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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem8 | Structured version Visualization version GIF version |
Description: Lemma for lcf1o 41534 and lcfr 41568. (Contributed by NM, 21-Feb-2015.) |
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 })) |
Ref | Expression |
---|---|
lcfrlem8 | ⊢ (𝜑 → (𝐽‘𝑋) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem8.x | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
2 | sneq 4641 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
3 | 2 | fveq2d 6911 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ( ⊥ ‘{𝑥}) = ( ⊥ ‘{𝑋})) |
4 | oveq2 7439 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑘 · 𝑥) = (𝑘 · 𝑋)) | |
5 | 4 | oveq2d 7447 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑤 + (𝑘 · 𝑥)) = (𝑤 + (𝑘 · 𝑋))) |
6 | 5 | eqeq2d 2746 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑋)))) |
7 | 3, 6 | rexeqbidv 3345 | . . . . 5 ⊢ (𝑥 = 𝑋 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) |
8 | 7 | riotabidv 7390 | . . . 4 ⊢ (𝑥 = 𝑋 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) |
9 | 8 | mpteq2dv 5250 | . . 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 7248 | . 2 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → (𝐽‘𝑋) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))) |
13 | 1, 12 | syl 17 | 1 ⊢ (𝜑 → (𝐽‘𝑋) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 {crab 3433 ∖ cdif 3960 {csn 4631 ↦ cmpt 5231 ‘cfv 6563 ℩crio 7387 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17486 LFnlclfn 39039 LKerclk 39067 LDualcld 39105 HLchlt 39332 LHypclh 39967 DVecHcdvh 41061 ocHcoch 41330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 |
This theorem is referenced by: lcfrlem9 41533 lcfrlem10 41535 lcfrlem11 41536 |
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