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Mirrors > Home > ILE Home > Th. List > grpidpropdg | Unicode version |
Description: If two structures have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, they have the same identity element. (Contributed by Mario Carneiro, 27-Nov-2014.) |
Ref | Expression |
---|---|
grpidpropd.1 | |
grpidpropd.2 | |
grpidproddg.k | |
grpidproddg.l | |
grpidpropd.3 |
Ref | Expression |
---|---|
grpidpropdg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpidpropd.3 | . . . . . . . . 9 | |
2 | 1 | eqeq1d 2179 | . . . . . . . 8 |
3 | 1 | oveqrspc2v 5880 | . . . . . . . . . . 11 |
4 | 3 | oveqrspc2v 5880 | . . . . . . . . . 10 |
5 | 4 | ancom2s 561 | . . . . . . . . 9 |
6 | 5 | eqeq1d 2179 | . . . . . . . 8 |
7 | 2, 6 | anbi12d 470 | . . . . . . 7 |
8 | 7 | anassrs 398 | . . . . . 6 |
9 | 8 | ralbidva 2466 | . . . . 5 |
10 | 9 | pm5.32da 449 | . . . 4 |
11 | grpidpropd.1 | . . . . . 6 | |
12 | 11 | eleq2d 2240 | . . . . 5 |
13 | 11 | raleqdv 2671 | . . . . 5 |
14 | 12, 13 | anbi12d 470 | . . . 4 |
15 | grpidpropd.2 | . . . . . 6 | |
16 | 15 | eleq2d 2240 | . . . . 5 |
17 | 15 | raleqdv 2671 | . . . . 5 |
18 | 16, 17 | anbi12d 470 | . . . 4 |
19 | 10, 14, 18 | 3bitr3d 217 | . . 3 |
20 | 19 | iotabidv 5181 | . 2 |
21 | grpidproddg.k | . . 3 | |
22 | eqid 2170 | . . . 4 | |
23 | eqid 2170 | . . . 4 | |
24 | eqid 2170 | . . . 4 | |
25 | 22, 23, 24 | grpidvalg 12627 | . . 3 |
26 | 21, 25 | syl 14 | . 2 |
27 | grpidproddg.l | . . 3 | |
28 | eqid 2170 | . . . 4 | |
29 | eqid 2170 | . . . 4 | |
30 | eqid 2170 | . . . 4 | |
31 | 28, 29, 30 | grpidvalg 12627 | . . 3 |
32 | 27, 31 | syl 14 | . 2 |
33 | 20, 26, 32 | 3eqtr4d 2213 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wral 2448 cio 5158 cfv 5198 (class class class)co 5853 cbs 12416 cplusg 12480 c0g 12596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-cnex 7865 ax-resscn 7866 ax-1re 7868 ax-addrcl 7871 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-iota 5160 df-fun 5200 df-fn 5201 df-fv 5206 df-riota 5809 df-ov 5856 df-inn 8879 df-ndx 12419 df-slot 12420 df-base 12422 df-0g 12598 |
This theorem is referenced by: mhmpropd 12689 grppropd 12724 |
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