cayhamlem2 - Metamath Proof Explorer

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Theorem cayhamlem2 21941

Description: Lemma for cayhamlem3 21944. (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 8595	. . . . . . . 8 ⊢ (𝐻 ∈ (𝐾 ↑_m ℕ₀) → 𝐻:ℕ₀⟶𝐾)
2	1	ffvelrnda 6943	. . . . . . 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 21477	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝐴))
7	6	3adant3 1130	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 = (Scalar‘𝐴))
8	7	fveq2d 6760	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘𝑅) = (Base‘(Scalar‘𝐴)))
9	4, 8	eqtr2id 2792	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝐴)) = 𝐾)
10	9	eleq2d 2824	. . . . . . 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 256	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → (𝐻‘𝐿) ∈ (Base‘(Scalar‘𝐴)))
13		eqid 2738	. . . . . 6 ⊢ (algSc‘𝐴) = (algSc‘𝐴)
14		eqid 2738	. . . . . 6 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴)
15		eqid 2738	. . . . . 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 20994	. . . . 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 2744	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → ((𝐻‘𝐿) ∗ 1 ) = ((algSc‘𝐴)‘(𝐻‘𝐿)))
21	20	oveq2d 7271	. 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 21501	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ AssAlg)
23	22	3adant3 1130	. . . 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		crngring 19710	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
26	25	anim2i 616	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
27	26	3adant3 1130	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
28	5	matring 21500	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
29		eqid 2738	. . . . . . 7 ⊢ (mulGrp‘𝐴) = (mulGrp‘𝐴)
30	29	ringmgp 19704	. . . . . 6 ⊢ (𝐴 ∈ Ring → (mulGrp‘𝐴) ∈ Mnd)
31	27, 28, 30	3syl 18	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (mulGrp‘𝐴) ∈ Mnd)
32	31	adantr 480	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → (mulGrp‘𝐴) ∈ Mnd)
33		simprr 769	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → 𝐿 ∈ ℕ₀)
34		simpl3 1191	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → 𝑀 ∈ 𝐵)
35		cayhamlem2.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
36	29, 35	mgpbas 19641	. . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝐴))
37		cayhamlem2.e	. . . . 5 ⊢ ↑ = (._g‘(mulGrp‘𝐴))
38	36, 37	mulgnn0cl 18635	. . . 4 ⊢ (((mulGrp‘𝐴) ∈ Mnd ∧ 𝐿 ∈ ℕ₀ ∧ 𝑀 ∈ 𝐵) → (𝐿 ↑ 𝑀) ∈ 𝐵)
39	32, 33, 34, 38	syl3anc 1369	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → (𝐿 ↑ 𝑀) ∈ 𝐵)
40		cayhamlem2.r	. . . 4 ⊢ · = (._r‘𝐴)
41	13, 14, 15, 35, 40, 16	asclmul2 21001	. . 3 ⊢ ((𝐴 ∈ AssAlg ∧ (𝐻‘𝐿) ∈ (Base‘(Scalar‘𝐴)) ∧ (𝐿 ↑ 𝑀) ∈ 𝐵) → ((𝐿 ↑ 𝑀) · ((algSc‘𝐴)‘(𝐻‘𝐿))) = ((𝐻‘𝐿) ∗ (𝐿 ↑ 𝑀)))
42	24, 12, 39, 41	syl3anc 1369	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → ((𝐿 ↑ 𝑀) · ((algSc‘𝐴)‘(𝐻‘𝐿))) = ((𝐻‘𝐿) ∗ (𝐿 ↑ 𝑀)))
43	21, 42	eqtr2d 2779	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 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 Fincfn 8691 ℕ₀cn0 12163 Basecbs 16840 ._rcmulr 16889 Scalarcsca 16891 ·_𝑠 cvsca 16892 Mndcmnd 18300 ._gcmg 18615 mulGrpcmgp 19635 1_rcur 19652 Ringcrg 19698 CRingccrg 19699 AssAlgcasa 20967 algSccascl 20969 Mat cmat 21464

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-ot 4567 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-fzo 13312 df-seq 13650 df-hash 13973 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-hom 16912 df-cco 16913 df-0g 17069 df-gsum 17070 df-prds 17075 df-pws 17077 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-ghm 18747 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-subrg 19937 df-lmod 20040 df-lss 20109 df-sra 20349 df-rgmod 20350 df-dsmm 20849 df-frlm 20864 df-assa 20970 df-ascl 20972 df-mamu 21443 df-mat 21465

This theorem is referenced by: cayhamlem3 21944

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