Theorem lfli 39685

Description: Property of a linear functional. (lnfnli 32243 analog.) (Contributed by NM, 16-Apr-2014.)

Hypotheses

Ref	Expression
lflset.v	⊢ 𝑉 = (Base‘𝑊)
lflset.a	⊢ + = (+g‘𝑊)
lflset.d	⊢ 𝐷 = (Scalar‘𝑊)
lflset.s	⊢ · = ( ·𝑠 ‘𝑊)
lflset.k	⊢ 𝐾 = (Base‘𝐷)
lflset.p	⊢ ⨣ = (+g‘𝐷)
lflset.t	⊢ × = (.r‘𝐷)
lflset.f	⊢ 𝐹 = (LFnl‘𝑊)

Assertion

Ref	Expression
lfli	⊢ ((𝑊 ∈ 𝑍 ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑌)))

Proof of Theorem lfli

Dummy variables 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lflset.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
2		lflset.a	. . . . 5 ⊢ + = (+g‘𝑊)
3		lflset.d	. . . . 5 ⊢ 𝐷 = (Scalar‘𝑊)
4		lflset.s	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
5		lflset.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐷)
6		lflset.p	. . . . 5 ⊢ ⨣ = (+g‘𝐷)
7		lflset.t	. . . . 5 ⊢ × = (.r‘𝐷)
8		lflset.f	. . . . 5 ⊢ 𝐹 = (LFnl‘𝑊)
9	1, 2, 3, 4, 5, 6, 7, 8	islfl 39684	. . . 4 ⊢ (𝑊 ∈ 𝑍 → (𝐺 ∈ 𝐹 ↔ (𝐺:𝑉⟶𝐾 ∧ ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))))
10	9	simplbda 503	. . 3 ⊢ ((𝑊 ∈ 𝑍 ∧ 𝐺 ∈ 𝐹) → ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))
11	10	3adant3 1145	. 2 ⊢ ((𝑊 ∈ 𝑍 ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))
12		oveq1 7403	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟 · 𝑥) = (𝑅 · 𝑥))
13	12	fvoveq1d 7418	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = (𝐺‘((𝑅 · 𝑥) + 𝑦)))
14		oveq1 7403	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟 × (𝐺‘𝑥)) = (𝑅 × (𝐺‘𝑥)))
15	14	oveq1d 7411	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) = ((𝑅 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))
16	13, 15	eqeq12d 2778	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) ↔ (𝐺‘((𝑅 · 𝑥) + 𝑦)) = ((𝑅 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦))))
17		oveq2 7404	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑅 · 𝑥) = (𝑅 · 𝑋))
18	17	fvoveq1d 7418	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐺‘((𝑅 · 𝑥) + 𝑦)) = (𝐺‘((𝑅 · 𝑋) + 𝑦)))
19		fveq2 6867	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
20	19	oveq2d 7412	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑅 × (𝐺‘𝑥)) = (𝑅 × (𝐺‘𝑋)))
21	20	oveq1d 7411	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑅 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑦)))
22	18, 21	eqeq12d 2778	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐺‘((𝑅 · 𝑥) + 𝑦)) = ((𝑅 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) ↔ (𝐺‘((𝑅 · 𝑋) + 𝑦)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑦))))
23		oveq2 7404	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑅 · 𝑋) + 𝑦) = ((𝑅 · 𝑋) + 𝑌))
24	23	fveq2d 6871	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐺‘((𝑅 · 𝑋) + 𝑦)) = (𝐺‘((𝑅 · 𝑋) + 𝑌)))
25		fveq2 6867	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝐺‘𝑦) = (𝐺‘𝑌))
26	25	oveq2d 7412	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑦)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑌)))
27	24, 26	eqeq12d 2778	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝐺‘((𝑅 · 𝑋) + 𝑦)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑦)) ↔ (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑌))))
28	16, 22, 27	rspc3v 3597	. . 3 ⊢ ((𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) → (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑌))))
29	28	3ad2ant3 1148	. 2 ⊢ ((𝑊 ∈ 𝑍 ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) → (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑌))))
30	11, 29	mpd 15	1 ⊢ ((𝑊 ∈ 𝑍 ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑌)))

Syntax hints:   → wi 4   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  +_gcplusg 17286  ._rcmulr 17287  Scalarcsca 17289   ·_𝑠 cvsca 17290  LFnlclfn 39681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-map 8810  df-lfl 39682

This theorem is referenced by:  lfl0  39689  lfladd  39690  lflsub  39691  lflmul  39692  lflnegcl  39699  lflvscl  39701  lkrlss  39719  hdmapln1  42530