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Mirrors > Home > MPE Home > Th. List > Mathboxes > prjspner | Structured version Visualization version GIF version |
Description: The relation used to define ℙ𝕣𝕠𝕛 (and indirectly ℙ𝕣𝕠𝕛n through df-prjspn 42504) is an equivalence relation. This is a lemma that converts the equivalence relation used in results like prjspertr 42494 and prjspersym 42496 (see prjspnerlem 42506). Several theorems are covered in one thanks to the theorems around df-er 8759. (Contributed by SN, 14-Aug-2023.) |
Ref | Expression |
---|---|
prjspner.e | ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))} |
prjspner.w | ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁)) |
prjspner.b | ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)}) |
prjspner.s | ⊢ 𝑆 = (Base‘𝐾) |
prjspner.x | ⊢ · = ( ·𝑠 ‘𝑊) |
prjspner.k | ⊢ (𝜑 → 𝐾 ∈ DivRing) |
Ref | Expression |
---|---|
prjspner | ⊢ (𝜑 → ∼ Er 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prjspner.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ DivRing) | |
2 | ovexd 7480 | . . . 4 ⊢ (𝜑 → (0...𝑁) ∈ V) | |
3 | prjspner.w | . . . . 5 ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁)) | |
4 | 3 | frlmlvec 21799 | . . . 4 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → 𝑊 ∈ LVec) |
5 | 1, 2, 4 | syl2anc 583 | . . 3 ⊢ (𝜑 → 𝑊 ∈ LVec) |
6 | eqid 2734 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))} = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))} | |
7 | prjspner.b | . . . 4 ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)}) | |
8 | eqid 2734 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
9 | prjspner.x | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
10 | eqid 2734 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
11 | 6, 7, 8, 9, 10 | prjsper 42497 | . . 3 ⊢ (𝑊 ∈ LVec → {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))} Er 𝐵) |
12 | 5, 11 | syl 17 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))} Er 𝐵) |
13 | prjspner.e | . . . 4 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))} | |
14 | prjspner.s | . . . 4 ⊢ 𝑆 = (Base‘𝐾) | |
15 | 13, 3, 7, 14, 9 | prjspnerlem 42506 | . . 3 ⊢ (𝐾 ∈ DivRing → ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}) |
16 | ereq1 8766 | . . 3 ⊢ ( ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))} → ( ∼ Er 𝐵 ↔ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))} Er 𝐵)) | |
17 | 1, 15, 16 | 3syl 18 | . 2 ⊢ (𝜑 → ( ∼ Er 𝐵 ↔ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))} Er 𝐵)) |
18 | 12, 17 | mpbird 257 | 1 ⊢ (𝜑 → ∼ Er 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2103 ∃wrex 3072 Vcvv 3482 ∖ cdif 3967 {csn 4648 {copab 5231 ‘cfv 6572 (class class class)co 7445 Er wer 8756 0cc0 11180 ...cfz 13563 Basecbs 17253 Scalarcsca 17309 ·𝑠 cvsca 17310 0gc0g 17494 DivRingcdr 20746 LVecclvec 21119 freeLMod cfrlm 21784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-tpos 8263 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-map 8882 df-ixp 8952 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-sup 9507 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-fz 13564 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-hom 17330 df-cco 17331 df-0g 17496 df-prds 17502 df-pws 17504 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-grp 18971 df-minusg 18972 df-sbg 18973 df-subg 19158 df-cmn 19819 df-abl 19820 df-mgp 20157 df-rng 20175 df-ur 20204 df-ring 20257 df-oppr 20355 df-dvdsr 20378 df-unit 20379 df-invr 20409 df-subrg 20592 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: prjspnssbas 42510 prjspnn0 42511 prjspner01 42514 prjspner1 42515 |
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