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Mirrors > Home > MPE Home > Th. List > issubm | Structured version Visualization version Unicode version |
Description: Expand definition of a submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.) |
Ref | Expression |
---|---|
issubm.b | |
issubm.z | |
issubm.p |
Ref | Expression |
---|---|
issubm | SubMnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6191 | . . . . . 6 | |
2 | 1 | pweqd 4163 | . . . . 5 |
3 | fveq2 6191 | . . . . . . 7 | |
4 | 3 | eleq1d 2686 | . . . . . 6 |
5 | fveq2 6191 | . . . . . . . . 9 | |
6 | 5 | oveqd 6667 | . . . . . . . 8 |
7 | 6 | eleq1d 2686 | . . . . . . 7 |
8 | 7 | 2ralbidv 2989 | . . . . . 6 |
9 | 4, 8 | anbi12d 747 | . . . . 5 |
10 | 2, 9 | rabeqbidv 3195 | . . . 4 |
11 | df-submnd 17336 | . . . 4 SubMnd | |
12 | fvex 6201 | . . . . . 6 | |
13 | 12 | pwex 4848 | . . . . 5 |
14 | 13 | rabex 4813 | . . . 4 |
15 | 10, 11, 14 | fvmpt 6282 | . . 3 SubMnd |
16 | 15 | eleq2d 2687 | . 2 SubMnd |
17 | eleq2 2690 | . . . . 5 | |
18 | eleq2 2690 | . . . . . . 7 | |
19 | 18 | raleqbi1dv 3146 | . . . . . 6 |
20 | 19 | raleqbi1dv 3146 | . . . . 5 |
21 | 17, 20 | anbi12d 747 | . . . 4 |
22 | 21 | elrab 3363 | . . 3 |
23 | issubm.b | . . . . . 6 | |
24 | 23 | sseq2i 3630 | . . . . 5 |
25 | issubm.z | . . . . . . 7 | |
26 | 25 | eleq1i 2692 | . . . . . 6 |
27 | issubm.p | . . . . . . . . 9 | |
28 | 27 | oveqi 6663 | . . . . . . . 8 |
29 | 28 | eleq1i 2692 | . . . . . . 7 |
30 | 29 | 2ralbii 2981 | . . . . . 6 |
31 | 26, 30 | anbi12i 733 | . . . . 5 |
32 | 24, 31 | anbi12i 733 | . . . 4 |
33 | 3anass 1042 | . . . 4 | |
34 | 12 | elpw2 4828 | . . . . 5 |
35 | 34 | anbi1i 731 | . . . 4 |
36 | 32, 33, 35 | 3bitr4ri 293 | . . 3 |
37 | 22, 36 | bitri 264 | . 2 |
38 | 16, 37 | syl6bb 276 | 1 SubMnd |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 crab 2916 wss 3574 cpw 4158 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 c0g 16100 cmnd 17294 SubMndcsubmnd 17334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-submnd 17336 |
This theorem is referenced by: issubm2 17348 issubmd 17349 submcl 17353 mhmima 17363 mhmeql 17364 submacs 17365 gsumwspan 17383 frmdsssubm 17398 issubg3 17612 cntzsubm 17768 oppgsubm 17792 lsmsubm 18068 issubrg3 18808 xrge0subm 19787 cnsubmlem 19794 nn0srg 19816 rge0srg 19817 efsubm 24297 iistmd 29948 isdomn3 37782 mon1psubm 37784 |
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