cpmatsubgpmat - Metamath Proof Explorer

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

Theorem cpmatsubgpmat 22767

Description: The set of all constant polynomial matrices over a ring 𝑅 is an additive subgroup of the ring of all polynomial matrices over the ring 𝑅. (Contributed by AV, 15-Nov-2019.)

**Hypotheses**
Ref	Expression
cpmatsrngpmat.s	⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
cpmatsrngpmat.p	⊢ 𝑃 = (Poly₁‘𝑅)
cpmatsrngpmat.c	⊢ 𝐶 = (𝑁 Mat 𝑃)

**Assertion**
Ref	Expression
cpmatsubgpmat	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶))

Proof of Theorem cpmatsubgpmat Dummy variables 𝑖 𝑗 𝑥 𝑦 𝑘 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cpmatsrngpmat.s	. . . 4 ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)
2		cpmatsrngpmat.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
3		cpmatsrngpmat.c	. . . 4 ⊢ 𝐶 = (𝑁 Mat 𝑃)
4		eqid 2761	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
5	1, 2, 3, 4	cpmat 22756	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = {𝑚 ∈ (Base‘𝐶) ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe₁‘(𝑖𝑚𝑗))‘𝑘) = (0_g‘𝑅)})
6		ssrab2 4031	. . 3 ⊢ {𝑚 ∈ (Base‘𝐶) ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe₁‘(𝑖𝑚𝑗))‘𝑘) = (0_g‘𝑅)} ⊆ (Base‘𝐶)
7	5, 6	eqsstrdi 3978	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ⊆ (Base‘𝐶))
8	1, 2, 3	1elcpmat 22762	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐶) ∈ 𝑆)
9	8	ne0d 4292	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ≠ ∅)
10	1, 2, 3	cpmatacl 22763	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+_g‘𝐶)𝑦) ∈ 𝑆)
11	1, 2, 3	cpmatinvcl 22764	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ((inv_g‘𝐶)‘𝑥) ∈ 𝑆)
12		r19.26 3121	. . 3 ⊢ (∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥(+_g‘𝐶)𝑦) ∈ 𝑆 ∧ ((inv_g‘𝐶)‘𝑥) ∈ 𝑆) ↔ (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+_g‘𝐶)𝑦) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ((inv_g‘𝐶)‘𝑥) ∈ 𝑆))
13	10, 11, 12	sylanbrc 592	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥(+_g‘𝐶)𝑦) ∈ 𝑆 ∧ ((inv_g‘𝐶)‘𝑥) ∈ 𝑆))
14	2, 3	pmatring 22739	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
15		ringgrp 20274	. . 3 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ Grp)
16		eqid 2761	. . . 4 ⊢ (+_g‘𝐶) = (+_g‘𝐶)
17		eqid 2761	. . . 4 ⊢ (inv_g‘𝐶) = (inv_g‘𝐶)
18	4, 16, 17	issubg2 19173	. . 3 ⊢ (𝐶 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐶) ↔ (𝑆 ⊆ (Base‘𝐶) ∧ 𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥(+_g‘𝐶)𝑦) ∈ 𝑆 ∧ ((inv_g‘𝐶)‘𝑥) ∈ 𝑆))))
19	14, 15, 18	3syl 18	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑆 ∈ (SubGrp‘𝐶) ↔ (𝑆 ⊆ (Base‘𝐶) ∧ 𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 (∀𝑦 ∈ 𝑆 (𝑥(+_g‘𝐶)𝑦) ∈ 𝑆 ∧ ((inv_g‘𝐶)‘𝑥) ∈ 𝑆))))
20	7, 9, 13, 19	mpbir3and 1355	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 {crab 3413 ⊆ wss 3902 ∅c0 4283 ‘cfv 6515 (class class class)co 7390 Fincfn 8920 ℕcn 12203 Basecbs 17235 +_gcplusg 17276 0_gc0g 17458 Grpcgrp 18965 inv_gcminusg 18966 SubGrpcsubg 19152 1_rcur 20217 Ringcrg 20269 Poly₁cpl1 22226 coe₁cco1 22227 Mat cmat 22454 ConstPolyMat ccpmat 22750

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-ot 4588 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-isom 6524 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7654 df-ofr 7655 df-om 7841 df-1st 7964 df-2nd 7965 df-supp 8134 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-2o 8431 df-er 8671 df-map 8803 df-pm 8804 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9301 df-sup 9381 df-oi 9451 df-card 9890 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-nn 12204 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12475 df-z 12562 df-dec 12682 df-uz 12833 df-fz 13506 df-fzo 13653 df-seq 14008 df-hash 14337 df-struct 17173 df-sets 17190 df-slot 17208 df-ndx 17220 df-base 17236 df-ress 17257 df-plusg 17289 df-mulr 17290 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-hom 17300 df-cco 17301 df-0g 17460 df-gsum 17461 df-prds 17466 df-pws 17468 df-mre 17604 df-mrc 17605 df-acs 17607 df-mgm 18664 df-sgrp 18743 df-mnd 18759 df-mhm 18807 df-submnd 18808 df-grp 18968 df-minusg 18969 df-sbg 18970 df-mulg 19100 df-subg 19155 df-ghm 19244 df-cntz 19347 df-cmn 19812 df-abl 19813 df-mgp 20177 df-rng 20189 df-ur 20218 df-srg 20223 df-ring 20271 df-subrng 20582 df-subrg 20606 df-lmod 20916 df-lss 20986 df-sra 21227 df-rgmod 21228 df-dsmm 21771 df-frlm 21786 df-ascl 21894 df-psr 21948 df-mvr 21949 df-mpl 21950 df-opsr 21952 df-psr1 22229 df-vr1 22230 df-ply1 22231 df-coe1 22232 df-mamu 22438 df-mat 22455 df-cpmat 22753

This theorem is referenced by: cpmatsrgpmat 22768 0elcpmat 22769 m2cpmghm 22791 chfacfisfcpmat 22902

W3C validator