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Theorem nmvs 24063
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 24062 . . . 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 7384 . . . . 5 (π‘₯ = 𝑋 β†’ (π‘β€˜(π‘₯ Β· 𝑦)) = (π‘β€˜(𝑋 Β· 𝑦)))
10 fveq2 6846 . . . . . 6 (π‘₯ = 𝑋 β†’ (π΄β€˜π‘₯) = (π΄β€˜π‘‹))
1110oveq1d 7376 . . . . 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 7369 . . . . . 6 (𝑦 = π‘Œ β†’ (𝑋 Β· 𝑦) = (𝑋 Β· π‘Œ))
1413fveq2d 6850 . . . . 5 (𝑦 = π‘Œ β†’ (π‘β€˜(𝑋 Β· 𝑦)) = (π‘β€˜(𝑋 Β· π‘Œ)))
15 fveq2 6846 . . . . . 6 (𝑦 = π‘Œ β†’ (π‘β€˜π‘¦) = (π‘β€˜π‘Œ))
1615oveq2d 7377 . . . . 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 3592 . . 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 3061  β€˜cfv 6500  (class class class)co 7361   Β· cmul 11064  Basecbs 17091  Scalarcsca 17144   ·𝑠 cvsca 17145  LModclmod 20365  normcnm 23955  NrmGrpcngp 23956  NrmRingcnrg 23958  NrmModcnlm 23959
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 5267
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 2941  df-ral 3062  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508  df-ov 7364  df-nlm 23965
This theorem is referenced by:  nlmdsdi  24068  nlmdsdir  24069  nlmmul0or  24070  lssnlm  24088  nmoleub2lem3  24501  nmoleub3  24505  ncvsprp  24539  cphnmvs  24577  nmmulg  32613
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