Theorem asclcom 50337

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 22204, assa2ass2 22205, and asclval 22220, 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 22224	. . 3 ⊢ (𝜑 → (𝐴‘𝐷) ∈ (Base‘𝑊))
8		eqid 2763	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
9		eqid 2763	. . . . 5 ⊢ (.r‘𝑊) = (.r‘𝑊)
10		eqid 2763	. . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
11	3, 4, 5, 8, 9, 10	asclmul1 22227	. . . 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 22228	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐶 ∈ 𝐵 ∧ (𝐴‘𝐷) ∈ (Base‘𝑊)) → ((𝐴‘𝐷)(.r‘𝑊)(𝐴‘𝐶)) = (𝐶( ·𝑠 ‘𝑊)(𝐴‘𝐷)))
13	11, 12	eqtr4d 2801	. . 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 1399	. 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 22229	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐴‘(𝐶 ∗ 𝐷)) = ((𝐴‘𝐶)(.r‘𝑊)(𝐴‘𝐷)))
17	1, 2, 6, 16	syl3anc 1399	. 2 ⊢ (𝜑 → (𝐴‘(𝐶 ∗ 𝐷)) = ((𝐴‘𝐶)(.r‘𝑊)(𝐴‘𝐷)))
18	3, 4, 5, 9, 15	ascldimul 22229	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐷 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) → (𝐴‘(𝐷 ∗ 𝐶)) = ((𝐴‘𝐷)(.r‘𝑊)(𝐴‘𝐶)))
19	1, 6, 2, 18	syl3anc 1399	. 2 ⊢ (𝜑 → (𝐴‘(𝐷 ∗ 𝐶)) = ((𝐴‘𝐷)(.r‘𝑊)(𝐴‘𝐶)))
20	14, 17, 19	3eqtr4d 2808	1 ⊢ (𝜑 → (𝐴‘(𝐶 ∗ 𝐷)) = (𝐴‘(𝐷 ∗ 𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  ._rcmulr 17461  Scalarcsca 17463   ·_𝑠 cvsca 17464  AssAlgcasa 22191  algSccascl 22193

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-er 8748  df-en 9005  df-dom 9006  df-sdom 9007  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-nn 12366  df-2 12435  df-sets 17374  df-slot 17392  df-ndx 17404  df-base 17420  df-plusg 17473  df-0g 17644  df-mgm 18848  df-sgrp 18940  df-mnd 18956  df-mgp 20394  df-ur 20441  df-ring 20494  df-lmod 21170  df-assa 22194  df-ascl 22196

This theorem is used by: (None)