cayhamlem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cayhamlem2		Structured version Visualization version GIF version

Theorem cayhamlem2 23208

Description: Lemma for cayhamlem3 23211. (Contributed by AV, 24-Nov-2019.)

**Hypotheses**
Ref	Expression
cayhamlem2.k	⊢ 𝐾 = (Base‘𝑅)
cayhamlem2.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
cayhamlem2.b	⊢ 𝐵 = (Base‘𝐴)
cayhamlem2.1	⊢ 1 = (1_r‘𝐴)
cayhamlem2.m	⊢ ∗ = ( ·_𝑠 ‘𝐴)
cayhamlem2.e	⊢ ↑ = (._g‘(mulGrp‘𝐴))
cayhamlem2.r	⊢ · = (._r‘𝐴)

**Assertion**
Ref	Expression
cayhamlem2	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → ((𝐻‘𝐿) ∗ (𝐿 ↑ 𝑀)) = ((𝐿 ↑ 𝑀) · ((𝐻‘𝐿) ∗ 1 )))

**Proof of Theorem cayhamlem2**
Step	Hyp	Ref	Expression
1		elmapi 8873	. . . . . . . 8 ⊢ (𝐻 ∈ (𝐾 ↑_m ℕ₀) → 𝐻:ℕ₀⟶𝐾)
2	1	ffvelcdmda 7085	. . . . . . 7 ⊢ ((𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀) → (𝐻‘𝐿) ∈ 𝐾)
3	2	adantl 487	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → (𝐻‘𝐿) ∈ 𝐾)
4		cayhamlem2.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑅)
5		cayhamlem2.a	. . . . . . . . . . . 12 ⊢ 𝐴 = (𝑁 Mat 𝑅)
6	5	matsca2 22741	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝐴))
7	6	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 = (Scalar‘𝐴))
8	7	fveq2d 6890	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘𝑅) = (Base‘(Scalar‘𝐴)))
9	4, 8	eqtr2id 2808	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝐴)) = 𝐾)
10	9	eleq2d 2846	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝐻‘𝐿) ∈ (Base‘(Scalar‘𝐴)) ↔ (𝐻‘𝐿) ∈ 𝐾))
11	10	adantr 486	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → ((𝐻‘𝐿) ∈ (Base‘(Scalar‘𝐴)) ↔ (𝐻‘𝐿) ∈ 𝐾))
12	3, 11	mpbird 260	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → (𝐻‘𝐿) ∈ (Base‘(Scalar‘𝐴)))
13		eqid 2760	. . . . . 6 ⊢ (algSc‘𝐴) = (algSc‘𝐴)
14		eqid 2760	. . . . . 6 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴)
15		eqid 2760	. . . . . 6 ⊢ (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
16		cayhamlem2.m	. . . . . 6 ⊢ ∗ = ( ·_𝑠 ‘𝐴)
17		cayhamlem2.1	. . . . . 6 ⊢ 1 = (1_r‘𝐴)
18	13, 14, 15, 16, 17	asclval 22193	. . . . 5 ⊢ ((𝐻‘𝐿) ∈ (Base‘(Scalar‘𝐴)) → ((algSc‘𝐴)‘(𝐻‘𝐿)) = ((𝐻‘𝐿) ∗ 1 ))
19	12, 18	syl 18	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → ((algSc‘𝐴)‘(𝐻‘𝐿)) = ((𝐻‘𝐿) ∗ 1 ))
20	19	eqcomd 2766	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → ((𝐻‘𝐿) ∗ 1 ) = ((algSc‘𝐴)‘(𝐻‘𝐿)))
21	20	oveq2d 7437	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → ((𝐿 ↑ 𝑀) · ((𝐻‘𝐿) ∗ 1 )) = ((𝐿 ↑ 𝑀) · ((algSc‘𝐴)‘(𝐻‘𝐿))))
22	5	matassa 22765	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ AssAlg)
23	22	3adant3 1150	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ AssAlg)
24	23	adantr 486	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → 𝐴 ∈ AssAlg)
25		eqid 2760	. . . . 5 ⊢ (mulGrp‘𝐴) = (mulGrp‘𝐴)
26		cayhamlem2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
27	25, 26	mgpbas 20371	. . . 4 ⊢ 𝐵 = (Base‘(mulGrp‘𝐴))
28		cayhamlem2.e	. . . 4 ⊢ ↑ = (._g‘(mulGrp‘𝐴))
29		crngring 20478	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
30	29	anim2i 629	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
31	30	3adant3 1150	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
32	5	matring 22764	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
33	25	ringmgp 20471	. . . . . 6 ⊢ (𝐴 ∈ Ring → (mulGrp‘𝐴) ∈ Mnd)
34	31, 32, 33	3syl 19	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝐴) ∈ Mnd)
35	34	adantr 486	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → (mulGrp‘𝐴) ∈ Mnd)
36		simprr 785	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → 𝐿 ∈ ℕ₀)
37		simpl3 1212	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → 𝑀 ∈ 𝐵)
38	27, 28, 35, 36, 37	mulgnn0cld 19311	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → (𝐿 ↑ 𝑀) ∈ 𝐵)
39		cayhamlem2.r	. . . 4 ⊢ · = (._r‘𝐴)
40	13, 14, 15, 26, 39, 16	asclmul2 22201	. . 3 ⊢ ((𝐴 ∈ AssAlg ∧ (𝐻‘𝐿) ∈ (Base‘(Scalar‘𝐴)) ∧ (𝐿 ↑ 𝑀) ∈ 𝐵) → ((𝐿 ↑ 𝑀) · ((algSc‘𝐴)‘(𝐻‘𝐿))) = ((𝐻‘𝐿) ∗ (𝐿 ↑ 𝑀)))
41	24, 12, 38, 40	syl3anc 1398	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → ((𝐿 ↑ 𝑀) · ((algSc‘𝐴)‘(𝐻‘𝐿))) = ((𝐻‘𝐿) ∗ (𝐿 ↑ 𝑀)))
42	21, 41	eqtr2d 2796	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → ((𝐻‘𝐿) ∗ (𝐿 ↑ 𝑀)) = ((𝐿 ↑ 𝑀) · ((𝐻‘𝐿) ∗ 1 )))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ‘cfv 6539 (class class class)co 7421 ↑_m cmap 8851 Fincfn 8977 ℕ₀cn0 12610 Basecbs 17393 ._rcmulr 17435 Scalarcsca 17437 ·_𝑠 cvsca 17438 Mndcmnd 18929 ._gcmg 19283 mulGrpcmgp 20366 1_rcur 20413 Ringcrg 20465 CRingccrg 20466 AssAlgcasa 22164 algSccascl 22166 Mat cmat 22728

