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Mirrors > Home > ILE Home > Th. List > mgpscag | Unicode version |
Description: The multiplication monoid has the same (if any) scalars as the original ring. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
Ref | Expression |
---|---|
mgpbas.1 |
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mgpsca.s |
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Ref | Expression |
---|---|
mgpscag |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mgpsca.s |
. 2
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2 | mulrslid 12605 |
. . . . 5
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3 | 2 | slotex 12503 |
. . . 4
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4 | scaslid 12626 |
. . . . 5
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5 | scandxnplusgndx 12628 |
. . . . 5
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6 | plusgslid 12586 |
. . . . . 6
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7 | 6 | simpri 113 |
. . . . 5
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8 | 4, 5, 7 | setsslnid 12528 |
. . . 4
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9 | 3, 8 | mpdan 421 |
. . 3
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10 | mgpbas.1 |
. . . . 5
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11 | eqid 2187 |
. . . . 5
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12 | 10, 11 | mgpvalg 13175 |
. . . 4
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13 | 12 | fveq2d 5531 |
. . 3
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14 | 9, 13 | eqtr4d 2223 |
. 2
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15 | 1, 14 | eqtrid 2232 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7916 ax-resscn 7917 ax-1cn 7918 ax-1re 7919 ax-icn 7920 ax-addcl 7921 ax-addrcl 7922 ax-mulcl 7923 ax-addcom 7925 ax-addass 7927 ax-i2m1 7930 ax-0lt1 7931 ax-0id 7933 ax-rnegex 7934 ax-pre-ltirr 7937 ax-pre-lttrn 7939 ax-pre-ltadd 7941 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-iota 5190 df-fun 5230 df-fn 5231 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8008 df-mnf 8009 df-ltxr 8011 df-inn 8934 df-2 8992 df-3 8993 df-4 8994 df-5 8995 df-ndx 12479 df-slot 12480 df-sets 12483 df-plusg 12564 df-mulr 12565 df-sca 12567 df-mgp 13173 |
This theorem is referenced by: (None) |
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