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Mirrors > Home > MPE Home > Th. List > Mathboxes > prjsperref | Structured version Visualization version GIF version |
Description: The relation in ℙ𝕣𝕠𝕛 is reflexive. (Contributed by Steven Nguyen, 30-Apr-2023.) |
Ref | Expression |
---|---|
prjsprel.1 | ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))} |
prjspertr.b | ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)}) |
prjspertr.s | ⊢ 𝑆 = (Scalar‘𝑉) |
prjspertr.x | ⊢ · = ( ·𝑠 ‘𝑉) |
prjspertr.k | ⊢ 𝐾 = (Base‘𝑆) |
Ref | Expression |
---|---|
prjsperref | ⊢ (𝑉 ∈ LMod → (𝑋 ∈ 𝐵 ↔ 𝑋 ∼ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prjspertr.s | . . . . . . 7 ⊢ 𝑆 = (Scalar‘𝑉) | |
2 | prjspertr.k | . . . . . . 7 ⊢ 𝐾 = (Base‘𝑆) | |
3 | eqid 2818 | . . . . . . 7 ⊢ (1r‘𝑆) = (1r‘𝑆) | |
4 | 1, 2, 3 | lmod1cl 19590 | . . . . . 6 ⊢ (𝑉 ∈ LMod → (1r‘𝑆) ∈ 𝐾) |
5 | 4 | adantr 481 | . . . . 5 ⊢ ((𝑉 ∈ LMod ∧ 𝑋 ∈ 𝐵) → (1r‘𝑆) ∈ 𝐾) |
6 | oveq1 7152 | . . . . . . 7 ⊢ (𝑚 = (1r‘𝑆) → (𝑚 · 𝑋) = ((1r‘𝑆) · 𝑋)) | |
7 | 6 | eqeq2d 2829 | . . . . . 6 ⊢ (𝑚 = (1r‘𝑆) → (𝑋 = (𝑚 · 𝑋) ↔ 𝑋 = ((1r‘𝑆) · 𝑋))) |
8 | 7 | adantl 482 | . . . . 5 ⊢ (((𝑉 ∈ LMod ∧ 𝑋 ∈ 𝐵) ∧ 𝑚 = (1r‘𝑆)) → (𝑋 = (𝑚 · 𝑋) ↔ 𝑋 = ((1r‘𝑆) · 𝑋))) |
9 | eldifi 4100 | . . . . . . . 8 ⊢ (𝑋 ∈ ((Base‘𝑉) ∖ {(0g‘𝑉)}) → 𝑋 ∈ (Base‘𝑉)) | |
10 | prjspertr.b | . . . . . . . 8 ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)}) | |
11 | 9, 10 | eleq2s 2928 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ (Base‘𝑉)) |
12 | eqid 2818 | . . . . . . . 8 ⊢ (Base‘𝑉) = (Base‘𝑉) | |
13 | prjspertr.x | . . . . . . . 8 ⊢ · = ( ·𝑠 ‘𝑉) | |
14 | 12, 1, 13, 3 | lmodvs1 19591 | . . . . . . 7 ⊢ ((𝑉 ∈ LMod ∧ 𝑋 ∈ (Base‘𝑉)) → ((1r‘𝑆) · 𝑋) = 𝑋) |
15 | 11, 14 | sylan2 592 | . . . . . 6 ⊢ ((𝑉 ∈ LMod ∧ 𝑋 ∈ 𝐵) → ((1r‘𝑆) · 𝑋) = 𝑋) |
16 | 15 | eqcomd 2824 | . . . . 5 ⊢ ((𝑉 ∈ LMod ∧ 𝑋 ∈ 𝐵) → 𝑋 = ((1r‘𝑆) · 𝑋)) |
17 | 5, 8, 16 | rspcedvd 3623 | . . . 4 ⊢ ((𝑉 ∈ LMod ∧ 𝑋 ∈ 𝐵) → ∃𝑚 ∈ 𝐾 𝑋 = (𝑚 · 𝑋)) |
18 | 17 | ex 413 | . . 3 ⊢ (𝑉 ∈ LMod → (𝑋 ∈ 𝐵 → ∃𝑚 ∈ 𝐾 𝑋 = (𝑚 · 𝑋))) |
19 | 18 | pm4.71d 562 | . 2 ⊢ (𝑉 ∈ LMod → (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑚 ∈ 𝐾 𝑋 = (𝑚 · 𝑋)))) |
20 | pm4.24 564 | . . . 4 ⊢ (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) | |
21 | 20 | anbi1i 623 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ ∃𝑚 ∈ 𝐾 𝑋 = (𝑚 · 𝑋)) ↔ ((𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝐾 𝑋 = (𝑚 · 𝑋))) |
22 | prjsprel.1 | . . . 4 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))} | |
23 | 22 | prjsprel 39132 | . . 3 ⊢ (𝑋 ∼ 𝑋 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝐾 𝑋 = (𝑚 · 𝑋))) |
24 | 21, 23 | bitr4i 279 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ ∃𝑚 ∈ 𝐾 𝑋 = (𝑚 · 𝑋)) ↔ 𝑋 ∼ 𝑋) |
25 | 19, 24 | syl6bb 288 | 1 ⊢ (𝑉 ∈ LMod → (𝑋 ∈ 𝐵 ↔ 𝑋 ∼ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 ∖ cdif 3930 {csn 4557 class class class wbr 5057 {copab 5119 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 Scalarcsca 16556 ·𝑠 cvsca 16557 0gc0g 16701 1rcur 19180 LModclmod 19563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 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 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mgp 19169 df-ur 19181 df-ring 19228 df-lmod 19565 |
This theorem is referenced by: prjsper 39136 |
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