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Mirrors > Home > MPE Home > Th. List > Mathboxes > prjspner01 | Structured version Visualization version GIF version |
Description: Any vector is equivalent to a vector whose zeroth coordinate is 0 or 1 (proof of the equivalence). (Contributed by SN, 13-Aug-2023.) |
Ref | Expression |
---|---|
prjspner01.e | ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))} |
prjspner01.f | ⊢ 𝐹 = (𝑏 ∈ 𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏))) |
prjspner01.b | ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)}) |
prjspner01.w | ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁)) |
prjspner01.t | ⊢ · = ( ·𝑠 ‘𝑊) |
prjspner01.s | ⊢ 𝑆 = (Base‘𝐾) |
prjspner01.0 | ⊢ 0 = (0g‘𝐾) |
prjspner01.i | ⊢ 𝐼 = (invr‘𝐾) |
prjspner01.k | ⊢ (𝜑 → 𝐾 ∈ DivRing) |
prjspner01.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
prjspner01.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
prjspner01 | ⊢ (𝜑 → 𝑋 ∼ (𝐹‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prjspner01.e | . . . . . . 7 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))} | |
2 | prjspner01.w | . . . . . . 7 ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁)) | |
3 | prjspner01.b | . . . . . . 7 ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)}) | |
4 | prjspner01.s | . . . . . . 7 ⊢ 𝑆 = (Base‘𝐾) | |
5 | prjspner01.t | . . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊) | |
6 | prjspner01.k | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ DivRing) | |
7 | 1, 2, 3, 4, 5, 6 | prjspner 39990 | . . . . . 6 ⊢ (𝜑 → ∼ Er 𝐵) |
8 | prjspner01.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | 7, 8 | erref 8325 | . . . . 5 ⊢ (𝜑 → 𝑋 ∼ 𝑋) |
10 | 9 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ (𝑋‘0) = 0 ) → 𝑋 ∼ 𝑋) |
11 | 7 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → ∼ Er 𝐵) |
12 | prjspner01.0 | . . . . . 6 ⊢ 0 = (0g‘𝐾) | |
13 | 6 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → 𝐾 ∈ DivRing) |
14 | 8 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → 𝑋 ∈ 𝐵) |
15 | ovexd 7191 | . . . . . . . . 9 ⊢ (𝜑 → (0...𝑁) ∈ V) | |
16 | 8, 3 | eleqtrdi 2862 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)})) |
17 | 16 | eldifad 3872 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊)) |
18 | eqid 2758 | . . . . . . . . . 10 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
19 | 2, 4, 18 | frlmbasf 20538 | . . . . . . . . 9 ⊢ (((0...𝑁) ∈ V ∧ 𝑋 ∈ (Base‘𝑊)) → 𝑋:(0...𝑁)⟶𝑆) |
20 | 15, 17, 19 | syl2anc 587 | . . . . . . . 8 ⊢ (𝜑 → 𝑋:(0...𝑁)⟶𝑆) |
21 | prjspner01.n | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
22 | 0elfz 13066 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁)) | |
23 | 21, 22 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 0 ∈ (0...𝑁)) |
24 | 20, 23 | ffvelrnd 6849 | . . . . . . 7 ⊢ (𝜑 → (𝑋‘0) ∈ 𝑆) |
25 | neqne 2959 | . . . . . . 7 ⊢ (¬ (𝑋‘0) = 0 → (𝑋‘0) ≠ 0 ) | |
26 | prjspner01.i | . . . . . . . 8 ⊢ 𝐼 = (invr‘𝐾) | |
27 | 4, 12, 26 | drnginvrcl 19600 | . . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (𝑋‘0) ∈ 𝑆 ∧ (𝑋‘0) ≠ 0 ) → (𝐼‘(𝑋‘0)) ∈ 𝑆) |
28 | 6, 24, 25, 27 | syl2an3an 1419 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → (𝐼‘(𝑋‘0)) ∈ 𝑆) |
29 | 4, 12, 26 | drnginvrn0 19601 | . . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (𝑋‘0) ∈ 𝑆 ∧ (𝑋‘0) ≠ 0 ) → (𝐼‘(𝑋‘0)) ≠ 0 ) |
