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Mirrors > Home > MPE Home > Th. List > lmicsym | Structured version Visualization version GIF version |
Description: Module isomorphism is symmetric. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
Ref | Expression |
---|---|
lmicsym | ⊢ (𝑅 ≃𝑚 𝑆 → 𝑆 ≃𝑚 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brlmic 19434 | . 2 ⊢ (𝑅 ≃𝑚 𝑆 ↔ (𝑅 LMIso 𝑆) ≠ ∅) | |
2 | n0 4162 | . . 3 ⊢ ((𝑅 LMIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑅 LMIso 𝑆)) | |
3 | lmimcnv 19433 | . . . . 5 ⊢ (𝑓 ∈ (𝑅 LMIso 𝑆) → ◡𝑓 ∈ (𝑆 LMIso 𝑅)) | |
4 | brlmici 19435 | . . . . 5 ⊢ (◡𝑓 ∈ (𝑆 LMIso 𝑅) → 𝑆 ≃𝑚 𝑅) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝑓 ∈ (𝑅 LMIso 𝑆) → 𝑆 ≃𝑚 𝑅) |
6 | 5 | exlimiv 2029 | . . 3 ⊢ (∃𝑓 𝑓 ∈ (𝑅 LMIso 𝑆) → 𝑆 ≃𝑚 𝑅) |
7 | 2, 6 | sylbi 209 | . 2 ⊢ ((𝑅 LMIso 𝑆) ≠ ∅ → 𝑆 ≃𝑚 𝑅) |
8 | 1, 7 | sylbi 209 | 1 ⊢ (𝑅 ≃𝑚 𝑆 → 𝑆 ≃𝑚 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1878 ∈ wcel 2164 ≠ wne 2999 ∅c0 4146 class class class wbr 4875 ◡ccnv 5345 (class class class)co 6910 LMIso clmim 19386 ≃𝑚 clmic 19387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-1st 7433 df-2nd 7434 df-1o 7831 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-grp 17786 df-ghm 18016 df-lmod 19228 df-lmhm 19388 df-lmim 19389 df-lmic 19390 |
This theorem is referenced by: lmisfree 20555 |
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