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Mirrors > Home > MPE Home > Th. List > lmicrcl | Structured version Visualization version Unicode version |
Description: Isomorphism implies the right side is a module. (Contributed by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
lmicrcl | 𝑚 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brlmic 19068 | . . 3 𝑚 LMIso | |
2 | n0 3931 | . . 3 LMIso LMIso | |
3 | 1, 2 | bitri 264 | . 2 𝑚 LMIso |
4 | lmimlmhm 19064 | . . . 4 LMIso LMHom | |
5 | lmhmlmod2 19032 | . . . 4 LMHom | |
6 | 4, 5 | syl 17 | . . 3 LMIso |
7 | 6 | exlimiv 1858 | . 2 LMIso |
8 | 3, 7 | sylbi 207 | 1 𝑚 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wex 1704 wcel 1990 wne 2794 c0 3915 class class class wbr 4653 (class class class)co 6650 clmod 18863 LMHom clmhm 19019 LMIso clmim 19020 𝑚 clmic 19021 |
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-8 1992 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 ax-un 6949 |
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-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 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-rn 5125 df-res 5126 df-ima 5127 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-1o 7560 df-lmhm 19022 df-lmim 19023 df-lmic 19024 |
This theorem is referenced by: (None) |
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