Theorem lcfrlem8 42574

Description: Lemma for lcf1o 42576 and lcfr 42610. (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 4594	. . . . . . 7 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋})
3	2	fveq2d 6881	. . . . . 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 2772	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑋))))
7	3, 6	rexeqbidv 3336	. . . . 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 5199	. . 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 7226	. 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 2145  ∃wrex 3087  {crab 3413   ∖ cdif 3896  {csn 4584   ↦ cmpt 5186  ‘cfv 6531  ℩crio 7368  (class class class)co 7412  Basecbs 17367  +_gcplusg 17408  Scalarcsca 17411   ·_𝑠 cvsca 17412  0_gc0g 17590  LFnlclfn 40082  LKerclk 40110  LDualcld 40148  HLchlt 40375  LHypclh 41009  DVecHcdvh 42103  ocHcoch 42372

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415

This theorem is used by:  lcfrlem9  42575  lcfrlem10  42577  lcfrlem11  42578