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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 20958 | . 2 ⊢ (𝑅 ≃𝑚 𝑆 ↔ (𝑅 LMIso 𝑆) ≠ ∅) | |
2 | n0 4348 | . . 3 ⊢ ((𝑅 LMIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑅 LMIso 𝑆)) | |
3 | lmimcnv 20957 | . . . . 5 ⊢ (𝑓 ∈ (𝑅 LMIso 𝑆) → ◡𝑓 ∈ (𝑆 LMIso 𝑅)) | |
4 | brlmici 20959 | . . . . 5 ⊢ (◡𝑓 ∈ (𝑆 LMIso 𝑅) → 𝑆 ≃𝑚 𝑅) | |
5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝑓 ∈ (𝑅 LMIso 𝑆) → 𝑆 ≃𝑚 𝑅) |
6 | 5 | exlimiv 1925 | . . 3 ⊢ (∃𝑓 𝑓 ∈ (𝑅 LMIso 𝑆) → 𝑆 ≃𝑚 𝑅) |
7 | 2, 6 | sylbi 216 | . 2 ⊢ ((𝑅 LMIso 𝑆) ≠ ∅ → 𝑆 ≃𝑚 𝑅) |
8 | 1, 7 | sylbi 216 | 1 ⊢ (𝑅 ≃𝑚 𝑆 → 𝑆 ≃𝑚 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1773 ∈ wcel 2098 ≠ wne 2936 ∅c0 4324 class class class wbr 5150 ◡ccnv 5679 (class class class)co 7424 LMIso clmim 20910 ≃𝑚 clmic 20911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 7997 df-2nd 7998 df-1o 8491 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-grp 18898 df-ghm 19173 df-lmod 20750 df-lmhm 20912 df-lmim 20913 df-lmic 20914 |
This theorem is referenced by: lmisfree 21781 |
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