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| Mirrors > Home > MPE Home > Th. List > Mathboxes > assalactf1o | Structured version Visualization version GIF version | ||
| Description: In an associative algebra 𝐴, left-multiplication by a fixed element of the algebra is bijective. See also lactlmhm 33685. (Contributed by Thierry Arnoux, 3-Aug-2025.) | 
| Ref | Expression | 
|---|---|
| lactlmhm.b | ⊢ 𝐵 = (Base‘𝐴) | 
| lactlmhm.m | ⊢ · = (.r‘𝐴) | 
| lactlmhm.f | ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐶 · 𝑥)) | 
| lactlmhm.a | ⊢ (𝜑 → 𝐴 ∈ AssAlg) | 
| assalactf1o.1 | ⊢ 𝐸 = (RLReg‘𝐴) | 
| assalactf1o.k | ⊢ 𝐾 = (Scalar‘𝐴) | 
| assalactf1o.2 | ⊢ (𝜑 → 𝐾 ∈ DivRing) | 
| assalactf1o.3 | ⊢ (𝜑 → (dim‘𝐴) ∈ ℕ0) | 
| assalactf1o.c | ⊢ (𝜑 → 𝐶 ∈ 𝐸) | 
| Ref | Expression | 
|---|---|
| assalactf1o | ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | lactlmhm.b | . 2 ⊢ 𝐵 = (Base‘𝐴) | |
| 2 | lactlmhm.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ AssAlg) | |
| 3 | assalmod 21880 | . . . 4 ⊢ (𝐴 ∈ AssAlg → 𝐴 ∈ LMod) | |
| 4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴 ∈ LMod) | 
| 5 | assalactf1o.2 | . . 3 ⊢ (𝜑 → 𝐾 ∈ DivRing) | |
| 6 | assalactf1o.k | . . . 4 ⊢ 𝐾 = (Scalar‘𝐴) | |
| 7 | 6 | islvec 21103 | . . 3 ⊢ (𝐴 ∈ LVec ↔ (𝐴 ∈ LMod ∧ 𝐾 ∈ DivRing)) | 
| 8 | 4, 5, 7 | sylanbrc 583 | . 2 ⊢ (𝜑 → 𝐴 ∈ LVec) | 
| 9 | assalactf1o.3 | . 2 ⊢ (𝜑 → (dim‘𝐴) ∈ ℕ0) | |
| 10 | lactlmhm.m | . . 3 ⊢ · = (.r‘𝐴) | |
| 11 | lactlmhm.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐶 · 𝑥)) | |
| 12 | assalactf1o.1 | . . . . 5 ⊢ 𝐸 = (RLReg‘𝐴) | |
| 13 | 12, 1 | rrgss 20702 | . . . 4 ⊢ 𝐸 ⊆ 𝐵 | 
| 14 | assalactf1o.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐸) | |
| 15 | 13, 14 | sselid 3981 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵) | 
| 16 | 1, 10, 11, 2, 15 | lactlmhm 33685 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝐴 LMHom 𝐴)) | 
| 17 | assaring 21881 | . . . . . . 7 ⊢ (𝐴 ∈ AssAlg → 𝐴 ∈ Ring) | |
| 18 | 2, 17 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Ring) | 
| 19 | 18 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ Ring) | 
| 20 | 15 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝐵) | 
| 21 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
| 22 | 1, 10, 19, 20, 21 | ringcld 20257 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐶 · 𝑥) ∈ 𝐵) | 
| 23 | 22 | ralrimiva 3146 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐶 · 𝑥) ∈ 𝐵) | 
| 24 | 18 | ringgrpd 20239 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Grp) | 
| 25 | 24 | ad3antrrr 730 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝐴 ∈ Grp) | 
| 26 | 21 | ad2antrr 726 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝑥 ∈ 𝐵) | 
| 27 | simplr 769 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝑦 ∈ 𝐵) | |
| 28 | 14 | ad3antrrr 730 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝐶 ∈ 𝐸) | 
| 29 | eqid 2737 | . . . . . . . . 9 ⊢ (-g‘𝐴) = (-g‘𝐴) | |
| 30 | 1, 29, 25, 26, 27 | grpsubcld 33045 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝑥(-g‘𝐴)𝑦) ∈ 𝐵) | 
| 31 | 18 | ad3antrrr 730 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝐴 ∈ Ring) | 
| 32 | 15 | ad3antrrr 730 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝐶 ∈ 𝐵) | 
| 33 | 1, 10, 29, 31, 32, 26, 27 | ringsubdi 20304 | . . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · (𝑥(-g‘𝐴)𝑦)) = ((𝐶 · 𝑥)(-g‘𝐴)(𝐶 · 𝑦))) | 
| 34 | 22 | ad2antrr 726 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · 𝑥) ∈ 𝐵) | 
| 35 | 1, 10, 31, 32, 27 | ringcld 20257 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · 𝑦) ∈ 𝐵) | 
