| Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem8 | Structured version Visualization version GIF version | ||
| Description: Lemma for lcf1o 41553 and lcfr 41587. (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 4636 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → {𝑥} = {𝑋}) | |
| 3 | 2 | fveq2d 6910 | . . . . . 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 2748 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑣 = (𝑤 + (𝑘 · 𝑥)) ↔ 𝑣 = (𝑤 + (𝑘 · 𝑋)))) |
| 7 | 3, 6 | rexeqbidv 3347 | . . . . 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 5244 | . . 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 1540 ∈ wcel 2108 ∃wrex 3070 {crab 3436 ∖ cdif 3948 {csn 4626 ↦ cmpt 5225 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 LFnlclfn 39058 LKerclk 39086 LDualcld 39124 HLchlt 39351 LHypclh 39986 DVecHcdvh 41080 ocHcoch 41349 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 |
| This theorem is referenced by: lcfrlem9 41552 lcfrlem10 41554 lcfrlem11 41555 |
| Copyright terms: Public domain | W3C validator |