cayhamlem2 - Metamath Proof Explorer

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Theorem cayhamlem2 22802

Description: Lemma for cayhamlem3 22805. (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 8781	. . . . . . . 8 ⊢ (𝐻 ∈ (𝐾 ↑_m ℕ₀) → 𝐻:ℕ₀⟶𝐾)
2	1	ffvelcdmda 7025	. . . . . . 7 ⊢ ((𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀) → (𝐻‘𝐿) ∈ 𝐾)
3	2	adantl 481	. . . . . 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 22338	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝐴))
7	6	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 = (Scalar‘𝐴))
8	7	fveq2d 6834	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘𝑅) = (Base‘(Scalar‘𝐴)))
9	4, 8	eqtr2id 2781	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝐴)) = 𝐾)
10	9	eleq2d 2819	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝐻‘𝐿) ∈ (Base‘(Scalar‘𝐴)) ↔ (𝐻‘𝐿) ∈ 𝐾))
11	10	adantr 480	. . . . . 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 257	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → (𝐻‘𝐿) ∈ (Base‘(Scalar‘𝐴)))
13		eqid 2733	. . . . . 6 ⊢ (algSc‘𝐴) = (algSc‘𝐴)
14		eqid 2733	. . . . . 6 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴)
15		eqid 2733	. . . . . 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 21821	. . . . 5 ⊢ ((𝐻‘𝐿) ∈ (Base‘(Scalar‘𝐴)) → ((algSc‘𝐴)‘(𝐻‘𝐿)) = ((𝐻‘𝐿) ∗ 1 ))
19	12, 18	syl 17	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → ((algSc‘𝐴)‘(𝐻‘𝐿)) = ((𝐻‘𝐿) ∗ 1 ))
20	19	eqcomd 2739	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → ((𝐻‘𝐿) ∗ 1 ) = ((algSc‘𝐴)‘(𝐻‘𝐿)))
21	20	oveq2d 7370	. 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 22362	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ AssAlg)
23	22	3adant3 1132	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐴 ∈ AssAlg)
24	23	adantr 480	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → 𝐴 ∈ AssAlg)
25		eqid 2733	. . . . 5 ⊢ (mulGrp‘𝐴) = (mulGrp‘𝐴)
26		cayhamlem2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
27	25, 26	mgpbas 20067	. . . 4 ⊢ 𝐵 = (Base‘(mulGrp‘𝐴))
28		cayhamlem2.e	. . . 4 ⊢ ↑ = (._g‘(mulGrp‘𝐴))
29		crngring 20167	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
30	29	anim2i 617	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
31	30	3adant3 1132	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
32	5	matring 22361	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
33	25	ringmgp 20161	. . . . . 6 ⊢ (𝐴 ∈ Ring → (mulGrp‘𝐴) ∈ Mnd)
34	31, 32, 33	3syl 18	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝐴) ∈ Mnd)
35	34	adantr 480	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → (mulGrp‘𝐴) ∈ Mnd)
36		simprr 772	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → 𝐿 ∈ ℕ₀)
37		simpl3 1194	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → 𝑀 ∈ 𝐵)
38	27, 28, 35, 36, 37	mulgnn0cld 19012	. . 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 21828	. . 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 1373	. 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 2769	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

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ‘cfv 6488 (class class class)co 7354 ↑_m cmap 8758 Fincfn 8877 ℕ₀cn0 12390 Basecbs 17124 ._rcmulr 17166 Scalarcsca 17168 ·_𝑠 cvsca 17169 Mndcmnd 18646 ._gcmg 18984 mulGrpcmgp 20062 1_rcur 20103 Ringcrg 20155 CRingccrg 20156 AssAlgcasa 21791 algSccascl 21793 Mat cmat 22325

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7676 ax-cnex 11071 ax-resscn 11072 ax-1cn 11073 ax-icn 11074 ax-addcl 11075 ax-addrcl 11076 ax-mulcl 11077 ax-mulrcl 11078 ax-mulcom 11079 ax-addass 11080 ax-mulass 11081 ax-distr 11082 ax-i2m1 11083 ax-1ne0 11084 ax-1rid 11085 ax-rnegex 11086 ax-rrecex 11087 ax-cnre 11088 ax-pre-lttri 11089 ax-pre-lttrn 11090 ax-pre-ltadd 11091 ax-pre-mulgt0 11092

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-ot 4586 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-isom 6497 df-riota 7311 df-ov 7357 df-oprab 7358 df-mpo 7359 df-of 7618 df-om 7805 df-1st 7929 df-2nd 7930 df-supp 8099 df-frecs 8219 df-wrecs 8250 df-recs 8299 df-rdg 8337 df-1o 8393 df-2o 8394 df-er 8630 df-map 8760 df-ixp 8830 df-en 8878 df-dom 8879 df-sdom 8880 df-fin 8881 df-fsupp 9255 df-sup 9335 df-oi 9405 df-card 9841 df-pnf 11157 df-mnf 11158 df-xr 11159 df-ltxr 11160 df-le 11161 df-sub 11355 df-neg 11356 df-nn 12135 df-2 12197 df-3 12198 df-4 12199 df-5 12200 df-6 12201 df-7 12202 df-8 12203 df-9 12204 df-n0 12391 df-z 12478 df-dec 12597 df-uz 12741 df-fz 13412 df-fzo 13559 df-seq 13913 df-hash 14242 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17125 df-ress 17146 df-plusg 17178 df-mulr 17179 df-sca 17181 df-vsca 17182 df-ip 17183 df-tset 17184 df-ple 17185 df-ds 17187 df-hom 17189 df-cco 17190 df-0g 17349 df-gsum 17350 df-prds 17355 df-pws 17357 df-mre 17492 df-mrc 17493 df-acs 17495 df-mgm 18552 df-sgrp 18631 df-mnd 18647 df-mhm 18695 df-submnd 18696 df-grp 18853 df-minusg 18854 df-sbg 18855 df-mulg 18985 df-subg 19040 df-ghm 19129 df-cntz 19233 df-cmn 19698 df-abl 19699 df-mgp 20063 df-rng 20075 df-ur 20104 df-ring 20157 df-cring 20158 df-subrg 20489 df-lmod 20799 df-lss 20869 df-sra 21111 df-rgmod 21112 df-dsmm 21673 df-frlm 21688 df-assa 21794 df-ascl 21796 df-mamu 22309 df-mat 22326

This theorem is referenced by: cayhamlem3 22805

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