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Mirrors > Home > MPE Home > Th. List > Mathboxes > lshpkrlem2 | Structured version Visualization version GIF version |
Description: Lemma for lshpkrex 33423. The value of tentative functional 𝐺 is a scalar. (Contributed by NM, 16-Jul-2014.) |
Ref | Expression |
---|---|
lshpkrlem.v | ⊢ 𝑉 = (Base‘𝑊) |
lshpkrlem.a | ⊢ + = (+g‘𝑊) |
lshpkrlem.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lshpkrlem.p | ⊢ ⊕ = (LSSum‘𝑊) |
lshpkrlem.h | ⊢ 𝐻 = (LSHyp‘𝑊) |
lshpkrlem.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lshpkrlem.u | ⊢ (𝜑 → 𝑈 ∈ 𝐻) |
lshpkrlem.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
lshpkrlem.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lshpkrlem.e | ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) |
lshpkrlem.d | ⊢ 𝐷 = (Scalar‘𝑊) |
lshpkrlem.k | ⊢ 𝐾 = (Base‘𝐷) |
lshpkrlem.t | ⊢ · = ( ·𝑠 ‘𝑊) |
lshpkrlem.o | ⊢ 0 = (0g‘𝐷) |
lshpkrlem.g | ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)))) |
Ref | Expression |
---|---|
lshpkrlem2 | ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lshpkrlem.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
2 | eqeq1 2614 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 = (𝑦 + (𝑘 · 𝑍)) ↔ 𝑋 = (𝑦 + (𝑘 · 𝑍)))) | |
3 | 2 | rexbidv 3034 | . . . . 5 ⊢ (𝑥 = 𝑋 → (∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)) ↔ ∃𝑦 ∈ 𝑈 𝑋 = (𝑦 + (𝑘 · 𝑍)))) |
4 | 3 | riotabidv 6513 | . . . 4 ⊢ (𝑥 = 𝑋 → (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍))) = (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑋 = (𝑦 + (𝑘 · 𝑍)))) |
5 | lshpkrlem.g | . . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑥 = (𝑦 + (𝑘 · 𝑍)))) | |
6 | riotaex 6515 | . . . 4 ⊢ (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑋 = (𝑦 + (𝑘 · 𝑍))) ∈ V | |
7 | 4, 5, 6 | fvmpt 6191 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝐺‘𝑋) = (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑋 = (𝑦 + (𝑘 · 𝑍)))) |
8 | 1, 7 | syl 17 | . 2 ⊢ (𝜑 → (𝐺‘𝑋) = (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑋 = (𝑦 + (𝑘 · 𝑍)))) |
9 | lshpkrlem.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
10 | lshpkrlem.a | . . . 4 ⊢ + = (+g‘𝑊) | |
11 | lshpkrlem.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
12 | lshpkrlem.p | . . . 4 ⊢ ⊕ = (LSSum‘𝑊) | |
13 | lshpkrlem.h | . . . 4 ⊢ 𝐻 = (LSHyp‘𝑊) | |
14 | lshpkrlem.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
15 | lshpkrlem.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐻) | |
16 | lshpkrlem.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
17 | lshpkrlem.e | . . . 4 ⊢ (𝜑 → (𝑈 ⊕ (𝑁‘{𝑍})) = 𝑉) | |
18 | lshpkrlem.d | . . . 4 ⊢ 𝐷 = (Scalar‘𝑊) | |
19 | lshpkrlem.k | . . . 4 ⊢ 𝐾 = (Base‘𝐷) | |
20 | lshpkrlem.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
21 | 9, 10, 11, 12, 13, 14, 15, 16, 1, 17, 18, 19, 20 | lshpsmreu 33414 | . . 3 ⊢ (𝜑 → ∃!𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑋 = (𝑦 + (𝑘 · 𝑍))) |
22 | riotacl 6525 | . . 3 ⊢ (∃!𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑋 = (𝑦 + (𝑘 · 𝑍)) → (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑋 = (𝑦 + (𝑘 · 𝑍))) ∈ 𝐾) | |
23 | 21, 22 | syl 17 | . 2 ⊢ (𝜑 → (℩𝑘 ∈ 𝐾 ∃𝑦 ∈ 𝑈 𝑋 = (𝑦 + (𝑘 · 𝑍))) ∈ 𝐾) |
24 | 8, 23 | eqeltrd 2688 | 1 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ∃!wreu 2898 {csn 4125 ↦ cmpt 4643 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Scalarcsca 15771 ·𝑠 cvsca 15772 0gc0g 15923 LSSumclsm 17872 LSpanclspn 18792 LVecclvec 18923 LSHypclsh 33280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-cntz 17573 df-lsm 17874 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-drng 18572 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lvec 18924 df-lshyp 33282 |
This theorem is referenced by: lshpkrlem4 33418 lshpkrlem5 33419 |
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