| 36 | simpr 484 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · 𝑥) = (𝐶 · 𝑦)) | |
| 37 | eqid 2737 | . . . . . . . . . . . 12 ⊢ (0g‘𝐴) = (0g‘𝐴) | |
| 38 | 1, 37, 29 | grpsubeq0 19044 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ Grp ∧ (𝐶 · 𝑥) ∈ 𝐵 ∧ (𝐶 · 𝑦) ∈ 𝐵) → (((𝐶 · 𝑥)(-g‘𝐴)(𝐶 · 𝑦)) = (0g‘𝐴) ↔ (𝐶 · 𝑥) = (𝐶 · 𝑦))) | 
| 39 | 38 | biimpar 477 | . . . . . . . . . 10 ⊢ (((𝐴 ∈ Grp ∧ (𝐶 · 𝑥) ∈ 𝐵 ∧ (𝐶 · 𝑦) ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → ((𝐶 · 𝑥)(-g‘𝐴)(𝐶 · 𝑦)) = (0g‘𝐴)) | 
| 40 | 25, 34, 35, 36, 39 | syl31anc 1375 | . . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → ((𝐶 · 𝑥)(-g‘𝐴)(𝐶 · 𝑦)) = (0g‘𝐴)) | 
| 41 | 33, 40 | eqtrd 2777 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · (𝑥(-g‘𝐴)𝑦)) = (0g‘𝐴)) | 
| 42 | 12, 1, 10, 37 | rrgeq0i 20699 | . . . . . . . . 9 ⊢ ((𝐶 ∈ 𝐸 ∧ (𝑥(-g‘𝐴)𝑦) ∈ 𝐵) → ((𝐶 · (𝑥(-g‘𝐴)𝑦)) = (0g‘𝐴) → (𝑥(-g‘𝐴)𝑦) = (0g‘𝐴))) | 
| 43 | 42 | imp 406 | . . . . . . . 8 ⊢ (((𝐶 ∈ 𝐸 ∧ (𝑥(-g‘𝐴)𝑦) ∈ 𝐵) ∧ (𝐶 · (𝑥(-g‘𝐴)𝑦)) = (0g‘𝐴)) → (𝑥(-g‘𝐴)𝑦) = (0g‘𝐴)) | 
| 44 | 28, 30, 41, 43 | syl21anc 838 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝑥(-g‘𝐴)𝑦) = (0g‘𝐴)) | 
| 45 | 1, 37, 29 | grpsubeq0 19044 | . . . . . . . 8 ⊢ ((𝐴 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥(-g‘𝐴)𝑦) = (0g‘𝐴) ↔ 𝑥 = 𝑦)) | 
| 46 | 45 | biimpa 476 | . . . . . . 7 ⊢ (((𝐴 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥(-g‘𝐴)𝑦) = (0g‘𝐴)) → 𝑥 = 𝑦) | 
| 47 | 25, 26, 27, 44, 46 | syl31anc 1375 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝑥 = 𝑦) | 
| 48 | 47 | ex 412 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝐶 · 𝑥) = (𝐶 · 𝑦) → 𝑥 = 𝑦)) | 
| 49 | 48 | anasss 466 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐶 · 𝑥) = (𝐶 · 𝑦) → 𝑥 = 𝑦)) | 
| 50 | 49 | ralrimivva 3202 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐶 · 𝑥) = (𝐶 · 𝑦) → 𝑥 = 𝑦)) | 
| 51 | oveq2 7439 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐶 · 𝑥) = (𝐶 · 𝑦)) | |
| 52 | 11, 51 | f1mpt 7281 | . . 3 ⊢ (𝐹:𝐵–1-1→𝐵 ↔ (∀𝑥 ∈ 𝐵 (𝐶 · 𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐶 · 𝑥) = (𝐶 · 𝑦) → 𝑥 = 𝑦))) | 
| 53 | 23, 50, 52 | sylanbrc 583 | . 2 ⊢ (𝜑 → 𝐹:𝐵–1-1→𝐵) | 
| 54 | 1, 8, 9, 16, 53 | lvecendof1f1o 33684 | 1 ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ↦ cmpt 5225 –1-1→wf1 6558 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ℕ0cn0 12526 Basecbs 17247 .rcmulr 17298 Scalarcsca 17300 0gc0g 17484 Grpcgrp 18951 -gcsg 18953 Ringcrg 20230 RLRegcrlreg 20691 DivRingcdr 20729 LModclmod 20858 LVecclvec 21101 AssAlgcasa 21870 dimcldim 33649 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-reg 9632 ax-inf2 9681 ax-ac2 10503 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-rpss 7743 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-r1 9804 df-rank 9805 df-dju 9941 df-card 9979 df-acn 9982 df-ac 10156 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-xadd 13155 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ocomp 17318 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-mri 17631 df-acs 17632 df-proset 18340 df-drs 18341 df-poset 18359 df-ipo 18573 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-nzr 20513 df-subrg 20570 df-rlreg 20694 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lmhm 21021 df-lmim 21022 df-lbs 21074 df-lvec 21102 df-sra 21172 df-rgmod 21173 df-dsmm 21752 df-frlm 21767 df-uvc 21803 df-lindf 21826 df-linds 21827 df-assa 21873 df-dim 33650 | 
| This theorem is referenced by: assarrginv 33687 | 
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