assalactf1o - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > assalactf1o		Structured version Visualization version GIF version

Theorem assalactf1o 33826

Description: In an associative algebra 𝐴, left-multiplication by a fixed element of the algebra is bijective. See also lactlmhm 33825. (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 21842	. . . 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 21101	. . 3 ⊢ (𝐴 ∈ LVec ↔ (𝐴 ∈ LMod ∧ 𝐾 ∈ DivRing))
8	4, 5, 7	sylanbrc 589	. 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 20681	. . . 4 ⊢ 𝐸 ⊆ 𝐵
14		assalactf1o.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐸)
15	13, 14	sselid 3920	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
16	1, 10, 11, 2, 15	lactlmhm 33825	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐴 LMHom 𝐴))
17		assaring 21843	. . . . . . 7 ⊢ (𝐴 ∈ AssAlg → 𝐴 ∈ Ring)
18	2, 17	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Ring)
19	18	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ Ring)
20	15	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝐵)
21		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
22	1, 10, 19, 20, 21	ringcld 20239	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐶 · 𝑥) ∈ 𝐵)
23	22	ralrimiva 3132	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐶 · 𝑥) ∈ 𝐵)
24	18	ringgrpd 20221	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Grp)
25	24	ad3antrrr 736	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝐴 ∈ Grp)
26	21	ad2antrr 732	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝑥 ∈ 𝐵)
27		simplr 774	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝑦 ∈ 𝐵)
28	14	ad3antrrr 736	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝐶 ∈ 𝐸)
29		eqid 2740	. . . . . . . . 9 ⊢ (-_g‘𝐴) = (-_g‘𝐴)
30	1, 29, 25, 26, 27	grpsubcld 33128	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝑥(-_g‘𝐴)𝑦) ∈ 𝐵)
31	18	ad3antrrr 736	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝐴 ∈ Ring)
32	15	ad3antrrr 736	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝐶 ∈ 𝐵)
33	1, 10, 29, 31, 32, 26, 27	ringsubdi 20286	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · (𝑥(-_g‘𝐴)𝑦)) = ((𝐶 · 𝑥)(-_g‘𝐴)(𝐶 · 𝑦)))
34	22	ad2antrr 732	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · 𝑥) ∈ 𝐵)
35	1, 10, 31, 32, 27	ringcld 20239	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · 𝑦) ∈ 𝐵)
36		simpr 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · 𝑥) = (𝐶 · 𝑦))
37		eqid 2740	. . . . . . . . . . . 12 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
38	1, 37, 29	grpsubeq0 19000	. . . . . . . . . . 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 478	. . . . . . . . . 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 1381	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → ((𝐶 · 𝑥)(-_g‘𝐴)(𝐶 · 𝑦)) = (0_g‘𝐴))
41	33, 40	eqtrd 2775	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝐶 · (𝑥(-_g‘𝐴)𝑦)) = (0_g‘𝐴))
42	12, 1, 10, 37	rrgeq0i 20678	. . . . . . . . 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 407	. . . . . . . 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 843	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → (𝑥(-_g‘𝐴)𝑦) = (0_g‘𝐴))
45	1, 37, 29	grpsubeq0 19000	. . . . . . . 8 ⊢ ((𝐴 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥(-_g‘𝐴)𝑦) = (0_g‘𝐴) ↔ 𝑥 = 𝑦))
46	45	biimpa 477	. . . . . . 7 ⊢ (((𝐴 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥(-_g‘𝐴)𝑦) = (0_g‘𝐴)) → 𝑥 = 𝑦)
47	25, 26, 27, 44, 46	syl31anc 1381	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (𝐶 · 𝑥) = (𝐶 · 𝑦)) → 𝑥 = 𝑦)
48	47	ex 413	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝐶 · 𝑥) = (𝐶 · 𝑦) → 𝑥 = 𝑦))
49	48	anasss 467	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐶 · 𝑥) = (𝐶 · 𝑦) → 𝑥 = 𝑦))
50	49	ralrimivva 3183	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝐶 · 𝑥) = (𝐶 · 𝑦) → 𝑥 = 𝑦))
51		oveq2 7371	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐶 · 𝑥) = (𝐶 · 𝑦))
52	11, 51	f1mpt 7212	. . 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 589	. 2 ⊢ (𝜑 → 𝐹:𝐵–1-1→𝐵)
54	1, 8, 9, 16, 53	lvecendof1f1o 33824	1 ⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ∀wral 3054 ↦ cmpt 5160 –1-1→wf1 6489 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7363 ℕ₀cn0 12435 Basecbs 17177 ._rcmulr 17219 Scalarcsca 17221 0_gc0g 17400 Grpcgrp 18907 -_gcsg 18909 Ringcrg 20212 RLRegcrlreg 20670 DivRingcdr 20708 LModclmod 20857 LVecclvec 21099 AssAlgcasa 21832 dimcldim 33790

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-reg 9504 ax-inf2 9560 ax-ac2 10383 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 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 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-rpss 7673 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-tpos 8173 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-oadd 8406 df-er 8640 df-map 8772 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-sup 9352 df-oi 9422 df-r1 9686 df-rank 9687 df-dju 9823 df-card 9861 df-acn 9864 df-ac 10036 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-xnn0 12509 df-z 12523 df-dec 12643 df-uz 12787 df-xadd 13062 df-fz 13460 df-fzo 13607 df-seq 13962 df-hash 14291 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ocomp 17239 df-ds 17240 df-hom 17242 df-cco 17243 df-0g 17402 df-gsum 17403 df-prds 17408 df-pws 17410 df-mre 17546 df-mrc 17547 df-mri 17548 df-acs 17549 df-proset 18258 df-drs 18259 df-poset 18277 df-ipo 18492 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-mhm 18749 df-submnd 18750 df-grp 18910 df-minusg 18911 df-sbg 18912 df-mulg 19042 df-subg 19097 df-ghm 19186 df-cntz 19290 df-lsm 19609 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-ring 20214 df-oppr 20315 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-nzr 20492 df-subrg 20549 df-rlreg 20673 df-drng 20710 df-lmod 20859 df-lss 20929 df-lsp 20969 df-lmhm 21019 df-lmim 21020 df-lbs 21072 df-lvec 21100 df-sra 21170 df-rgmod 21171 df-dsmm 21714 df-frlm 21729 df-uvc 21765 df-lindf 21788 df-linds 21789 df-assa 21835 df-dim 33791

This theorem is referenced by: assarrginv 33827

W3C validator