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Mirrors > Home > MPE Home > Th. List > grpoass | Structured version Unicode version |
Description: A group operation is associative. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpfo.1 |
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Ref | Expression |
---|---|
grpoass |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpfo.1 |
. . . . 5
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2 | 1 | isgrpo 23820 |
. . . 4
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3 | 2 | ibi 241 |
. . 3
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4 | 3 | simp2d 1001 |
. 2
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5 | oveq1 6199 |
. . . . 5
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6 | 5 | oveq1d 6207 |
. . . 4
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7 | oveq1 6199 |
. . . 4
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8 | 6, 7 | eqeq12d 2473 |
. . 3
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9 | oveq2 6200 |
. . . . 5
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10 | 9 | oveq1d 6207 |
. . . 4
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11 | oveq1 6199 |
. . . . 5
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12 | 11 | oveq2d 6208 |
. . . 4
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13 | 10, 12 | eqeq12d 2473 |
. . 3
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14 | oveq2 6200 |
. . . 4
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15 | oveq2 6200 |
. . . . 5
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16 | 15 | oveq2d 6208 |
. . . 4
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17 | 14, 16 | eqeq12d 2473 |
. . 3
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18 | 8, 13, 17 | rspc3v 3181 |
. 2
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19 | 4, 18 | mpan9 469 |
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 1952 ax-ext 2430 ax-sep 4513 ax-nul 4521 ax-pr 4631 ax-un 6474 |
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 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-ral 2800 df-rex 2801 df-rab 2804 df-v 3072 df-sbc 3287 df-csb 3389 df-dif 3431 df-un 3433 df-in 3435 df-ss 3442 df-nul 3738 df-if 3892 df-sn 3978 df-pr 3980 df-op 3984 df-uni 4192 df-iun 4273 df-br 4393 df-opab 4451 df-mpt 4452 df-id 4736 df-xp 4946 df-rel 4947 df-cnv 4948 df-co 4949 df-dm 4950 df-rn 4951 df-iota 5481 df-fun 5520 df-fn 5521 df-f 5522 df-fo 5524 df-fv 5526 df-ov 6195 df-grpo 23815 |
This theorem is referenced by: grpoidinvlem1 23828 grpoidinvlem2 23829 grpoidinvlem4 23831 grporcan 23845 grpoinvid1 23854 grpoinvid2 23855 grpolcan 23857 grpo2grp 23858 grpoasscan1 23861 grpoasscan2 23862 grpoinvop 23865 grpomuldivass 23873 grponpcan 23876 grpopnpcan2 23877 gxcom 23893 gxnn0add 23898 ablo32 23910 ablo4 23911 issubgoi 23934 ghgrp 23992 rngoaass 24017 vcaass 24076 vcm 24086 nvass 24137 |
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