assalactf1o - Mathbox for Thierry Arnoux

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Theorem assalactf1o 33812

Description: In an associative algebra 𝐴, left-multiplication by a fixed element of the algebra is bijective. See also lactlmhm 33811. (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 21827	. . . 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 21068	. . 3 ⊢ (𝐴 ∈ LVec ↔ (𝐴 ∈ LMod ∧ 𝐾 ∈ DivRing))
8	4, 5, 7	sylanbrc 584	. 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 20647	. . . 4 ⊢ 𝐸 ⊆ 𝐵
14		assalactf1o.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐸)
15	13, 14	sselid 3933	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
16	1, 10, 11, 2, 15	lactlmhm 33811	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐴 LMHom 𝐴))
17		assaring 21828	. . . . . . 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 20207	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐶 · 𝑥) ∈ 𝐵)
23	22	ralrimiva 3130	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐶 · 𝑥) ∈ 𝐵)
24	18	ringgrpd 20189	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Grp)
25	24	ad3antrrr 731	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝐴 ∈ Grp)
26	21	ad2antrr 727	. . . . . . 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 731	. . . . . . . 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 33133	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝑥(-_g‘𝐴)𝑦) ∈ 𝐵)
31	18	ad3antrrr 731	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝐴 ∈ Ring)
32	15	ad3antrrr 731	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝐶 ∈ 𝐵)
33	1, 10, 29, 31, 32, 26, 27	ringsubdi 20254	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · (𝑥(-_g‘𝐴)𝑦)) = ((𝐶 · 𝑥)(-_g‘𝐴)(𝐶 · 𝑦)))
34	22	ad2antrr 727	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · 𝑥) ∈ 𝐵)
35	1, 10, 31, 32, 27	ringcld 20207	. . . . . . . . . 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 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
38	1, 37, 29	grpsubeq0 18968	. . . . . . . . . . 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 1376	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → ((𝐶 · 𝑥)(-_g‘𝐴)(𝐶 · 𝑦)) = (0_g‘𝐴))
41	33, 40	eqtrd 2772	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · (𝑥(-_g‘𝐴)𝑦)) = (0_g‘𝐴))
42	12, 1, 10, 37	rrgeq0i 20644	. . . . . . . . 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 838	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝑥(-_g‘𝐴)𝑦) = (0_g‘𝐴))
45	1, 37, 29	grpsubeq0 18968	. . . . . . . 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 1376	. . . . . 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 3181	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐶 · 𝑥) = (𝐶 · 𝑦) → 𝑥 = 𝑦))
51		oveq2 7376	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐶 · 𝑥) = (𝐶 · 𝑦))
52	11, 51	f1mpt 7217	. . 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 584	. 2 ⊢ (𝜑 → 𝐹:𝐵–1-1→𝐵)
54	1, 8, 9, 16, 53	lvecendof1f1o 33810	1 ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ↦ cmpt 5181 –1-1→wf1 6497 –1-1-onto→wf1o 6499 ‘cfv 6500 (class class class)co 7368 ℕ₀cn0 12413 Basecbs 17148 ._rcmulr 17190 Scalarcsca 17192 0_gc0g 17371 Grpcgrp 18875 -_gcsg 18877 Ringcrg 20180 RLRegcrlreg 20636 DivRingcdr 20674 LModclmod 20823 LVecclvec 21066 AssAlgcasa 21817 dimcldim 33775

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-reg 9509 ax-inf2 9562 ax-ac2 10385 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-rpss 7678 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-sup 9357 df-oi 9427 df-r1 9688 df-rank 9689 df-dju 9825 df-card 9863 df-acn 9866 df-ac 10038 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-xnn0 12487 df-z 12501 df-dec 12620 df-uz 12764 df-xadd 13039 df-fz 13436 df-fzo 13583 df-seq 13937 df-hash 14266 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ocomp 17210 df-ds 17211 df-hom 17213 df-cco 17214 df-0g 17373 df-gsum 17374 df-prds 17379 df-pws 17381 df-mre 17517 df-mrc 17518 df-mri 17519 df-acs 17520 df-proset 18229 df-drs 18230 df-poset 18248 df-ipo 18463 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19010 df-subg 19065 df-ghm 19154 df-cntz 19258 df-lsm 19577 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-nzr 20458 df-subrg 20515 df-rlreg 20639 df-drng 20676 df-lmod 20825 df-lss 20895 df-lsp 20935 df-lmhm 20986 df-lmim 20987 df-lbs 21039 df-lvec 21067 df-sra 21137 df-rgmod 21138 df-dsmm 21699 df-frlm 21714 df-uvc 21750 df-lindf 21773 df-linds 21774 df-assa 21820 df-dim 33776

This theorem is referenced by: assarrginv 33813

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