assalactf1o - Mathbox for Thierry Arnoux

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Theorem assalactf1o 34310

Description: In an associative algebra 𝐴, left-multiplication by a fixed element of the algebra is bijective. See also lactlmhm 34309. (Contributed by Thierry Arnoux, 3-Aug-2025.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
assalactf1o	⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐵)

Distinct variable groups: 𝑥, · 𝑥,𝐴 𝑥,𝐵 𝑥,𝐶 𝜑,𝑥 Allowed substitution hints: 𝐸(𝑥) 𝐹(𝑥) 𝐾(𝑥)

Proof of Theorem assalactf1o Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lactlmhm.b	. 2 ⊢ 𝐵 = (Base‘𝐴)
2		lactlmhm.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ AssAlg)
3		assalmod 22169	. . . 4 ⊢ (𝐴 ∈ AssAlg → 𝐴 ∈ LMod)
4	2, 3	syl 18	. . 3 ⊢ (𝜑 → 𝐴 ∈ LMod)
5		assalactf1o.2	. . 3 ⊢ (𝜑 → 𝐾 ∈ DivRing)
6		assalactf1o.k	. . . 4 ⊢ 𝐾 = (Scalar‘𝐴)
7	6	islvec 21380	. . 3 ⊢ (𝐴 ∈ LVec ↔ (𝐴 ∈ LMod ∧ 𝐾 ∈ DivRing))
8	4, 5, 7	sylanbrc 595	. 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 20955	. . . 4 ⊢ 𝐸 ⊆ 𝐵
14		assalactf1o.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐸)
15	13, 14	sselid 3906	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
16	1, 10, 11, 2, 15	lactlmhm 34309	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐴 LMHom 𝐴))
17		assaring 22170	. . . . . . 7 ⊢ (𝐴 ∈ AssAlg → 𝐴 ∈ Ring)
18	2, 17	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Ring)
19	18	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ Ring)
20	15	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝐵)
21		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
22	1, 10, 19, 20, 21	ringcld 20485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐶 · 𝑥) ∈ 𝐵)
23	22	ralrimiva 3154	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐶 · 𝑥) ∈ 𝐵)
24	18	ringgrpd 20470	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Grp)
25	24	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝐴 ∈ Grp)
26	21	ad2antrr 739	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝑥 ∈ 𝐵)
27		simplr 781	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝑦 ∈ 𝐵)
28	14	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝐶 ∈ 𝐸)
29		eqid 2760	. . . . . . . . 9 ⊢ (-_g‘𝐴) = (-_g‘𝐴)
30	1, 29, 25, 26, 27	grpsubcld 33645	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝑥(-_g‘𝐴)𝑦) ∈ 𝐵)
31	18	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝐴 ∈ Ring)
32	15	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝐶 ∈ 𝐵)
33	1, 10, 29, 31, 32, 26, 27	ringsubdi 20539	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · (𝑥(-_g‘𝐴)𝑦)) = ((𝐶 · 𝑥)(-_g‘𝐴)(𝐶 · 𝑦)))
34	22	ad2antrr 739	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · 𝑥) ∈ 𝐵)
35	1, 10, 31, 32, 27	ringcld 20485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · 𝑦) ∈ 𝐵)
36		simpr 490	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · 𝑥) = (𝐶 · 𝑦))
37		eqid 2760	. . . . . . . . . . . 12 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
38	1, 37, 29	grpsubeq0 19237	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Grp ∧ (𝐶 · 𝑥) ∈ 𝐵 ∧ (𝐶 · 𝑦) ∈ 𝐵) → (((𝐶 · 𝑥)(-_g‘𝐴)(𝐶 · 𝑦)) = (0_g‘𝐴) ↔ (𝐶 · 𝑥) = (𝐶 · 𝑦)))
39	38	biimpar 483	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Grp ∧ (𝐶 · 𝑥) ∈ 𝐵 ∧ (𝐶 · 𝑦) ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → ((𝐶 · 𝑥)(-_g‘𝐴)(𝐶 · 𝑦)) = (0_g‘𝐴))
40	25, 34, 35, 36, 39	syl31anc 1400	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → ((𝐶 · 𝑥)(-_g‘𝐴)(𝐶 · 𝑦)) = (0_g‘𝐴))
41	33, 40	eqtrd 2795	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · (𝑥(-_g‘𝐴)𝑦)) = (0_g‘𝐴))
42	12, 1, 10, 37	rrgeq0i 20952	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝐸 ∧ (𝑥(-_g‘𝐴)𝑦) ∈ 𝐵) → ((𝐶 · (𝑥(-_g‘𝐴)𝑦)) = (0_g‘𝐴) → (𝑥(-_g‘𝐴)𝑦) = (0_g‘𝐴)))
43	42	imp 412	. . . . . . . 8 ⊢ (((𝐶 ∈ 𝐸 ∧ (𝑥(-_g‘𝐴)𝑦) ∈ 𝐵) ∧ (𝐶 · (𝑥(-_g‘𝐴)𝑦)) = (0_g‘𝐴)) → (𝑥(-_g‘𝐴)𝑦) = (0_g‘𝐴))
44	28, 30, 41, 43	syl21anc 851	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝑥(-_g‘𝐴)𝑦) = (0_g‘𝐴))
45	1, 37, 29	grpsubeq0 19237	. . . . . . . 8 ⊢ ((𝐴 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥(-_g‘𝐴)𝑦) = (0_g‘𝐴) ↔ 𝑥 = 𝑦))
46	45	biimpa 482	. . . . . . 7 ⊢ (((𝐴 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥(-_g‘𝐴)𝑦) = (0_g‘𝐴)) → 𝑥 = 𝑦)
47	25, 26, 27, 44, 46	syl31anc 1400	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝑥 = 𝑦)
48	47	ex 418	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝐶 · 𝑥) = (𝐶 · 𝑦) → 𝑥 = 𝑦))
49	48	anasss 472	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐶 · 𝑥) = (𝐶 · 𝑦) → 𝑥 = 𝑦))
50	49	ralrimivva 3205	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐶 · 𝑥) = (𝐶 · 𝑦) → 𝑥 = 𝑦))
51		oveq2 7428	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐶 · 𝑥) = (𝐶 · 𝑦))
52	11, 51	f1mpt 7265	. . 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 595	. 2 ⊢ (𝜑 → 𝐹:𝐵–1-1→𝐵)
54	1, 8, 9, 16, 53	lvecendof1f1o 34308	1 ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∀wral 3076 ↦ cmpt 5185 –1-1→wf1 6535 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7420 ℕ₀cn0 12607 Basecbs 17388 ._rcmulr 17430 Scalarcsca 17432 0_gc0g 17611 Grpcgrp 19145 -_gcsg 19147 Ringcrg 20460 RLRegcrlreg 20944 DivRingcdr 20981 LModclmod 21136 LVecclvec 21378 AssAlgcasa 22159 dimcldim 34274

