assalactf1o - Mathbox for Thierry Arnoux

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Theorem assalactf1o 33792

Description: In an associative algebra 𝐴, left-multiplication by a fixed element of the algebra is bijective. See also lactlmhm 33791. (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 21815	. . . 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 21056	. . 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 20635	. . . 4 ⊢ 𝐸 ⊆ 𝐵
14		assalactf1o.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐸)
15	13, 14	sselid 3931	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
16	1, 10, 11, 2, 15	lactlmhm 33791	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐴 LMHom 𝐴))
17		assaring 21816	. . . . . . 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 20195	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐶 · 𝑥) ∈ 𝐵)
23	22	ralrimiva 3128	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐶 · 𝑥) ∈ 𝐵)
24	18	ringgrpd 20177	. . . . . . . 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 768	. . . . . . 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 2736	. . . . . . . . 9 ⊢ (-_g‘𝐴) = (-_g‘𝐴)
30	1, 29, 25, 26, 27	grpsubcld 33123	. . . . . . . 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 20242	. . . . . . . . 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 20195	. . . . . . . . . 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 2736	. . . . . . . . . . . 12 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
38	1, 37, 29	grpsubeq0 18956	. . . . . . . . . . 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 477	. . . . . . . . . 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 1375	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → ((𝐶 · 𝑥)(-_g‘𝐴)(𝐶 · 𝑦)) = (0_g‘𝐴))
41	33, 40	eqtrd 2771	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · (𝑥(-_g‘𝐴)𝑦)) = (0_g‘𝐴))
42	12, 1, 10, 37	rrgeq0i 20632	. . . . . . . . 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 406	. . . . . . . 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 837	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝑥(-_g‘𝐴)𝑦) = (0_g‘𝐴))
45	1, 37, 29	grpsubeq0 18956	. . . . . . . 8 ⊢ ((𝐴 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥(-_g‘𝐴)𝑦) = (0_g‘𝐴) ↔ 𝑥 = 𝑦))
46	45	biimpa 476	. . . . . . 7 ⊢ (((𝐴 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥(-_g‘𝐴)𝑦) = (0_g‘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 3179	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐶 · 𝑥) = (𝐶 · 𝑦) → 𝑥 = 𝑦))
51		oveq2 7366	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐶 · 𝑥) = (𝐶 · 𝑦))
52	11, 51	f1mpt 7207	. . 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 33790	1 ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∀wral 3051 ↦ cmpt 5179 –1-1→wf1 6489 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7358 ℕ₀cn0 12401 Basecbs 17136 ._rcmulr 17178 Scalarcsca 17180 0_gc0g 17359 Grpcgrp 18863 -_gcsg 18865 Ringcrg 20168 RLRegcrlreg 20624 DivRingcdr 20662 LModclmod 20811 LVecclvec 21054 AssAlgcasa 21805 dimcldim 33755

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-reg 9497 ax-inf2 9550 ax-ac2 10373 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-rpss 7668 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-tpos 8168 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-er 8635 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-sup 9345 df-oi 9415 df-r1 9676 df-rank 9677 df-dju 9813 df-card 9851 df-acn 9854 df-ac 10026 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-xnn0 12475 df-z 12489 df-dec 12608 df-uz 12752 df-xadd 13027 df-fz 13424 df-fzo 13571 df-seq 13925 df-hash 14254 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ocomp 17198 df-ds 17199 df-hom 17201 df-cco 17202 df-0g 17361 df-gsum 17362 df-prds 17367 df-pws 17369 df-mre 17505 df-mrc 17506 df-mri 17507 df-acs 17508 df-proset 18217 df-drs 18218 df-poset 18236 df-ipo 18451 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18708 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18998 df-subg 19053 df-ghm 19142 df-cntz 19246 df-lsm 19565 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-nzr 20446 df-subrg 20503 df-rlreg 20627 df-drng 20664 df-lmod 20813 df-lss 20883 df-lsp 20923 df-lmhm 20974 df-lmim 20975 df-lbs 21027 df-lvec 21055 df-sra 21125 df-rgmod 21126 df-dsmm 21687 df-frlm 21702 df-uvc 21738 df-lindf 21761 df-linds 21762 df-assa 21808 df-dim 33756

This theorem is referenced by: assarrginv 33793

W3C validator