30 | 6, 24, 25, 29 | syl2an3an 1419 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → (𝐼‘(𝑋‘0)) ≠ 0 ) |
31 | 1, 2, 3, 4, 5, 12, 13, 14, 28, 30 | prjspnvs 39991 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → ((𝐼‘(𝑋‘0)) · 𝑋) ∼ 𝑋) |
32 | 11, 31 | ersym 8317 | . . . 4 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → 𝑋 ∼ ((𝐼‘(𝑋‘0)) · 𝑋)) |
33 | 10, 32 | ifpimpda 1078 | . . 3 ⊢ (𝜑 → if-((𝑋‘0) = 0 , 𝑋 ∼ 𝑋, 𝑋 ∼ ((𝐼‘(𝑋‘0)) · 𝑋))) |
34 | brif2 39741 | . . 3 ⊢ (𝑋 ∼ if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)) ↔ if-((𝑋‘0) = 0 , 𝑋 ∼ 𝑋, 𝑋 ∼ ((𝐼‘(𝑋‘0)) · 𝑋))) | |
35 | 33, 34 | sylibr 237 | . 2 ⊢ (𝜑 → 𝑋 ∼ if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋))) |
36 | prjspner01.f | . . 3 ⊢ 𝐹 = (𝑏 ∈ 𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏))) | |
37 | fveq1 6662 | . . . . 5 ⊢ (𝑏 = 𝑋 → (𝑏‘0) = (𝑋‘0)) | |
38 | 37 | eqeq1d 2760 | . . . 4 ⊢ (𝑏 = 𝑋 → ((𝑏‘0) = 0 ↔ (𝑋‘0) = 0 )) |
39 | id 22 | . . . 4 ⊢ (𝑏 = 𝑋 → 𝑏 = 𝑋) | |
40 | 37 | fveq2d 6667 | . . . . 5 ⊢ (𝑏 = 𝑋 → (𝐼‘(𝑏‘0)) = (𝐼‘(𝑋‘0))) |
41 | 40, 39 | oveq12d 7174 | . . . 4 ⊢ (𝑏 = 𝑋 → ((𝐼‘(𝑏‘0)) · 𝑏) = ((𝐼‘(𝑋‘0)) · 𝑋)) |
42 | 38, 39, 41 | ifbieq12d 4451 | . . 3 ⊢ (𝑏 = 𝑋 → if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)) = if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋))) |
43 | ovexd 7191 | . . . 4 ⊢ (𝜑 → ((𝐼‘(𝑋‘0)) · 𝑋) ∈ V) | |
44 | 8, 43 | ifexd 4471 | . . 3 ⊢ (𝜑 → if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)) ∈ V) |
45 | 36, 42, 8, 44 | fvmptd3 6787 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) = if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋))) |
46 | 35, 45 | breqtrrd 5064 | 1 ⊢ (𝜑 → 𝑋 ∼ (𝐹‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 if-wif 1058 = wceq 1538 ∈ wcel 2111 ≠ wne 2951 ∃wrex 3071 Vcvv 3409 ∖ cdif 3857 ifcif 4423 {csn 4525 class class class wbr 5036 {copab 5098 ↦ cmpt 5116 ⟶wf 6336 ‘cfv 6340 (class class class)co 7156 Er wer 8302 0cc0 10588 ℕ0cn0 11947 ...cfz 12952 Basecbs 16554 ·𝑠 cvsca 16640 0gc0g 16784 invrcinvr 19505 DivRingcdr 19583 freeLMod cfrlm 20524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ifp 1059 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7586 df-1st 7699 df-2nd 7700 df-supp 7842 df-tpos 7908 df-wrecs 7963 df-recs 8024 df-rdg 8062 df-1o 8118 df-er 8305 df-map 8424 df-ixp 8493 df-en 8541 df-dom 8542 df-sdom 8543 df-fin 8544 df-fsupp 8880 df-sup 8952 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924 df-nn 11688 df-2 11750 df-3 11751 df-4 11752 df-5 11753 df-6 11754 df-7 11755 df-8 11756 df-9 11757 df-n0 11948 df-z 12034 df-dec 12151 df-uz 12296 df-fz 12953 df-struct 16556 df-ndx 16557 df-slot 16558 df-base 16560 df-sets 16561 df-ress 16562 df-plusg 16649 df-mulr 16650 df-sca 16652 df-vsca 16653 df-ip 16654 df-tset 16655 df-ple 16656 df-ds 16658 df-hom 16660 df-cco 16661 df-0g 16786 df-prds 16792 df-pws 16794 df-mgm 17931 df-sgrp 17980 df-mnd 17991 df-grp 18185 df-minusg 18186 df-sbg 18187 df-subg 18356 df-mgp 19321 df-ur 19333 df-ring 19380 df-oppr 19457 df-dvdsr 19475 df-unit 19476 df-invr 19506 df-drng 19585 df-subrg 19614 df-lmod 19717 df-lss 19785 df-lvec 19956 df-sra 20025 df-rgmod 20026 df-dsmm 20510 df-frlm 20525 |
This theorem is referenced by: prjspner1 39995 |
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