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-reg 9587 ax-inf2 9643 ax-ac2 10542 ax-cnex 11257 ax-resscn 11258 ax-1cn 11259 ax-icn 11260 ax-addcl 11261 ax-addrcl 11262 ax-mulcl 11263 ax-mulrcl 11264 ax-mulcom 11265 ax-addass 11266 ax-mulass 11267 ax-distr 11268 ax-i2m1 11269 ax-1ne0 11270 ax-1rid 11271 ax-rnegex 11272 ax-rrecex 11273 ax-cnre 11274 ax-pre-lttri 11275 ax-pre-lttrn 11276 ax-pre-ltadd 11277 ax-pre-mulgt0 11278

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-rpss 7739 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-tpos 8244 df-frecs 8300 df-wrecs 8331 df-recs 8380 df-rdg 8419 df-1o 8477 df-2o 8478 df-oadd 8481 df-er 8718 df-map 8850 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9355 df-sup 9435 df-oi 9505 df-r1 9769 df-rank 9770 df-scott 9930 df-dju 9983 df-card 10021 df-acn 10024 df-ac 10196 df-pnf 11346 df-mnf 11347 df-xr 11348 df-ltxr 11349 df-le 11350 df-sub 11544 df-neg 11545 df-nn 12337 df-2 12406 df-3 12407 df-4 12408 df-5 12409 df-6 12410 df-7 12411 df-8 12412 df-9 12413 df-n0 12608 df-xnn0 12681 df-z 12695 df-dec 12816 df-uz 12967 df-xadd 13243 df-fz 13641 df-fzo 13790 df-seq 14146 df-hash 14476 df-struct 17326 df-sets 17343 df-slot 17361 df-ndx 17373 df-base 17389 df-ress 17410 df-plusg 17442 df-mulr 17443 df-sca 17445 df-vsca 17446 df-ip 17447 df-tset 17448 df-ple 17449 df-ocomp 17450 df-ds 17451 df-hom 17453 df-cco 17454 df-0g 17613 df-gsum 17614 df-prds 17619 df-pws 17621 df-mre 17757 df-mrc 17758 df-mri 17759 df-acs 17760 df-proset 18469 df-drs 18470 df-poset 18488 df-ipo 18703 df-mgm 18817 df-sgrp 18909 df-mnd 18925 df-mhm 18979 df-submnd 18980 df-grp 19148 df-minusg 19149 df-sbg 19150 df-mulg 19279 df-subg 19334 df-ghm 19429 df-cntz 19532 df-lsm 19851 df-cmn 19997 df-abl 19998 df-mgp 20362 df-rng 20376 df-ur 20409 df-ring 20462 df-oppr 20568 df-dvdsr 20588 df-unit 20589 df-invr 20619 df-nzr 20764 df-subrg 20823 df-rlreg 20947 df-drng 20983 df-lmod 21138 df-lss 21208 df-lsp 21248 df-lmhm 21298 df-lmim 21299 df-lbs 21351 df-lvec 21379 df-sra 21449 df-rgmod 21450 df-dsmm 22039 df-frlm 22054 df-uvc 22090 df-lindf 22113 df-linds 22114 df-assa 22162 df-dim 34275

This theorem is used by: assarrginv 34311

W3C validator