assalactf1o - Mathbox for Thierry Arnoux

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Theorem assalactf1o 34147

Description: In an associative algebra 𝐴, left-multiplication by a fixed element of the algebra is bijective. See also lactlmhm 34146. (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 22079	. . . 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 21292	. . 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 20868	. . . 4 ⊢ 𝐸 ⊆ 𝐵
14		assalactf1o.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐸)
15	13, 14	sselid 3932	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
16	1, 10, 11, 2, 15	lactlmhm 34146	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐴 LMHom 𝐴))
17		assaring 22080	. . . . . . 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 20400	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐶 · 𝑥) ∈ 𝐵)
23	22	ralrimiva 3156	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐶 · 𝑥) ∈ 𝐵)
24	18	ringgrpd 20385	. . . . . . . 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 2762	. . . . . . . . 9 ⊢ (-_g‘𝐴) = (-_g‘𝐴)
30	1, 29, 25, 26, 27	grpsubcld 33483	. . . . . . . 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 20453	. . . . . . . . 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 20400	. . . . . . . . . 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 2762	. . . . . . . . . . . 12 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
38	1, 37, 29	grpsubeq0 19153	. . . . . . . . . . 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 2797	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · (𝑥(-_g‘𝐴)𝑦)) = (0_g‘𝐴))
42	12, 1, 10, 37	rrgeq0i 20865	. . . . . . . . 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 19153	. . . . . . . 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 3207	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐶 · 𝑥) = (𝐶 · 𝑦) → 𝑥 = 𝑦))
51		oveq2 7424	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐶 · 𝑥) = (𝐶 · 𝑦))
52	11, 51	f1mpt 7261	. . 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 34145	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 1570 ∈ wcel 2145 ∀wral 3078 ↦ cmpt 5190 –1-1→wf1 6534 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7416 ℕ₀cn0 12531 Basecbs 17305 ._rcmulr 17347 Scalarcsca 17349 0_gc0g 17528 Grpcgrp 19061 -_gcsg 19063 Ringcrg 20376 RLRegcrlreg 20857 DivRingcdr 20894 LModclmod 21048 LVecclvec 21290 AssAlgcasa 22069 dimcldim 34111

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-reg 9567 ax-inf2 9623 ax-ac2 10468 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7681 df-rpss 7727 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-tpos 8227 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-oadd 8462 df-er 8699 df-map 8831 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-sup 9415 df-oi 9485 df-r1 9749 df-rank 9750 df-scott 9871 df-dju 9909 df-card 9947 df-acn 9950 df-ac 10122 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-xnn0 12605 df-z 12619 df-dec 12740 df-uz 12891 df-xadd 13166 df-fz 13564 df-fzo 13712 df-seq 14068 df-hash 14397 df-struct 17243 df-sets 17260 df-slot 17278 df-ndx 17290 df-base 17306 df-ress 17327 df-plusg 17359 df-mulr 17360 df-sca 17362 df-vsca 17363 df-ip 17364 df-tset 17365 df-ple 17366 df-ocomp 17367 df-ds 17368 df-hom 17370 df-cco 17371 df-0g 17530 df-gsum 17531 df-prds 17536 df-pws 17538 df-mre 17674 df-mrc 17675 df-mri 17676 df-acs 17677 df-proset 18386 df-drs 18387 df-poset 18405 df-ipo 18620 df-mgm 18734 df-sgrp 18825 df-mnd 18841 df-mhm 18895 df-submnd 18896 df-grp 19064 df-minusg 19065 df-sbg 19066 df-mulg 19195 df-subg 19250 df-ghm 19345 df-cntz 19448 df-lsm 19767 df-cmn 19913 df-abl 19914 df-mgp 20278 df-rng 20292 df-ur 20325 df-ring 20378 df-oppr 20482 df-dvdsr 20502 df-unit 20503 df-invr 20533 df-nzr 20677 df-subrg 20736 df-rlreg 20860 df-drng 20896 df-lmod 21050 df-lss 21120 df-lsp 21160 df-lmhm 21210 df-lmim 21211 df-lbs 21263 df-lvec 21291 df-sra 21361 df-rgmod 21362 df-dsmm 21949 df-frlm 21964 df-uvc 22000 df-lindf 22023 df-linds 22024 df-assa 22072 df-dim 34112

This theorem is used by: assarrginv 34148

W3C validator