Theorem lfli 36199

Description: Property of a linear functional. (lnfnli 29819 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 36198	. . . 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 502	. . 3 ⊢ ((𝑊 ∈ 𝑍 ∧ 𝐺 ∈ 𝐹) → ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))
11	10	3adant3 1128	. 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 7165	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟 · 𝑥) = (𝑅 · 𝑥))
13	12	fvoveq1d 7180	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝐺‘((𝑟 · 𝑥) + 𝑦)) = (𝐺‘((𝑅 · 𝑥) + 𝑦)))
14		oveq1 7165	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟 × (𝐺‘𝑥)) = (𝑅 × (𝐺‘𝑥)))
15	14	oveq1d 7173	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) = ((𝑅 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)))
16	13, 15	eqeq12d 2839	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝐺‘((𝑟 · 𝑥) + 𝑦)) = ((𝑟 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) ↔ (𝐺‘((𝑅 · 𝑥) + 𝑦)) = ((𝑅 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦))))
17		oveq2 7166	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑅 · 𝑥) = (𝑅 · 𝑋))
18	17	fvoveq1d 7180	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐺‘((𝑅 · 𝑥) + 𝑦)) = (𝐺‘((𝑅 · 𝑋) + 𝑦)))
19		fveq2 6672	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
20	19	oveq2d 7174	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑅 × (𝐺‘𝑥)) = (𝑅 × (𝐺‘𝑋)))
21	20	oveq1d 7173	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑅 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑦)))
22	18, 21	eqeq12d 2839	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐺‘((𝑅 · 𝑥) + 𝑦)) = ((𝑅 × (𝐺‘𝑥)) ⨣ (𝐺‘𝑦)) ↔ (𝐺‘((𝑅 · 𝑋) + 𝑦)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑦))))
23		oveq2 7166	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑅 · 𝑋) + 𝑦) = ((𝑅 · 𝑋) + 𝑌))
24	23	fveq2d 6676	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐺‘((𝑅 · 𝑋) + 𝑦)) = (𝐺‘((𝑅 · 𝑋) + 𝑌)))
25		fveq2 6672	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝐺‘𝑦) = (𝐺‘𝑌))
26	25	oveq2d 7174	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑦)) = ((𝑅 × (𝐺‘𝑋)) ⨣ (𝐺‘𝑌)))
27	24, 26	eqeq12d 2839	. . . 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 3638	. . 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 1131	. 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 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  +_gcplusg 16567  ._rcmulr 16568  Scalarcsca 16570   ·_𝑠 cvsca 16571  LFnlclfn 36195

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-map 8410  df-lfl 36196

This theorem is referenced by:  lfl0  36203  lfladd  36204  lflsub  36205  lflmul  36206  lflnegcl  36213  lflvscl  36215  lkrlss  36233  hdmapln1  39044