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Mirrors > Home > ILE Home > Th. List > ringidvalg | Unicode version |
Description: The value of the unity element of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
Ref | Expression |
---|---|
ringidval.g |
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ringidval.u |
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Ref | Expression |
---|---|
ringidvalg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2748 |
. . 3
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2 | df-ur 12956 |
. . . . 5
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3 | 2 | fveq1i 5511 |
. . . 4
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4 | fnmgp 12946 |
. . . . 5
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5 | fvco2 5580 |
. . . . 5
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6 | 4, 5 | mpan 424 |
. . . 4
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7 | 3, 6 | eqtrid 2222 |
. . 3
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8 | 1, 7 | syl 14 |
. 2
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9 | ringidval.u |
. 2
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10 | ringidval.g |
. . 3
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11 | 10 | fveq2i 5513 |
. 2
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12 | 8, 9, 11 | 3eqtr4g 2235 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-fv 5219 df-ov 5871 df-oprab 5872 df-mpo 5873 df-inn 8896 df-2 8954 df-3 8955 df-ndx 12435 df-slot 12436 df-sets 12439 df-plusg 12518 df-mulr 12519 df-mgp 12945 df-ur 12956 |
This theorem is referenced by: dfur2g 12958 srgidcl 12972 srgidmlem 12974 issrgid 12977 srgpcomp 12986 srg1expzeq1 12991 ringidcl 13016 ringidmlem 13018 isringid 13021 oppr1g 13064 unitsubm 13100 |
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