Theorem lmhmlin 20971

Description: A homomorphism of left modules is 𝐾-linear. (Contributed by Stefan O'Rear, 1-Jan-2015.)

Hypotheses

Ref	Expression
lmhmlin.k	⊢ 𝐾 = (Scalar‘𝑆)
lmhmlin.b	⊢ 𝐵 = (Base‘𝐾)
lmhmlin.e	⊢ 𝐸 = (Base‘𝑆)
lmhmlin.m	⊢ · = ( ·𝑠 ‘𝑆)
lmhmlin.n	⊢ × = ( ·𝑠 ‘𝑇)

Assertion

Ref	Expression
lmhmlin	⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐸) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹‘𝑌)))

Proof of Theorem lmhmlin

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmhmlin.k	. . . . . 6 ⊢ 𝐾 = (Scalar‘𝑆)
2		eqid 2733	. . . . . 6 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
3		lmhmlin.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
4		lmhmlin.e	. . . . . 6 ⊢ 𝐸 = (Base‘𝑆)
5		lmhmlin.m	. . . . . 6 ⊢ · = ( ·𝑠 ‘𝑆)
6		lmhmlin.n	. . . . . 6 ⊢ × = ( ·𝑠 ‘𝑇)
7	1, 2, 3, 4, 5, 6	islmhm 20963	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalar‘𝑇) = 𝐾 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹‘𝑏)))))
8	7	simprbi 496	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalar‘𝑇) = 𝐾 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹‘𝑏))))
9	8	simp3d 1144	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹‘𝑏)))
10		fvoveq1 7375	. . . . 5 ⊢ (𝑎 = 𝑋 → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑋 · 𝑏)))
11		oveq1 7359	. . . . 5 ⊢ (𝑎 = 𝑋 → (𝑎 × (𝐹‘𝑏)) = (𝑋 × (𝐹‘𝑏)))
12	10, 11	eqeq12d 2749	. . . 4 ⊢ (𝑎 = 𝑋 → ((𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹‘𝑏)) ↔ (𝐹‘(𝑋 · 𝑏)) = (𝑋 × (𝐹‘𝑏))))
13		oveq2 7360	. . . . . 6 ⊢ (𝑏 = 𝑌 → (𝑋 · 𝑏) = (𝑋 · 𝑌))
14	13	fveq2d 6832	. . . . 5 ⊢ (𝑏 = 𝑌 → (𝐹‘(𝑋 · 𝑏)) = (𝐹‘(𝑋 · 𝑌)))
15		fveq2 6828	. . . . . 6 ⊢ (𝑏 = 𝑌 → (𝐹‘𝑏) = (𝐹‘𝑌))
16	15	oveq2d 7368	. . . . 5 ⊢ (𝑏 = 𝑌 → (𝑋 × (𝐹‘𝑏)) = (𝑋 × (𝐹‘𝑌)))
17	14, 16	eqeq12d 2749	. . . 4 ⊢ (𝑏 = 𝑌 → ((𝐹‘(𝑋 · 𝑏)) = (𝑋 × (𝐹‘𝑏)) ↔ (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹‘𝑌))))
18	12, 17	rspc2v 3584	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐸) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹‘𝑏)) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹‘𝑌))))
19	9, 18	syl5com 31	. 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐸) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹‘𝑌))))
20	19	3impib 1116	1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐸) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ‘cfv 6486  (class class class)co 7352  Basecbs 17122  Scalarcsca 17166   ·_𝑠 cvsca 17167   GrpHom cghm 19126  LModclmod 20795   LMHom clmhm 20955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-lmhm 20958

This theorem is referenced by:  islmhm2  20974  lmhmco  20979  lmhmplusg  20980  lmhmvsca  20981  lmhmf1o  20982  lmhmima  20983  lmhmpreima  20984  reslmhm  20988  reslmhm2  20989  reslmhm2b  20990  lmhmeql  20991  ipass  21584  lindfmm  21766  nmoleub2lem3  25043  nmoleub3  25047  lmhmimasvsca  33027  lmhmqusker  33389  mendassa  43307