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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 42606 | . . . . . 6 ⊢ (𝜑 → ∼ Er 𝐵) |
8 | prjspner01.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | 7, 8 | erref 8764 | . . . . 5 ⊢ (𝜑 → 𝑋 ∼ 𝑋) |
10 | 9 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑋‘0) = 0 ) → 𝑋 ∼ 𝑋) |
11 | 7 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → ∼ Er 𝐵) |
12 | prjspner01.0 | . . . . . 6 ⊢ 0 = (0g‘𝐾) | |
13 | 6 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → 𝐾 ∈ DivRing) |
14 | 8 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → 𝑋 ∈ 𝐵) |
15 | ovexd 7466 | . . . . . . . . 9 ⊢ (𝜑 → (0...𝑁) ∈ V) | |
16 | 8, 3 | eleqtrdi 2849 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ((Base‘𝑊) ∖ {(0g‘𝑊)})) |
17 | 16 | eldifad 3975 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊)) |
18 | eqid 2735 | . . . . . . . . . 10 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
19 | 2, 4, 18 | frlmbasf 21798 | . . . . . . . . 9 ⊢ (((0...𝑁) ∈ V ∧ 𝑋 ∈ (Base‘𝑊)) → 𝑋:(0...𝑁)⟶𝑆) |
20 | 15, 17, 19 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → 𝑋:(0...𝑁)⟶𝑆) |
21 | prjspner01.n | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
22 | 0elfz 13661 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁)) | |
23 | 21, 22 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 0 ∈ (0...𝑁)) |
24 | 20, 23 | ffvelcdmd 7105 | . . . . . . 7 ⊢ (𝜑 → (𝑋‘0) ∈ 𝑆) |
25 | neqne 2946 | . . . . . . 7 ⊢ (¬ (𝑋‘0) = 0 → (𝑋‘0) ≠ 0 ) | |
26 | prjspner01.i | . . . . . . . 8 ⊢ 𝐼 = (invr‘𝐾) | |
27 | 4, 12, 26 | drnginvrcl 20770 | . . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (𝑋‘0) ∈ 𝑆 ∧ (𝑋‘0) ≠ 0 ) → (𝐼‘(𝑋‘0)) ∈ 𝑆) |
28 | 6, 24, 25, 27 | syl2an3an 1421 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → (𝐼‘(𝑋‘0)) ∈ 𝑆) |
29 | 4, 12, 26 | drnginvrn0 20771 | . . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (𝑋‘0) ∈ 𝑆 ∧ (𝑋‘0) ≠ 0 ) → (𝐼‘(𝑋‘0)) ≠ 0 ) |
30 | 6, 24, 25, 29 | syl2an3an 1421 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → (𝐼‘(𝑋‘0)) ≠ 0 ) |
31 | 1, 2, 3, 4, 5, 12, 13, 14, 28, 30 | prjspnvs 42607 | . . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → ((𝐼‘(𝑋‘0)) · 𝑋) ∼ 𝑋) |
32 | 11, 31 | ersym 8756 | . . . 4 ⊢ ((𝜑 ∧ ¬ (𝑋‘0) = 0 ) → 𝑋 ∼ ((𝐼‘(𝑋‘0)) · 𝑋)) |
33 | 10, 32 | ifpimpda 1080 | . . 3 ⊢ (𝜑 → if-((𝑋‘0) = 0 , 𝑋 ∼ 𝑋, 𝑋 ∼ ((𝐼‘(𝑋‘0)) · 𝑋))) |
34 | brif2 42242 | . . 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 234 | . 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 6906 | . . . . 5 ⊢ (𝑏 = 𝑋 → (𝑏‘0) = (𝑋‘0)) | |
38 | 37 | eqeq1d 2737 | . . . 4 ⊢ (𝑏 = 𝑋 → ((𝑏‘0) = 0 ↔ (𝑋‘0) = 0 )) |
39 | id 22 | . . . 4 ⊢ (𝑏 = 𝑋 → 𝑏 = 𝑋) | |
40 | 37 | fveq2d 6911 | . . . . 5 ⊢ (𝑏 = 𝑋 → (𝐼‘(𝑏‘0)) = (𝐼‘(𝑋‘0))) |
41 | 40, 39 | oveq12d 7449 | . . . 4 ⊢ (𝑏 = 𝑋 → ((𝐼‘(𝑏‘0)) · 𝑏) = ((𝐼‘(𝑋‘0)) · 𝑋)) |
42 | 38, 39, 41 | ifbieq12d 4559 | . . 3 ⊢ (𝑏 = 𝑋 → if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)) = if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋))) |
43 | ovexd 7466 | . . . 4 ⊢ (𝜑 → ((𝐼‘(𝑋‘0)) · 𝑋) ∈ V) | |
44 | 8, 43 | ifexd 4579 | . . 3 ⊢ (𝜑 → if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋)) ∈ V) |
45 | 36, 42, 8, 44 | fvmptd3 7039 | . 2 ⊢ (𝜑 → (𝐹‘𝑋) = if((𝑋‘0) = 0 , 𝑋, ((𝐼‘(𝑋‘0)) · 𝑋))) |
46 | 35, 45 | breqtrrd 5176 | 1 ⊢ (𝜑 → 𝑋 ∼ (𝐹‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 if-wif 1062 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 Vcvv 3478 ∖ cdif 3960 ifcif 4531 {csn 4631 class class class wbr 5148 {copab 5210 ↦ cmpt 5231 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Er wer 8741 0cc0 11153 ℕ0cn0 12524 ...cfz 13544 Basecbs 17245 ·𝑠 cvsca 17302 0gc0g 17486 invrcinvr 20404 DivRingcdr 20746 freeLMod cfrlm 21784 |
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-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 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-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 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-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 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-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 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 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-prds 17494 df-pws 17496 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-subrg 20587 df-drng 20748 df-lmod 20877 df-lss 20948 df-lvec 21120 df-sra 21190 df-rgmod 21191 df-dsmm 21770 df-frlm 21785 |
This theorem is referenced by: prjspner1 42613 |
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