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Mirrors > Home > MPE Home > Th. List > Mathboxes > prjspnval2 | Structured version Visualization version GIF version |
Description: Value of the n-dimensional projective space function, expanded. (Contributed by Steven Nguyen, 15-Jul-2023.) |
Ref | Expression |
---|---|
prjspnval2.e | ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))} |
prjspnval2.w | ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁)) |
prjspnval2.b | ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)}) |
prjspnval2.s | ⊢ 𝑆 = (Base‘𝐾) |
prjspnval2.x | ⊢ · = ( ·𝑠 ‘𝑊) |
Ref | Expression |
---|---|
prjspnval2 | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (𝐵 / ∼ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prjspnval 40435 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁)))) | |
2 | prjspnval2.w | . . . . 5 ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁)) | |
3 | 2 | fveq2i 6771 | . . . 4 ⊢ (ℙ𝕣𝕠𝕛‘𝑊) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) |
4 | ovex 7301 | . . . . . . 7 ⊢ (0...𝑁) ∈ V | |
5 | 2 | frlmlvec 20949 | . . . . . . 7 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → 𝑊 ∈ LVec) |
6 | 4, 5 | mpan2 687 | . . . . . 6 ⊢ (𝐾 ∈ DivRing → 𝑊 ∈ LVec) |
7 | prjspnval2.b | . . . . . . 7 ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)}) | |
8 | prjspnval2.x | . . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊) | |
9 | eqid 2739 | . . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
10 | eqid 2739 | . . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
11 | 7, 8, 9, 10 | prjspval 40422 | . . . . . 6 ⊢ (𝑊 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑊) = (𝐵 / {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))})) |
12 | 6, 11 | syl 17 | . . . . 5 ⊢ (𝐾 ∈ DivRing → (ℙ𝕣𝕠𝕛‘𝑊) = (𝐵 / {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))})) |
13 | prjspnval2.e | . . . . . . 7 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))} | |
14 | prjspnval2.s | . . . . . . 7 ⊢ 𝑆 = (Base‘𝐾) | |
15 | 13, 2, 7, 14, 8 | prjspnerlem 40436 | . . . . . 6 ⊢ (𝐾 ∈ DivRing → ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}) |
16 | 15 | qseq2d 8529 | . . . . 5 ⊢ (𝐾 ∈ DivRing → (𝐵 / ∼ ) = (𝐵 / {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))})) |
17 | 12, 16 | eqtr4d 2782 | . . . 4 ⊢ (𝐾 ∈ DivRing → (ℙ𝕣𝕠𝕛‘𝑊) = (𝐵 / ∼ )) |
18 | 3, 17 | eqtr3id 2793 | . . 3 ⊢ (𝐾 ∈ DivRing → (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) = (𝐵 / ∼ )) |
19 | 18 | adantl 481 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) = (𝐵 / ∼ )) |
20 | 1, 19 | eqtrd 2779 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (𝐵 / ∼ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 ∃wrex 3066 Vcvv 3430 ∖ cdif 3888 {csn 4566 {copab 5140 ‘cfv 6430 (class class class)co 7268 / cqs 8471 0cc0 10855 ℕ0cn0 12216 ...cfz 13221 Basecbs 16893 Scalarcsca 16946 ·𝑠 cvsca 16947 0gc0g 17131 DivRingcdr 19972 LVecclvec 20345 freeLMod cfrlm 20934 ℙ𝕣𝕠𝕛cprjsp 40420 ℙ𝕣𝕠𝕛ncprjspn 40433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-ec 8474 df-qs 8478 df-map 8591 df-ixp 8660 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-sup 9162 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-fz 13222 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-sca 16959 df-vsca 16960 df-ip 16961 df-tset 16962 df-ple 16963 df-ds 16965 df-hom 16967 df-cco 16968 df-0g 17133 df-prds 17139 df-pws 17141 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-grp 18561 df-minusg 18562 df-sbg 18563 df-subg 18733 df-mgp 19702 df-ur 19719 df-ring 19766 df-drng 19974 df-subrg 20003 df-lmod 20106 df-lss 20175 df-lvec 20346 df-sra 20415 df-rgmod 20416 df-dsmm 20920 df-frlm 20935 df-prjsp 40421 df-prjspn 40434 |
This theorem is referenced by: 0prjspn 40445 |
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