Theorem lmhmlin 19360

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 2803	. . . . . 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 19352	. . . . 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 491	. . . 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 1175	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐸 (𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹‘𝑏)))
10		fvoveq1 6905	. . . . 5 ⊢ (𝑎 = 𝑋 → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑋 · 𝑏)))
11		oveq1 6889	. . . . 5 ⊢ (𝑎 = 𝑋 → (𝑎 × (𝐹‘𝑏)) = (𝑋 × (𝐹‘𝑏)))
12	10, 11	eqeq12d 2818	. . . 4 ⊢ (𝑎 = 𝑋 → ((𝐹‘(𝑎 · 𝑏)) = (𝑎 × (𝐹‘𝑏)) ↔ (𝐹‘(𝑋 · 𝑏)) = (𝑋 × (𝐹‘𝑏))))
13		oveq2 6890	. . . . . 6 ⊢ (𝑏 = 𝑌 → (𝑋 · 𝑏) = (𝑋 · 𝑌))
14	13	fveq2d 6419	. . . . 5 ⊢ (𝑏 = 𝑌 → (𝐹‘(𝑋 · 𝑏)) = (𝐹‘(𝑋 · 𝑌)))
15		fveq2 6415	. . . . . 6 ⊢ (𝑏 = 𝑌 → (𝐹‘𝑏) = (𝐹‘𝑌))
16	15	oveq2d 6898	. . . . 5 ⊢ (𝑏 = 𝑌 → (𝑋 × (𝐹‘𝑏)) = (𝑋 × (𝐹‘𝑌)))
17	14, 16	eqeq12d 2818	. . . 4 ⊢ (𝑏 = 𝑌 → ((𝐹‘(𝑋 · 𝑏)) = (𝑋 × (𝐹‘𝑏)) ↔ (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹‘𝑌))))
18	12, 17	rspc2v 3514	. . 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 1145	1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐸) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 385   ∧ w3a 1108   = wceq 1653   ∈ wcel 2157  ∀wral 3093  ‘cfv 6105  (class class class)co 6882  Basecbs 16188  Scalarcsca 16274   ·_𝑠 cvsca 16275   GrpHom cghm 17974  LModclmod 19185   LMHom clmhm 19344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2379  ax-ext 2781  ax-sep 4979  ax-nul 4987  ax-pow 5039  ax-pr 5101

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2593  df-eu 2611  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ral 3098  df-rex 3099  df-rab 3102  df-v 3391  df-sbc 3638  df-dif 3776  df-un 3778  df-in 3780  df-ss 3787  df-nul 4120  df-if 4282  df-sn 4373  df-pr 4375  df-op 4379  df-uni 4633  df-br 4848  df-opab 4910  df-id 5224  df-xp 5322  df-rel 5323  df-cnv 5324  df-co 5325  df-dm 5326  df-iota 6068  df-fun 6107  df-fv 6113  df-ov 6885  df-oprab 6886  df-mpt2 6887  df-lmhm 19347

This theorem is referenced by:  islmhm2  19363  lmhmco  19368  lmhmplusg  19369  lmhmvsca  19370  lmhmf1o  19371  lmhmima  19372  lmhmpreima  19373  reslmhm  19377  reslmhm2  19378  reslmhm2b  19379  lmhmeql  19380  ipass  20318  lindfmm  20495  nmoleub2lem3  23246  nmoleub3  23250  mendassa  38553