Theorem asclcom 49495

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 21853, assa2ass2 21854, and asclval 21869, 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 21873	. . 3 ⊢ (𝜑 → (𝐴‘𝐷) ∈ (Base‘𝑊))
8		eqid 2737	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
9		eqid 2737	. . . . 5 ⊢ (.r‘𝑊) = (.r‘𝑊)
10		eqid 2737	. . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
11	3, 4, 5, 8, 9, 10	asclmul1 21876	. . . 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 21877	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐶 ∈ 𝐵 ∧ (𝐴‘𝐷) ∈ (Base‘𝑊)) → ((𝐴‘𝐷)(.r‘𝑊)(𝐴‘𝐶)) = (𝐶( ·𝑠 ‘𝑊)(𝐴‘𝐷)))
13	11, 12	eqtr4d 2775	. . 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 1374	. 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 21878	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐴‘(𝐶 ∗ 𝐷)) = ((𝐴‘𝐶)(.r‘𝑊)(𝐴‘𝐷)))
17	1, 2, 6, 16	syl3anc 1374	. 2 ⊢ (𝜑 → (𝐴‘(𝐶 ∗ 𝐷)) = ((𝐴‘𝐶)(.r‘𝑊)(𝐴‘𝐷)))
18	3, 4, 5, 9, 15	ascldimul 21878	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐷 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵) → (𝐴‘(𝐷 ∗ 𝐶)) = ((𝐴‘𝐷)(.r‘𝑊)(𝐴‘𝐶)))
19	1, 6, 2, 18	syl3anc 1374	. 2 ⊢ (𝜑 → (𝐴‘(𝐷 ∗ 𝐶)) = ((𝐴‘𝐷)(.r‘𝑊)(𝐴‘𝐶)))
20	14, 17, 19	3eqtr4d 2782	1 ⊢ (𝜑 → (𝐴‘(𝐶 ∗ 𝐷)) = (𝐴‘(𝐷 ∗ 𝐶)))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  ._rcmulr 17212  Scalarcsca 17214   ·_𝑠 cvsca 17215  AssAlgcasa 21840  algSccascl 21842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-plusg 17224  df-0g 17395  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mgp 20113  df-ur 20154  df-ring 20207  df-lmod 20848  df-assa 21843  df-ascl 21845

This theorem is referenced by: (None)