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Mirrors > Home > MPE Home > Th. List > subgores | Structured version Visualization version Unicode version |
Description: A subgroup operation is the restriction of its parent group operation to its underlying set. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
subgores.1 |
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Ref | Expression |
---|---|
subgores |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issubgo 26024 |
. . . . 5
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2 | 1 | simp1bi 1022 |
. . . 4
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3 | eqid 2450 |
. . . . 5
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4 | 3 | grpofo 25920 |
. . . 4
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5 | fofun 5792 |
. . . 4
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6 | 2, 4, 5 | 3syl 18 |
. . 3
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7 | 1 | simp3bi 1024 |
. . 3
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8 | 1 | simp2bi 1023 |
. . . . 5
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9 | subgores.1 |
. . . . . 6
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10 | 9 | grpofo 25920 |
. . . . 5
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11 | fof 5791 |
. . . . 5
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12 | 8, 10, 11 | 3syl 18 |
. . . 4
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13 | fdm 5731 |
. . . 4
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14 | eqimss2 3484 |
. . . 4
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15 | 12, 13, 14 | 3syl 18 |
. . 3
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16 | fun2ssres 5622 |
. . 3
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17 | 6, 7, 15, 16 | syl3anc 1267 |
. 2
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18 | fofn 5793 |
. . 3
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19 | fnresdm 5683 |
. . 3
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20 | 8, 10, 18, 19 | 4syl 19 |
. 2
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21 | 17, 20 | eqtr2d 2485 |
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 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-ral 2741 df-rex 2742 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-op 3974 df-uni 4198 df-iun 4279 df-br 4402 df-opab 4461 df-mpt 4462 df-id 4748 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-fo 5587 df-fv 5589 df-ov 6291 df-grpo 25912 df-subgo 26023 |
This theorem is referenced by: subgoov 26026 subgornss 26027 |
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