Theorem lfli 39108

Description: Property of a linear functional. (lnfnli 32020 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 39107	. . . 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 499	. . 3 ⊢ ((𝑊 ∈ 𝑍 ∧ 𝐺 ∈ 𝐹) → ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))
11	10	3adant3 1132	. 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 7353	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟 · 𝑥) = (𝑅 · 𝑥))
13	12	fvoveq1d 7368	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = (𝐺‘((𝑅 · 𝑥) + 𝑦)))
14		oveq1 7353	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟 × (𝐺‘𝑥)) = (𝑅 × (𝐺‘𝑥)))
15	14	oveq1d 7361	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) = ((𝑅 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))
16	13, 15	eqeq12d 2747	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) ↔ (𝐺‘((𝑅 · 𝑥) + 𝑦)) = ((𝑅 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦))))
17		oveq2 7354	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑅 · 𝑥) = (𝑅 · 𝑋))
18	17	fvoveq1d 7368	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐺‘((𝑅 · 𝑥) + 𝑦)) = (𝐺‘((𝑅 · 𝑋) + 𝑦)))
19		fveq2 6822	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
20	19	oveq2d 7362	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑅 × (𝐺‘𝑥)) = (𝑅 × (𝐺‘𝑋)))
21	20	oveq1d 7361	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑅 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑦)))
22	18, 21	eqeq12d 2747	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐺‘((𝑅 · 𝑥) + 𝑦)) = ((𝑅 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) ↔ (𝐺‘((𝑅 · 𝑋) + 𝑦)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑦))))
23		oveq2 7354	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑅 · 𝑋) + 𝑦) = ((𝑅 · 𝑋) + 𝑌))
24	23	fveq2d 6826	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐺‘((𝑅 · 𝑋) + 𝑦)) = (𝐺‘((𝑅 · 𝑋) + 𝑌)))
25		fveq2 6822	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝐺‘𝑦) = (𝐺‘𝑌))
26	25	oveq2d 7362	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑦)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑌)))
27	24, 26	eqeq12d 2747	. . . 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 3588	. . 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 1135	. 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 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  +_gcplusg 17161  ._rcmulr 17162  Scalarcsca 17164   ·_𝑠 cvsca 17165  LFnlclfn 39104

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-map 8752  df-lfl 39105

This theorem is referenced by:  lfl0  39112  lfladd  39113  lflsub  39114  lflmul  39115  lflnegcl  39122  lflvscl  39124  lkrlss  39142  hdmapln1  41953