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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 39342 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁)))) | |
2 | simpr 487 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → 𝐾 ∈ DivRing) | |
3 | ovexd 7184 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (0...𝑁) ∈ V) | |
4 | prjspnval2.w | . . . . . . 7 ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁)) | |
5 | 4 | frlmlvec 20900 | . . . . . 6 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → 𝑊 ∈ LVec) |
6 | 2, 3, 5 | syl2anc 586 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → 𝑊 ∈ LVec) |
7 | prjspnval2.b | . . . . . 6 ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)}) | |
8 | prjspnval2.x | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑊) | |
9 | eqid 2820 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
10 | eqid 2820 | . . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
11 | 7, 8, 9, 10 | prjspval 39329 | . . . . 5 ⊢ (𝑊 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑊) = (𝐵 / {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))})) |
12 | 6, 11 | syl 17 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (ℙ𝕣𝕠𝕛‘𝑊) = (𝐵 / {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))})) |
13 | prjspnval2.s | . . . . . . . . 9 ⊢ 𝑆 = (Base‘𝐾) | |
14 | 4 | frlmsca 20892 | . . . . . . . . . . 11 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → 𝐾 = (Scalar‘𝑊)) |
15 | 2, 3, 14 | syl2anc 586 | . . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → 𝐾 = (Scalar‘𝑊)) |
16 | 15 | fveq2d 6667 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (Base‘𝐾) = (Base‘(Scalar‘𝑊))) |
17 | 13, 16 | syl5req 2868 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (Base‘(Scalar‘𝑊)) = 𝑆) |
18 | 17 | rexeqdv 3415 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦) ↔ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))) |
19 | 18 | anbi2d 630 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦)))) |
20 | 19 | opabbidv 5125 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))} = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}) |
21 | 20 | qseq2d 8339 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝐵 / {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}) = (𝐵 / {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))})) |
22 | 12, 21 | eqtrd 2855 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (ℙ𝕣𝕠𝕛‘𝑊) = (𝐵 / {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))})) |
23 | 4 | eqcomi 2829 | . . . 4 ⊢ (𝐾 freeLMod (0...𝑁)) = 𝑊 |
24 | 23 | fveq2i 6666 | . . 3 ⊢ (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) = (ℙ𝕣𝕠𝕛‘𝑊) |
25 | prjspnval2.e | . . . 4 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))} | |
26 | 25 | qseq2i 8338 | . . 3 ⊢ (𝐵 / ∼ ) = (𝐵 / {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}) |
27 | 22, 24, 26 | 3eqtr4g 2880 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))) = (𝐵 / ∼ )) |
28 | 1, 27 | eqtrd 2855 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (𝐵 / ∼ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3138 Vcvv 3491 ∖ cdif 3926 {csn 4560 {copab 5121 ‘cfv 6348 (class class class)co 7149 / cqs 8281 0cc0 10530 ℕ0cn0 11891 ...cfz 12889 Basecbs 16478 Scalarcsca 16563 ·𝑠 cvsca 16564 0gc0g 16708 DivRingcdr 19497 LVecclvec 19869 freeLMod cfrlm 20885 ℙ𝕣𝕠𝕛cprjsp 39327 ℙ𝕣𝕠𝕛ncprjspn 39340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-ec 8284 df-qs 8288 df-map 8401 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-sup 8899 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12890 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-sca 16576 df-vsca 16577 df-ip 16578 df-tset 16579 df-ple 16580 df-ds 16582 df-hom 16584 df-cco 16585 df-0g 16710 df-prds 16716 df-pws 16718 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-grp 18101 df-minusg 18102 df-sbg 18103 df-subg 18271 df-mgp 19235 df-ur 19247 df-ring 19294 df-drng 19499 df-subrg 19528 df-lmod 19631 df-lss 19699 df-lvec 19870 df-sra 19939 df-rgmod 19940 df-dsmm 20871 df-frlm 20886 df-prjsp 39328 df-prjspn 39341 |
This theorem is referenced by: 0prjspn 39346 |
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