Theorem asclcom 49039

Description: Scalars are commutative after being lifted.

However, the scalars themselves are not necessarily commutative if the algebra is not a faithful module. For example, Let 𝐹 be the 2 by 2 upper triangular matrix algebra over a commutative ring 𝑊. It is provable that 𝐹 is in general non-commutative. Define scalar multiplication 𝐶 · 𝑋 as multipying the top-left entry, which is a "vector" element of 𝑊, of the "scalar" 𝐶, which is now an upper triangular matrix, with the "vector" 𝑋 ∈ (Base‘𝑊).

Equivalently, the algebra scalar lifting function is not necessarily injective unless the algebra is faithful. Therefore, all "scalar injection" was renamed.

Alternate proof involves assa2ass 21798, assa2ass2 21799, and asclval 21815, by setting 𝑋 and 𝑌 the multiplicative identity of the algebra.

(Contributed by Zhi Wang, 11-Sep-2025.)

Hypotheses

Ref	Expression
asclelbas.a	⊢ 𝐴 = (algSc‘𝑊)
asclelbas.f	⊢ 𝐹 = (Scalar‘𝑊)
asclelbas.b	⊢ 𝐵 = (Base‘𝐹)
asclelbas.w	⊢ (𝜑 → 𝑊 ∈ AssAlg)
asclelbas.c	⊢ (𝜑 → 𝐶 ∈ 𝐵)
asclcom.m	⊢ ∗ = (.r‘𝐹)
asclcom.d	⊢ (𝜑 → 𝐷 ∈ 𝐵)

Assertion

Ref	Expression
asclcom	⊢ (𝜑 → (𝐴‘(𝐶 ∗ 𝐷)) = (𝐴‘(𝐷 ∗ 𝐶)))

Proof of Theorem asclcom

Step	Hyp	Ref	Expression
1		asclelbas.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ AssAlg)
2		asclelbas.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
3		asclelbas.a	. . . 4 ⊢ 𝐴 = (algSc‘𝑊)
4		asclelbas.f	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
5		asclelbas.b	. . . 4 ⊢ 𝐵 = (Base‘𝐹)
6		asclcom.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝐵)
7	3, 4, 5, 1, 6	asclelbas 49036	. . 3 ⊢ (𝜑 → (𝐴‘𝐷) ∈ (Base‘𝑊))
8		eqid 2731	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
9		eqid 2731	. . . . 5 ⊢ (.r‘𝑊) = (.r‘𝑊)
10		eqid 2731	. . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
11	3, 4, 5, 8, 9, 10	asclmul1 21821	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐶 ∈ 𝐵 ∧ (𝐴‘𝐷) ∈ (Base‘𝑊)) → ((𝐴‘𝐶)(.r‘𝑊)(𝐴‘𝐷)) = (𝐶( ·𝑠 ‘𝑊)(𝐴‘𝐷)))
12	3, 4, 5, 8, 9, 10	asclmul2 21822	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐶 ∈ 𝐵 ∧ (𝐴‘𝐷) ∈ (Base‘𝑊)) → ((𝐴‘𝐷)(.r‘𝑊)(𝐴‘𝐶)) = (𝐶( ·𝑠 ‘𝑊)(𝐴‘𝐷)))
13	11, 12	eqtr4d 2769	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐶 ∈ 𝐵 ∧ (𝐴‘𝐷) ∈ (Base‘𝑊)) → ((𝐴‘𝐶)(.r‘𝑊)(𝐴‘𝐷)) = ((𝐴‘𝐷)(.r‘𝑊)(𝐴‘𝐶)))
14	1, 2, 7, 13	syl3anc 1373	. 2 ⊢ (𝜑 → ((𝐴‘𝐶)(.r‘𝑊)(𝐴‘𝐷)) = ((𝐴‘𝐷)(.r‘𝑊)(𝐴‘𝐶)))
15		asclcom.m	. . . 4 ⊢ ∗ = (.r‘𝐹)
16	3, 4, 5, 9, 15	ascldimul 21823	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐴‘(𝐶 ∗ 𝐷)) = ((𝐴‘𝐶)(.r‘𝑊)(𝐴‘𝐷)))
17	1, 2, 6, 16	syl3anc 1373	. 2 ⊢ (𝜑 → (𝐴‘(𝐶 ∗ 𝐷)) = ((𝐴‘𝐶)(.r‘𝑊)(𝐴‘𝐷)))
18	3, 4, 5, 9, 15	ascldimul 21823	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐷 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) → (𝐴‘(𝐷 ∗ 𝐶)) = ((𝐴‘𝐷)(.r‘𝑊)(𝐴‘𝐶)))
19	1, 6, 2, 18	syl3anc 1373	. 2 ⊢ (𝜑 → (𝐴‘(𝐷 ∗ 𝐶)) = ((𝐴‘𝐷)(.r‘𝑊)(𝐴‘𝐶)))
20	14, 17, 19	3eqtr4d 2776	1 ⊢ (𝜑 → (𝐴‘(𝐶 ∗ 𝐷)) = (𝐴‘(𝐷 ∗ 𝐶)))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ‘cfv 6481  (class class class)co 7346  Basecbs 17117  ._rcmulr 17159  Scalarcsca 17161   ·_𝑠 cvsca 17162  AssAlgcasa 21785  algSccascl 21787

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-nn 12123  df-2 12185  df-sets 17072  df-slot 17090  df-ndx 17102  df-base 17118  df-plusg 17171  df-0g 17342  df-mgm 18545  df-sgrp 18624  df-mnd 18640  df-mgp 20057  df-ur 20098  df-ring 20151  df-lmod 20793  df-assa 21788  df-ascl 21790

This theorem is referenced by: (None)