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Mirrors > Home > MPE Home > Th. List > Mathboxes > prjspnfv01 | Structured version Visualization version GIF version |
Description: Any vector is equivalent to a vector whose zeroth coordinate is 0 or 1 (proof of the value of the zeroth coordinate). (Contributed by SN, 13-Aug-2023.) |
Ref | Expression |
---|---|
prjspnfv01.f | ⊢ 𝐹 = (𝑏 ∈ 𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏))) |
prjspnfv01.b | ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)}) |
prjspnfv01.w | ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁)) |
prjspnfv01.t | ⊢ · = ( ·𝑠 ‘𝑊) |
prjspnfv01.0 | ⊢ 0 = (0g‘𝐾) |
prjspnfv01.1 | ⊢ 1 = (1r‘𝐾) |
prjspnfv01.i | ⊢ 𝐼 = (invr‘𝐾) |
prjspnfv01.k | ⊢ (𝜑 → 𝐾 ∈ DivRing) |
prjspnfv01.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
prjspnfv01.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
prjspnfv01 | ⊢ (𝜑 → ((𝐹‘𝑋)‘0) = if((𝑋‘0) = 0 , 0 , 1 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prjspnfv01.f | . . . 4 ⊢ 𝐹 = (𝑏 ∈ 𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏))) | |
2 | fveq1 6765 | . . . . . 6 ⊢ (𝑏 = 𝑋 → (𝑏‘0) = (𝑋‘0)) | |
3 | 2 | eqeq1d 2740 | . . . . 5 ⊢ (𝑏 = 𝑋 → ((𝑏‘0) = 0 ↔ (𝑋‘0) = 0 )) |
4 | id 22 | . . . . 5 ⊢ (𝑏 = 𝑋 → 𝑏 = 𝑋) | |
5 | 2 | fveq2d 6770 | . . . . . 6 ⊢ (𝑏 = 𝑋 → (𝐼‘(𝑏‘0)) = (𝐼‘(𝑋‘0))) |
6 | 5, 4 | oveq12d 7285 | . . . . 5 ⊢ (𝑏 = 𝑋 → ((𝐼‘(𝑏‘0)) · 𝑏) = ((𝐼‘(𝑋‘0)) · 𝑋)) |
7 | 3, 4, 6 | ifbieq12d 4487 | . . . 4 ⊢ (𝑏 = 𝑋 → if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)) = if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋))) |
8 | prjspnfv01.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | ovexd 7302 | . . . . 5 ⊢ (𝜑 → ((𝐼‘(𝑋‘0)) · 𝑋) ∈ V) | |
10 | 8, 9 | ifexd 4507 | . . . 4 ⊢ (𝜑 → if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)) ∈ V) |
11 | 1, 7, 8, 10 | fvmptd3 6890 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) = if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋))) |
12 | 11 | fveq1d 6768 | . 2 ⊢ (𝜑 → ((𝐹‘𝑋)‘0) = (if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋))‘0)) |
13 | iffv 6783 | . . 3 ⊢ (if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋))‘0) = if((𝑋‘0) = 0 , (𝑋‘0), (((𝐼‘(𝑋‘0)) · 𝑋)‘0)) | |
14 | 13 | a1i 11 | . 2 ⊢ (𝜑 → (if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋))‘0) = if((𝑋‘0) = 0 , (𝑋‘0), (((𝐼‘(𝑋‘0)) · 𝑋)‘0))) |
15 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ (𝑋‘0) = 0 ) → (𝑋‘0) = 0 ) | |
16 | prjspnfv01.w | . . . . 5 ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁)) | |
17 | eqid 2738 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
18 | eqid 2738 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
19 | ovexd 7302 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → (0...𝑁) ∈ V) | |
20 | prjspnfv01.k | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ DivRing) | |
21 | ovexd 7302 | . . . . . . . 8 ⊢ (𝜑 → (0...𝑁) ∈ V) | |
22 | prjspnfv01.b | . . . . . . . . . 10 ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)}) | |
23 | 8, 22 | eleqtrdi 2849 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)})) |
24 | 23 | eldifad 3898 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊)) |