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-ot 4858 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-supp 8181 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-2o 8481 df-er 8721 df-map 8853 df-ixp 8930 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-fsupp 9358 df-sup 9438 df-oi 9508 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-7 12414 df-8 12415 df-9 12416 df-n0 12611 df-z 12698 df-dec 12819 df-uz 12970 df-fz 13644 df-fzo 13793 df-seq 14149 df-hash 14479 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-sca 17450 df-vsca 17451 df-ip 17452 df-tset 17453 df-ple 17454 df-ds 17456 df-hom 17458 df-cco 17459 df-0g 17618 df-gsum 17619 df-prds 17624 df-pws 17626 df-mre 17762 df-mrc 17763 df-acs 17765 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-mhm 18984 df-submnd 18985 df-grp 19153 df-minusg 19154 df-sbg 19155 df-mulg 19284 df-subg 19339 df-ghm 19434 df-cntz 19537 df-cmn 20002 df-abl 20003 df-mgp 20367 df-rng 20381 df-ur 20414 df-ring 20467 df-cring 20468 df-subrg 20828 df-lmod 21143 df-lss 21213 df-sra 21454 df-rgmod 21455 df-dsmm 22044 df-frlm 22059 df-assa 22167 df-ascl 22169 df-mamu 22712 df-mat 22729

This theorem is used by: cayhamlem3 23211

W3C validator