Theorem lfli 40258

Description: Property of a linear functional. (lnfnli 32654 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 40257	. . . 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 505	. . 3 ⊢ ((𝑊 ∈ 𝑍 ∧ 𝐺 ∈ 𝐹) → ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))
11	10	3adant3 1150	. 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 7434	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟 · 𝑥) = (𝑅 · 𝑥))
13	12	fvoveq1d 7449	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = (𝐺‘((𝑅 · 𝑥) + 𝑦)))
14		oveq1 7434	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟 × (𝐺‘𝑥)) = (𝑅 × (𝐺‘𝑥)))
15	14	oveq1d 7442	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) = ((𝑅 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))
16	13, 15	eqeq12d 2779	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) ↔ (𝐺‘((𝑅 · 𝑥) + 𝑦)) = ((𝑅 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦))))
17		oveq2 7435	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑅 · 𝑥) = (𝑅 · 𝑋))
18	17	fvoveq1d 7449	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐺‘((𝑅 · 𝑥) + 𝑦)) = (𝐺‘((𝑅 · 𝑋) + 𝑦)))
19		fveq2 6893	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
20	19	oveq2d 7443	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑅 × (𝐺‘𝑥)) = (𝑅 × (𝐺‘𝑋)))
21	20	oveq1d 7442	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑅 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑦)))
22	18, 21	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐺‘((𝑅 · 𝑥) + 𝑦)) = ((𝑅 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) ↔ (𝐺‘((𝑅 · 𝑋) + 𝑦)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑦))))
23		oveq2 7435	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑅 · 𝑋) + 𝑦) = ((𝑅 · 𝑋) + 𝑌))
24	23	fveq2d 6897	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐺‘((𝑅 · 𝑋) + 𝑦)) = (𝐺‘((𝑅 · 𝑋) + 𝑌)))
25		fveq2 6893	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝐺‘𝑦) = (𝐺‘𝑌))
26	25	oveq2d 7443	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑦)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑌)))
27	24, 26	eqeq12d 2779	. . . 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 3594	. . 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 1153	. 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 16	1 ⊢ ((𝑊 ∈ 𝑍 ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐺‘((𝑅 · 𝑋) + 𝑌)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑌)))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  ∀wral 3079  ⟶wf 6542  ‘cfv 6546  (class class class)co 7427  Basecbs 17399  +_gcplusg 17440  ._rcmulr 17441  Scalarcsca 17443   ·_𝑠 cvsca 17444  LFnlclfn 40254

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-sbc 3742  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-map 8860  df-lfl 40255

This theorem is used by:  lfl0  40262  lfladd  40263  lflsub  40264  lflmul  40265  lflnegcl  40272  lflvscl  40274  lkrlss  40292  hdmapln1  43103