25 | 16, 18, 17 | frlmbasf 20977 | . . . . . . . 8 ⊢ (((0...𝑁) ∈ V ∧ 𝑋 ∈ (Base‘𝑊)) → 𝑋:(0...𝑁)⟶(Base‘𝐾)) |
26 | 21, 24, 25 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → 𝑋:(0...𝑁)⟶(Base‘𝐾)) |
27 | prjspnfv01.n | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
28 | 0elfz 13363 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁)) | |
29 | 27, 28 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ (0...𝑁)) |
30 | 26, 29 | ffvelrnd 6954 | . . . . . 6 ⊢ (𝜑 → (𝑋‘0) ∈ (Base‘𝐾)) |
31 | neqne 2951 | . . . . . 6 ⊢ (¬ (𝑋‘0) = 0 → (𝑋‘0) ≠ 0 ) | |
32 | prjspnfv01.0 | . . . . . . 7 ⊢ 0 = (0g‘𝐾) | |
33 | prjspnfv01.i | . . . . . . 7 ⊢ 𝐼 = (invr‘𝐾) | |
34 | 18, 32, 33 | drnginvrcl 20018 | . . . . . 6 ⊢ ((𝐾 ∈ DivRing ∧ (𝑋‘0) ∈ (Base‘𝐾) ∧ (𝑋‘0) ≠ 0 ) → (𝐼‘(𝑋‘0)) ∈ (Base‘𝐾)) |
35 | 20, 30, 31, 34 | syl2an3an 1421 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → (𝐼‘(𝑋‘0)) ∈ (Base‘𝐾)) |
36 | 24 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → 𝑋 ∈ (Base‘𝑊)) |
37 | 29 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → 0 ∈ (0...𝑁)) |
38 | prjspnfv01.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
39 | eqid 2738 | . . . . 5 ⊢ (.r‘𝐾) = (.r‘𝐾) | |
40 | 16, 17, 18, 19, 35, 36, 37, 38, 39 | frlmvscaval 20985 | . . . 4 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → (((𝐼‘(𝑋‘0)) · 𝑋)‘0) = ((𝐼‘(𝑋‘0))(.r‘𝐾)(𝑋‘0))) |
41 | prjspnfv01.1 | . . . . . 6 ⊢ 1 = (1r‘𝐾) | |
42 | 18, 32, 39, 41, 33 | drnginvrl 20020 | . . . . 5 ⊢ ((𝐾 ∈ DivRing ∧ (𝑋‘0) ∈ (Base‘𝐾) ∧ (𝑋‘0) ≠ 0 ) → ((𝐼‘(𝑋‘0))(.r‘𝐾)(𝑋‘0)) = 1 ) |
43 | 20, 30, 31, 42 | syl2an3an 1421 | . . . 4 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → ((𝐼‘(𝑋‘0))(.r‘𝐾)(𝑋‘0)) = 1 ) |
44 | 40, 43 | eqtrd 2778 | . . 3 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → (((𝐼‘(𝑋‘0)) · 𝑋)‘0) = 1 ) |
45 | 15, 44 | ifeq12da 4492 | . 2 ⊢ (𝜑 → if((𝑋‘0) = 0 , (𝑋‘0), (((𝐼‘(𝑋‘0)) · 𝑋)‘0)) = if((𝑋‘0) = 0 , 0 , 1 )) |
46 | 12, 14, 45 | 3eqtrd 2782 | 1 ⊢ (𝜑 → ((𝐹‘𝑋)‘0) = if((𝑋‘0) = 0 , 0 , 1 )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 Vcvv 3429 ∖ cdif 3883 ifcif 4459 {csn 4561 ↦ cmpt 5156 ⟶wf 6422 ‘cfv 6426 (class class class)co 7267 0cc0 10881 ℕ0cn0 12243 ...cfz 13249 Basecbs 16922 .rcmulr 16973 ·𝑠 cvsca 16976 0gc0g 17160 1rcur 19747 invrcinvr 19923 DivRingcdr 20001 freeLMod cfrlm 20963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-of 7523 df-om 7703 df-1st 7820 df-2nd 7821 df-supp 7965 df-tpos 8029 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-map 8604 df-ixp 8673 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-fsupp 9116 df-sup 9188 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-uz 12593 df-fz 13250 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-sca 16988 df-vsca 16989 df-ip 16990 df-tset 16991 df-ple 16992 df-ds 16994 df-hom 16996 df-cco 16997 df-0g 17162 df-prds 17168 df-pws 17170 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-grp 18590 df-minusg 18591 df-mgp 19731 df-ur 19748 df-ring 19795 df-oppr 19872 df-dvdsr 19893 df-unit 19894 df-invr 19924 df-drng 20003 df-sra 20444 df-rgmod 20445 df-dsmm 20949 df-frlm 20964 |
This theorem is referenced by: (None) |
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