Theorem asclcom 49637

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 21973, assa2ass2 21974, and asclval 21989, by setting 𝑋 and 𝑌 the multiplicative identity of the algebra.

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

Hypotheses

Ref	Expression
asclelbasALT.a	⊢ 𝐴 = (algSc‘𝑊)
asclelbasALT.f	⊢ 𝐹 = (Scalar‘𝑊)
asclelbasALT.b	⊢ 𝐵 = (Base‘𝐹)
asclelbasALT.w	⊢ (𝜑 → 𝑊 ∈ AssAlg)
asclelbasALT.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		asclelbasALT.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ AssAlg)
2		asclelbasALT.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
3		asclelbasALT.a	. . . 4 ⊢ 𝐴 = (algSc‘𝑊)
4		asclelbasALT.f	. . . 4 ⊢ 𝐹 = (Scalar‘𝑊)
5		asclelbasALT.b	. . . 4 ⊢ 𝐵 = (Base‘𝐹)
6		asclcom.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝐵)
7	3, 4, 5, 1, 6	asclelbas 21993	. . 3 ⊢ (𝜑 → (𝐴‘𝐷) ∈ (Base‘𝑊))
8		eqid 2765	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
9		eqid 2765	. . . . 5 ⊢ (.r‘𝑊) = (.r‘𝑊)
10		eqid 2765	. . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
11	3, 4, 5, 8, 9, 10	asclmul1 21996	. . . 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 21997	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐶 ∈ 𝐵 ∧ (𝐴‘𝐷) ∈ (Base‘𝑊)) → ((𝐴‘𝐷)(.r‘𝑊)(𝐴‘𝐶)) = (𝐶( ·𝑠 ‘𝑊)(𝐴‘𝐷)))
13	11, 12	eqtr4d 2803	. . 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 1394	. 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 21998	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐴‘(𝐶 ∗ 𝐷)) = ((𝐴‘𝐶)(.r‘𝑊)(𝐴‘𝐷)))
17	1, 2, 6, 16	syl3anc 1394	. 2 ⊢ (𝜑 → (𝐴‘(𝐶 ∗ 𝐷)) = ((𝐴‘𝐶)(.r‘𝑊)(𝐴‘𝐷)))
18	3, 4, 5, 9, 15	ascldimul 21998	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐷 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) → (𝐴‘(𝐷 ∗ 𝐶)) = ((𝐴‘𝐷)(.r‘𝑊)(𝐴‘𝐶)))
19	1, 6, 2, 18	syl3anc 1394	. 2 ⊢ (𝜑 → (𝐴‘(𝐷 ∗ 𝐶)) = ((𝐴‘𝐷)(.r‘𝑊)(𝐴‘𝐶)))
20	14, 17, 19	3eqtr4d 2810	1 ⊢ (𝜑 → (𝐴‘(𝐶 ∗ 𝐷)) = (𝐴‘(𝐷 ∗ 𝐶)))

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  ._rcmulr 17301  Scalarcsca 17303   ·_𝑠 cvsca 17304  AssAlgcasa 21960  algSccascl 21962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-plusg 17313  df-0g 17484  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-mgp 20208  df-ur 20255  df-ring 20308  df-lmod 20952  df-assa 21963  df-ascl 21965

This theorem is referenced by: (None)