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Theorem lmicrcl 19052
Description: Isomorphism implies the right side is a module. (Contributed by Mario Carneiro, 6-May-2015.)
Assertion
Ref Expression
lmicrcl (𝑅𝑚 𝑆𝑆 ∈ LMod)

Proof of Theorem lmicrcl
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 brlmic 19049 . . 3 (𝑅𝑚 𝑆 ↔ (𝑅 LMIso 𝑆) ≠ ∅)
2 n0 3923 . . 3 ((𝑅 LMIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑅 LMIso 𝑆))
31, 2bitri 264 . 2 (𝑅𝑚 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 LMIso 𝑆))
4 lmimlmhm 19045 . . . 4 (𝑓 ∈ (𝑅 LMIso 𝑆) → 𝑓 ∈ (𝑅 LMHom 𝑆))
5 lmhmlmod2 19013 . . . 4 (𝑓 ∈ (𝑅 LMHom 𝑆) → 𝑆 ∈ LMod)
64, 5syl 17 . . 3 (𝑓 ∈ (𝑅 LMIso 𝑆) → 𝑆 ∈ LMod)
76exlimiv 1856 . 2 (∃𝑓 𝑓 ∈ (𝑅 LMIso 𝑆) → 𝑆 ∈ LMod)
83, 7sylbi 207 1 (𝑅𝑚 𝑆𝑆 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1702  wcel 1988  wne 2791  c0 3907   class class class wbr 4644  (class class class)co 6635  LModclmod 18844   LMHom clmhm 19000   LMIso clmim 19001  𝑚 clmic 19002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-1o 7545  df-lmhm 19003  df-lmim 19004  df-lmic 19005
This theorem is referenced by: (None)
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