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Mirrors > Home > MPE Home > Th. List > Mathboxes > prjspnvs | Structured version Visualization version GIF version |
Description: A nonzero multiple of a vector is equivalent to the vector. This converts the equivalence relation used in prjspvs 42597 (see prjspnerlem 42604). (Contributed by SN, 8-Aug-2024.) |
Ref | Expression |
---|---|
prjspnvs.e | ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))} |
prjspnvs.w | ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁)) |
prjspnvs.b | ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)}) |
prjspnvs.s | ⊢ 𝑆 = (Base‘𝐾) |
prjspnvs.x | ⊢ · = ( ·𝑠 ‘𝑊) |
prjspnvs.0 | ⊢ 0 = (0g‘𝐾) |
prjspnvs.k | ⊢ (𝜑 → 𝐾 ∈ DivRing) |
prjspnvs.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
prjspnvs.2 | ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
prjspnvs.3 | ⊢ (𝜑 → 𝐶 ≠ 0 ) |
Ref | Expression |
---|---|
prjspnvs | ⊢ (𝜑 → (𝐶 · 𝑋) ∼ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prjspnvs.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ DivRing) | |
2 | ovexd 7466 | . . . 4 ⊢ (𝜑 → (0...𝑁) ∈ V) | |
3 | prjspnvs.w | . . . . 5 ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁)) | |
4 | 3 | frlmlvec 21799 | . . . 4 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → 𝑊 ∈ LVec) |
5 | 1, 2, 4 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝑊 ∈ LVec) |
6 | prjspnvs.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
7 | prjspnvs.2 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
8 | prjspnvs.3 | . . . . . 6 ⊢ (𝜑 → 𝐶 ≠ 0 ) | |
9 | nelsn 4671 | . . . . . 6 ⊢ (𝐶 ≠ 0 → ¬ 𝐶 ∈ { 0 }) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → ¬ 𝐶 ∈ { 0 }) |
11 | 7, 10 | eldifd 3974 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝑆 ∖ { 0 })) |
12 | prjspnvs.s | . . . . . 6 ⊢ 𝑆 = (Base‘𝐾) | |
13 | 3 | frlmsca 21791 | . . . . . . . 8 ⊢ ((𝐾 ∈ DivRing ∧ (0...𝑁) ∈ V) → 𝐾 = (Scalar‘𝑊)) |
14 | 1, 2, 13 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → 𝐾 = (Scalar‘𝑊)) |
15 | 14 | fveq2d 6911 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝑊))) |
16 | 12, 15 | eqtrid 2787 | . . . . 5 ⊢ (𝜑 → 𝑆 = (Base‘(Scalar‘𝑊))) |
17 | prjspnvs.0 | . . . . . . 7 ⊢ 0 = (0g‘𝐾) | |
18 | 14 | fveq2d 6911 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝐾) = (0g‘(Scalar‘𝑊))) |
19 | 17, 18 | eqtrid 2787 | . . . . . 6 ⊢ (𝜑 → 0 = (0g‘(Scalar‘𝑊))) |
20 | 19 | sneqd 4643 | . . . . 5 ⊢ (𝜑 → { 0 } = {(0g‘(Scalar‘𝑊))}) |
21 | 16, 20 | difeq12d 4137 | . . . 4 ⊢ (𝜑 → (𝑆 ∖ { 0 }) = ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))})) |
22 | 11, 21 | eleqtrd 2841 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))})) |
23 | eqid 2735 | . . . 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))} | |
24 | prjspnvs.b | . . . 4 ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)}) | |
25 | eqid 2735 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
26 | prjspnvs.x | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
27 | eqid 2735 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
28 | eqid 2735 | . . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
29 | 23, 24, 25, 26, 27, 28 | prjspvs 42597 | . . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝐶 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))})) → (𝐶 · 𝑋){〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}𝑋) |
30 | 5, 6, 22, 29 | syl3anc 1370 | . 2 ⊢ (𝜑 → (𝐶 · 𝑋){〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}𝑋) |
31 | prjspnvs.e | . . . . 5 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))} | |
32 | 31, 3, 24, 12, 26 | prjspnerlem 42604 | . . . 4 ⊢ (𝐾 ∈ DivRing → ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}) |
33 | 1, 32 | syl 17 | . . 3 ⊢ (𝜑 → ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}) |
34 | 33 | breqd 5159 | . 2 ⊢ (𝜑 → ((𝐶 · 𝑋) ∼ 𝑋 ↔ (𝐶 · 𝑋){〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))}𝑋)) |
35 | 30, 34 | mpbird 257 | 1 ⊢ (𝜑 → (𝐶 · 𝑋) ∼ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 Vcvv 3478 ∖ cdif 3960 {csn 4631 class class class wbr 5148 {copab 5210 ‘cfv 6563 (class class class)co 7431 0cc0 11153 ...cfz 13544 Basecbs 17245 Scalarcsca 17301 ·𝑠 cvsca 17302 0gc0g 17486 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 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-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-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-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: prjspner01 42612 |
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