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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 33662. (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 21898 | . . . 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 21121 | . . 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 20719 | . . . 4 ⊢ 𝐸 ⊆ 𝐵 |
14 | assalactf1o.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐸) | |
15 | 13, 14 | sselid 3993 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
16 | 1, 10, 11, 2, 15 | lactlmhm 33662 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝐴 LMHom 𝐴)) |
17 | assaring 21899 | . . . . . . 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 20277 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐶 · 𝑥) ∈ 𝐵) |
23 | 22 | ralrimiva 3144 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐶 · 𝑥) ∈ 𝐵) |
24 | 18 | ringgrpd 20260 | . . . . . . . 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 2735 | . . . . . . . . 9 ⊢ (-g‘𝐴) = (-g‘𝐴) | |
30 | 1, 29, 25, 26, 27 | grpsubcld 33028 | . . . . . . . 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 20321 | . . . . . . . . 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 20277 | . . . . . . . . . 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 2735 | . . . . . . . . . . . 12 ⊢ (0g‘𝐴) = (0g‘𝐴) | |
38 | 1, 37, 29 | grpsubeq0 19057 | . . . . . . . . . . 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 1372 | . . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → ((𝐶 · 𝑥)(-g‘𝐴)(𝐶 · 𝑦)) = (0g‘𝐴)) |
41 | 33, 40 | eqtrd 2775 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · (𝑥(-g‘𝐴)𝑦)) = (0g‘𝐴)) |
42 | 12, 1, 10, 37 | rrgeq0i 20716 | . . . . . . . . 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 19057 | . . . . . . . 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 1372 | . . . . . 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 3200 | . . 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 33661 | 1 ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ↦ cmpt 5231 –1-1→wf1 6560 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 ℕ0cn0 12524 Basecbs 17245 .rcmulr 17299 Scalarcsca 17301 0gc0g 17486 Grpcgrp 18964 -gcsg 18966 Ringcrg 20251 RLRegcrlreg 20708 DivRingcdr 20746 LModclmod 20875 LVecclvec 21119 AssAlgcasa 21888 dimcldim 33626 |
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-reg 9630 ax-inf2 9679 ax-ac2 10501 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-int 4952 df-iun 4998 df-iin 4999 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-se 5642 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-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-rpss 7742 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-r1 9802 df-rank 9803 df-dju 9939 df-card 9977 df-acn 9980 df-ac 10154 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-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-xadd 13153 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 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-ocomp 17319 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-mri 17633 df-acs 17634 df-proset 18352 df-drs 18353 df-poset 18371 df-ipo 18586 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-lsm 19669 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-nzr 20530 df-subrg 20587 df-rlreg 20711 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lmhm 21039 df-lmim 21040 df-lbs 21092 df-lvec 21120 df-sra 21190 df-rgmod 21191 df-dsmm 21770 df-frlm 21785 df-uvc 21821 df-lindf 21844 df-linds 21845 df-assa 21891 df-dim 33627 |
This theorem is referenced by: assarrginv 33664 |
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