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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ofaddmndmap | Structured version Unicode version |
Description: The function operation applied to the addition for functions (with the same domain) into a monoid is a function (with the same domain) into the monoid. (Contributed by AV, 6-Apr-2019.) |
Ref | Expression |
---|---|
ofaddmndmap.r |
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ofaddmndmap.p |
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Ref | Expression |
---|---|
ofaddmndmap |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 991 |
. . . 4
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2 | simprl 755 |
. . . 4
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3 | simprr 756 |
. . . 4
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4 | ofaddmndmap.r |
. . . . 5
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5 | ofaddmndmap.p |
. . . . 5
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6 | 4, 5 | mndcl 15540 |
. . . 4
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7 | 1, 2, 3, 6 | syl3anc 1219 |
. . 3
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8 | elmapi 7345 |
. . . . 5
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9 | 8 | adantr 465 |
. . . 4
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10 | 9 | 3ad2ant3 1011 |
. . 3
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11 | elmapi 7345 |
. . . . 5
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12 | 11 | adantl 466 |
. . . 4
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13 | 12 | 3ad2ant3 1011 |
. . 3
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14 | simp2 989 |
. . 3
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15 | inidm 3668 |
. . 3
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16 | 7, 10, 13, 14, 14, 15 | off 6445 |
. 2
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17 | fvex 5810 |
. . . 4
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18 | 4, 17 | eqeltri 2538 |
. . 3
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19 | elmapg 7338 |
. . 3
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20 | 18, 14, 19 | sylancr 663 |
. 2
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21 | 16, 20 | mpbird 232 |
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 1955 ax-ext 2432 ax-rep 4512 ax-sep 4522 ax-nul 4530 ax-pow 4579 ax-pr 4640 ax-un 6483 |
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 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-ral 2804 df-rex 2805 df-reu 2806 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3397 df-dif 3440 df-un 3442 df-in 3444 df-ss 3451 df-nul 3747 df-if 3901 df-pw 3971 df-sn 3987 df-pr 3989 df-op 3993 df-uni 4201 df-iun 4282 df-br 4402 df-opab 4460 df-mpt 4461 df-id 4745 df-xp 4955 df-rel 4956 df-cnv 4957 df-co 4958 df-dm 4959 df-rn 4960 df-res 4961 df-ima 4962 df-iota 5490 df-fun 5529 df-fn 5530 df-f 5531 df-f1 5532 df-fo 5533 df-f1o 5534 df-fv 5535 df-ov 6204 df-oprab 6205 df-mpt2 6206 df-of 6431 df-1st 6688 df-2nd 6689 df-map 7327 df-mnd 15535 |
This theorem is referenced by: lincsumcl 31098 |
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