Theorem asclcom 49505

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 21845, assa2ass2 21846, and asclval 21861, 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 21865	. . 3 ⊢ (𝜑 → (𝐴‘𝐷) ∈ (Base‘𝑊))
8		eqid 2740	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
9		eqid 2740	. . . . 5 ⊢ (.r‘𝑊) = (.r‘𝑊)
10		eqid 2740	. . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
11	3, 4, 5, 8, 9, 10	asclmul1 21868	. . . 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 21869	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐶 ∈ 𝐵 ∧ (𝐴‘𝐷) ∈ (Base‘𝑊)) → ((𝐴‘𝐷)(.r‘𝑊)(𝐴‘𝐶)) = (𝐶( ·𝑠 ‘𝑊)(𝐴‘𝐷)))
13	11, 12	eqtr4d 2778	. . 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 1379	. 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 21870	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐴‘(𝐶 ∗ 𝐷)) = ((𝐴‘𝐶)(.r‘𝑊)(𝐴‘𝐷)))
17	1, 2, 6, 16	syl3anc 1379	. 2 ⊢ (𝜑 → (𝐴‘(𝐶 ∗ 𝐷)) = ((𝐴‘𝐶)(.r‘𝑊)(𝐴‘𝐷)))
18	3, 4, 5, 9, 15	ascldimul 21870	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐷 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) → (𝐴‘(𝐷 ∗ 𝐶)) = ((𝐴‘𝐷)(.r‘𝑊)(𝐴‘𝐶)))
19	1, 6, 2, 18	syl3anc 1379	. 2 ⊢ (𝜑 → (𝐴‘(𝐷 ∗ 𝐶)) = ((𝐴‘𝐷)(.r‘𝑊)(𝐴‘𝐶)))
20	14, 17, 19	3eqtr4d 2785	1 ⊢ (𝜑 → (𝐴‘(𝐶 ∗ 𝐷)) = (𝐴‘(𝐷 ∗ 𝐶)))

Syntax hints:   → wi 4   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  ._rcmulr 17219  Scalarcsca 17221   ·_𝑠 cvsca 17222  AssAlgcasa 21832  algSccascl 21834

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-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-op 4569  df-uni 4846  df-iun 4930  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-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-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  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-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-plusg 17231  df-0g 17402  df-mgm 18606  df-sgrp 18685  df-mnd 18701  df-mgp 20120  df-ur 20161  df-ring 20214  df-lmod 20859  df-assa 21835  df-ascl 21837

This theorem is referenced by: (None)