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Theorem nmvs 24193
Description: Defining property of a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
isnlm.v 𝑉 = (Baseβ€˜π‘Š)
isnlm.n 𝑁 = (normβ€˜π‘Š)
isnlm.s Β· = ( ·𝑠 β€˜π‘Š)
isnlm.f 𝐹 = (Scalarβ€˜π‘Š)
isnlm.k 𝐾 = (Baseβ€˜πΉ)
isnlm.a 𝐴 = (normβ€˜πΉ)
Assertion
Ref Expression
nmvs ((π‘Š ∈ NrmMod ∧ 𝑋 ∈ 𝐾 ∧ π‘Œ ∈ 𝑉) β†’ (π‘β€˜(𝑋 Β· π‘Œ)) = ((π΄β€˜π‘‹) Β· (π‘β€˜π‘Œ)))

Proof of Theorem nmvs
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnlm.v . . . . 5 𝑉 = (Baseβ€˜π‘Š)
2 isnlm.n . . . . 5 𝑁 = (normβ€˜π‘Š)
3 isnlm.s . . . . 5 Β· = ( ·𝑠 β€˜π‘Š)
4 isnlm.f . . . . 5 𝐹 = (Scalarβ€˜π‘Š)
5 isnlm.k . . . . 5 𝐾 = (Baseβ€˜πΉ)
6 isnlm.a . . . . 5 𝐴 = (normβ€˜πΉ)
71, 2, 3, 4, 5, 6isnlm 24192 . . . 4 (π‘Š ∈ NrmMod ↔ ((π‘Š ∈ NrmGrp ∧ π‘Š ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝑉 (π‘β€˜(π‘₯ Β· 𝑦)) = ((π΄β€˜π‘₯) Β· (π‘β€˜π‘¦))))
87simprbi 498 . . 3 (π‘Š ∈ NrmMod β†’ βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝑉 (π‘β€˜(π‘₯ Β· 𝑦)) = ((π΄β€˜π‘₯) Β· (π‘β€˜π‘¦)))
9 fvoveq1 7432 . . . . 5 (π‘₯ = 𝑋 β†’ (π‘β€˜(π‘₯ Β· 𝑦)) = (π‘β€˜(𝑋 Β· 𝑦)))
10 fveq2 6892 . . . . . 6 (π‘₯ = 𝑋 β†’ (π΄β€˜π‘₯) = (π΄β€˜π‘‹))
1110oveq1d 7424 . . . . 5 (π‘₯ = 𝑋 β†’ ((π΄β€˜π‘₯) Β· (π‘β€˜π‘¦)) = ((π΄β€˜π‘‹) Β· (π‘β€˜π‘¦)))
129, 11eqeq12d 2749 . . . 4 (π‘₯ = 𝑋 β†’ ((π‘β€˜(π‘₯ Β· 𝑦)) = ((π΄β€˜π‘₯) Β· (π‘β€˜π‘¦)) ↔ (π‘β€˜(𝑋 Β· 𝑦)) = ((π΄β€˜π‘‹) Β· (π‘β€˜π‘¦))))
13 oveq2 7417 . . . . . 6 (𝑦 = π‘Œ β†’ (𝑋 Β· 𝑦) = (𝑋 Β· π‘Œ))
1413fveq2d 6896 . . . . 5 (𝑦 = π‘Œ β†’ (π‘β€˜(𝑋 Β· 𝑦)) = (π‘β€˜(𝑋 Β· π‘Œ)))
15 fveq2 6892 . . . . . 6 (𝑦 = π‘Œ β†’ (π‘β€˜π‘¦) = (π‘β€˜π‘Œ))
1615oveq2d 7425 . . . . 5 (𝑦 = π‘Œ β†’ ((π΄β€˜π‘‹) Β· (π‘β€˜π‘¦)) = ((π΄β€˜π‘‹) Β· (π‘β€˜π‘Œ)))
1714, 16eqeq12d 2749 . . . 4 (𝑦 = π‘Œ β†’ ((π‘β€˜(𝑋 Β· 𝑦)) = ((π΄β€˜π‘‹) Β· (π‘β€˜π‘¦)) ↔ (π‘β€˜(𝑋 Β· π‘Œ)) = ((π΄β€˜π‘‹) Β· (π‘β€˜π‘Œ))))
1812, 17rspc2v 3623 . . 3 ((𝑋 ∈ 𝐾 ∧ π‘Œ ∈ 𝑉) β†’ (βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝑉 (π‘β€˜(π‘₯ Β· 𝑦)) = ((π΄β€˜π‘₯) Β· (π‘β€˜π‘¦)) β†’ (π‘β€˜(𝑋 Β· π‘Œ)) = ((π΄β€˜π‘‹) Β· (π‘β€˜π‘Œ))))
198, 18syl5com 31 . 2 (π‘Š ∈ NrmMod β†’ ((𝑋 ∈ 𝐾 ∧ π‘Œ ∈ 𝑉) β†’ (π‘β€˜(𝑋 Β· π‘Œ)) = ((π΄β€˜π‘‹) Β· (π‘β€˜π‘Œ))))
20193impib 1117 1 ((π‘Š ∈ NrmMod ∧ 𝑋 ∈ 𝐾 ∧ π‘Œ ∈ 𝑉) β†’ (π‘β€˜(𝑋 Β· π‘Œ)) = ((π΄β€˜π‘‹) Β· (π‘β€˜π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  β€˜cfv 6544  (class class class)co 7409   Β· cmul 11115  Basecbs 17144  Scalarcsca 17200   ·𝑠 cvsca 17201  LModclmod 20471  normcnm 24085  NrmGrpcngp 24086  NrmRingcnrg 24088  NrmModcnlm 24089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-nlm 24095
This theorem is referenced by:  nlmdsdi  24198  nlmdsdir  24199  nlmmul0or  24200  lssnlm  24218  nmoleub2lem3  24631  nmoleub3  24635  ncvsprp  24669  cphnmvs  24707  nmmulg  32948
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