Theorem lcfrlem8 42364

Description: Lemma for lcf1o 42366 and lcfr 42400. (Contributed by NM, 21-Feb-2015.)

Hypotheses

Ref	Expression
lcf1o.h	⊢ 𝐻 = (LHyp‘𝐾)
lcf1o.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lcf1o.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcf1o.v	⊢ 𝑉 = (Base‘𝑈)
lcf1o.a	⊢ + = (+g‘𝑈)
lcf1o.t	⊢ · = ( ·𝑠 ‘𝑈)
lcf1o.s	⊢ 𝑆 = (Scalar‘𝑈)
lcf1o.r	⊢ 𝑅 = (Base‘𝑆)
lcf1o.z	⊢ 0 = (0g‘𝑈)
lcf1o.f	⊢ 𝐹 = (LFnl‘𝑈)
lcf1o.l	⊢ 𝐿 = (LKer‘𝑈)
lcf1o.d	⊢ 𝐷 = (LDual‘𝑈)
lcf1o.q	⊢ 𝑄 = (0g‘𝐷)
lcf1o.c	⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
lcf1o.j	⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
lcflo.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lcfrlem8.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))

Assertion

Ref	Expression
lcfrlem8	⊢ (𝜑 → (𝐽‘𝑋) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))))

Distinct variable groups:   𝑥,𝑤, ⊥   𝑥, 0   𝑥,𝑣,𝑉   𝑥, ·   𝑣,𝑘,𝑤,𝑥,𝑋   𝑥, +   𝑥,𝑅

Allowed substitution hints:   𝜑(𝑥, 𝑤, 𝑣, 𝑓, 𝑘)   𝐶(𝑥, 𝑤, 𝑣, 𝑓, 𝑘)   𝐷(𝑥, 𝑤, 𝑣, 𝑓, 𝑘)   + (𝑤, 𝑣, 𝑓, 𝑘)   𝑄(𝑥, 𝑤, 𝑣, 𝑓, 𝑘)   𝑅(𝑤, 𝑣, 𝑓, 𝑘)   𝑆(𝑥, 𝑤, 𝑣, 𝑓, 𝑘)   · (𝑤, 𝑣, 𝑓, 𝑘)   𝑈(𝑥, 𝑤, 𝑣, 𝑓, 𝑘)   𝐹(𝑥, 𝑤, 𝑣, 𝑓, 𝑘)   𝐻(𝑥, 𝑤, 𝑣, 𝑓, 𝑘)   𝐽(𝑥, 𝑤, 𝑣, 𝑓, 𝑘)   𝐾(𝑥, 𝑤, 𝑣, 𝑓, 𝑘)   𝐿(𝑥, 𝑤, 𝑣, 𝑓, 𝑘)   ⊥ (𝑣, 𝑓, 𝑘)   𝑉(𝑤, 𝑓, 𝑘)   𝑊(𝑥, 𝑤, 𝑣, 𝑓, 𝑘)   𝑋(𝑓)   0 (𝑤, 𝑣, 𝑓, 𝑘)

Proof of Theorem lcfrlem8

Step	Hyp	Ref	Expression
1		lcfrlem8.x	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
2		sneq 4604	. . . . . . 7 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
3	2	fveq2d 6892	. . . . . 6 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘{𝑥}) = ( ⊥ ‘{𝑋}))
4		oveq2 7431	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑘 · 𝑥) = (𝑘 · 𝑋))
5	4	oveq2d 7439	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑤 + (𝑘 · 𝑥)) = (𝑤 + (𝑘 · 𝑋)))
6	5	eqeq2d 2777	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑋))))
7	3, 6	rexeqbidv 3342	. . . . 5 ⊢ (𝑥 = 𝑋 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))
8	7	riotabidv 7382	. . . 4 ⊢ (𝑥 = 𝑋 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))
9	8	mpteq2dv 5210	. . 3 ⊢ (𝑥 = 𝑋 → (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))))
10		lcf1o.j	. . 3 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
11		lcf1o.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
12	9, 10, 11	mptfvmpt 7233	. 2 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → (𝐽‘𝑋) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))))
13	1, 12	syl 18	1 ⊢ (𝜑 → (𝐽‘𝑋) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3092  {crab 3419   ∖ cdif 3905  {csn 4594   ↦ cmpt 5197  ‘cfv 6543  ℩crio 7379  (class class class)co 7423  Basecbs 17294  +_gcplusg 17335  Scalarcsca 17338   ·_𝑠 cvsca 17339  0_gc0g 17517  LFnlclfn 39872  LKerclk 39900  LDualcld 39938  HLchlt 40165  LHypclh 40799  DVecHcdvh 41893  ocHcoch 42162

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426

This theorem is used by:  lcfrlem9  42365  lcfrlem10  42367  lcfrlem11  42368