Theorem lcfrlem8 42301

Description: Lemma for lcf1o 42303 and lcfr 42337. (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 4600	. . . . . . 7 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
3	2	fveq2d 6887	. . . . . 6 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘{𝑥}) = ( ⊥ ‘{𝑋}))
4		oveq2 7420	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑘 · 𝑥) = (𝑘 · 𝑋))
5	4	oveq2d 7428	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑤 + (𝑘 · 𝑥)) = (𝑤 + (𝑘 · 𝑋)))
6	5	eqeq2d 2774	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑋))))
7	3, 6	rexeqbidv 3339	. . . . 5 ⊢ (𝑥 = 𝑋 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))
8	7	riotabidv 7371	. . . 4 ⊢ (𝑥 = 𝑋 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))
9	8	mpteq2dv 5206	. . 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 7228	. 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)))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  {crab 3416   ∖ cdif 3903  {csn 4590   ↦ cmpt 5193  ‘cfv 6538  ℩crio 7368  (class class class)co 7412  Basecbs 17270  +_gcplusg 17311  Scalarcsca 17314   ·_𝑠 cvsca 17315  0_gc0g 17493  LFnlclfn 39809  LKerclk 39837  LDualcld 39875  HLchlt 40102  LHypclh 40736  DVecHcdvh 41830  ocHcoch 42099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  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-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415

This theorem is referenced by:  lcfrlem9  42302  lcfrlem10  42304  lcfrlem11  42305