
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23 | omllat 33193 |
. . . 4
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24 | 23 | 3ad2ant1 1009 |
. . 3
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25 | 6, 19 | latjcl 15323 |
. . . 4
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26 | 24, 3, 4, 25 | syl3anc 1219 |
. . 3
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27 | 6, 20 | latmcom 15347 |
. . 3
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28 | 24, 2, 26, 27 | syl3anc 1219 |
. 2
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29 | 6, 7, 20 | latleeqm2 15352 |
. . . . 5
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30 | 24, 3, 2, 29 | syl3anc 1219 |
. . . 4
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31 | 5, 30 | mpbid 210 |
. . 3
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32 | 6, 20 | latmcom 15347 |
. . . 4
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33 | 24, 2, 4, 32 | syl3anc 1219 |
. . 3
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34 | 31, 33 | oveq12d 6208 |
. 2
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35 | 22, 28, 34 | 3eqtr3rd 2501 |
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 1952 ax-ext 2430 ax-rep 4501 ax-sep 4511 ax-nul 4519 ax-pow 4568 ax-pr 4629 ax-un 6472 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-ral 2800 df-rex 2801 df-reu 2802 df-rab 2804 df-v 3070 df-sbc 3285 df-csb 3387 df-dif 3429 df-un 3431 df-in 3433 df-ss 3440 df-nul 3736 df-if 3890 df-pw 3960 df-sn 3976 df-pr 3978 df-op 3982 df-uni 4190 df-iun 4271 df-br 4391 df-opab 4449 df-mpt 4450 df-id 4734 df-xp 4944 df-rel 4945 df-cnv 4946 df-co 4947 df-dm 4948 df-rn 4949 df-res 4950 df-ima 4951 df-iota 5479 df-fun 5518 df-fn 5519 df-f 5520 df-f1 5521 df-fo 5522 df-f1o 5523 df-fv 5524 df-riota 6151 df-ov 6193 df-oprab 6194 df-poset 15218 df-lub 15246 df-glb 15247 df-join 15248 df-meet 15249 df-p0 15311 df-lat 15318 df-oposet 33127 df-cmtN 33128 df-ol 33129 df-oml 33130 |
This theorem is referenced by: omlspjN 33212 |
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