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Mirrors > Home > ILE Home > Th. List > 0mhm | Unicode version |
Description: The constant zero linear function between two monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
0mhm.z |
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0mhm.b |
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Ref | Expression |
---|---|
0mhm |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 |
. 2
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2 | eqid 2193 |
. . . . . 6
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3 | 0mhm.z |
. . . . . 6
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4 | 2, 3 | mndidcl 13011 |
. . . . 5
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5 | 4 | adantl 277 |
. . . 4
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6 | fconst6g 5452 |
. . . 4
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7 | 5, 6 | syl 14 |
. . 3
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8 | simpr 110 |
. . . . . . 7
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9 | eqid 2193 |
. . . . . . . . 9
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10 | 2, 9, 3 | mndlid 13016 |
. . . . . . . 8
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11 | 10 | eqcomd 2199 |
. . . . . . 7
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12 | 8, 4, 11 | syl2anc2 412 |
. . . . . 6
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13 | 12 | adantr 276 |
. . . . 5
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14 | fn0g 12958 |
. . . . . . . . 9
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15 | 8 | elexd 2773 |
. . . . . . . . 9
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16 | funfvex 5571 |
. . . . . . . . . 10
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17 | 16 | funfni 5354 |
. . . . . . . . 9
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18 | 14, 15, 17 | sylancr 414 |
. . . . . . . 8
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19 | 3, 18 | eqeltrid 2280 |
. . . . . . 7
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20 | 19 | adantr 276 |
. . . . . 6
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21 | 0mhm.b |
. . . . . . . . 9
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22 | eqid 2193 |
. . . . . . . . 9
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23 | 21, 22 | mndcl 13004 |
. . . . . . . 8
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24 | 23 | 3expb 1206 |
. . . . . . 7
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25 | 24 | adantlr 477 |
. . . . . 6
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26 | fvconst2g 5772 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 20, 25, 26 | syl2anc 411 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | simprl 529 |
. . . . . . 7
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29 | fvconst2g 5772 |
. . . . . . 7
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30 | 20, 28, 29 | syl2anc 411 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | simprr 531 |
. . . . . . 7
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32 | fvconst2g 5772 |
. . . . . . 7
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33 | 20, 31, 32 | syl2anc 411 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 30, 33 | oveq12d 5936 |
. . . . 5
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35 | 13, 27, 34 | 3eqtr4d 2236 |
. . . 4
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36 | 35 | ralrimivva 2576 |
. . 3
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37 | eqid 2193 |
. . . . . 6
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38 | 21, 37 | mndidcl 13011 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | 38 | adantr 276 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | fvconst2g 5772 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
41 | 19, 39, 40 | syl2anc 411 |
. . 3
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42 | 7, 36, 41 | 3jca 1179 |
. 2
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43 | 21, 2, 22, 9, 37, 3 | ismhm 13033 |
. 2
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44 | 1, 42, 43 | sylanbrc 417 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1re 7966 ax-addrcl 7969 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-map 6704 df-inn 8983 df-2 9041 df-ndx 12621 df-slot 12622 df-base 12624 df-plusg 12708 df-0g 12869 df-mgm 12939 df-sgrp 12985 df-mnd 12998 df-mhm 13031 |
This theorem is referenced by: 0ghm 13328 |
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