ressply10g - Mathbox for Thierry Arnoux

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Theorem ressply10g 32651

Description: A restricted polynomial algebra has the same group identity (zero polynomial). (Contributed by Thierry Arnoux, 20-Jan-2025.)

**Hypotheses**
Ref	Expression
ressply.1	⊢ 𝑆 = (Poly₁‘𝑅)
ressply.2	⊢ 𝐻 = (𝑅 ↾_s 𝑇)
ressply.3	⊢ 𝑈 = (Poly₁‘𝐻)
ressply.4	⊢ 𝐵 = (Base‘𝑈)
ressply.5	⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))
ressply10g.6	⊢ 𝑍 = (0_g‘𝑆)

**Assertion**
Ref	Expression
ressply10g	⊢ (𝜑 → 𝑍 = (0_g‘𝑈))

**Proof of Theorem ressply10g**
Step	Hyp	Ref	Expression
1		ressply.1	. . . . 5 ⊢ 𝑆 = (Poly₁‘𝑅)
2		eqid 2732	. . . . 5 ⊢ (algSc‘𝑆) = (algSc‘𝑆)
3		ressply.2	. . . . 5 ⊢ 𝐻 = (𝑅 ↾_s 𝑇)
4		ressply.3	. . . . 5 ⊢ 𝑈 = (Poly₁‘𝐻)
5		ressply.5	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))
6		eqid 2732	. . . . 5 ⊢ (algSc‘𝑈) = (algSc‘𝑈)
7	1, 2, 3, 4, 5, 6	subrg1ascl 21780	. . . 4 ⊢ (𝜑 → (algSc‘𝑈) = ((algSc‘𝑆) ↾ 𝑇))
8	7	fveq1d 6893	. . 3 ⊢ (𝜑 → ((algSc‘𝑈)‘(0_g‘𝐻)) = (((algSc‘𝑆) ↾ 𝑇)‘(0_g‘𝐻)))
9		eqid 2732	. . . 4 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
10		eqid 2732	. . . 4 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
11	3	subrgring 20321	. . . . 5 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐻 ∈ Ring)
12	5, 11	syl 17	. . . 4 ⊢ (𝜑 → 𝐻 ∈ Ring)
13	4, 6, 9, 10, 12	ply1ascl0 32647	. . 3 ⊢ (𝜑 → ((algSc‘𝑈)‘(0_g‘𝐻)) = (0_g‘𝑈))
14		eqid 2732	. . . . . . 7 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
15	3, 14	subrg0 20325	. . . . . 6 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (0_g‘𝑅) = (0_g‘𝐻))
16	5, 15	syl 17	. . . . 5 ⊢ (𝜑 → (0_g‘𝑅) = (0_g‘𝐻))
17		subrgsubg 20324	. . . . . 6 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑇 ∈ (SubGrp‘𝑅))
18	14	subg0cl 19013	. . . . . 6 ⊢ (𝑇 ∈ (SubGrp‘𝑅) → (0_g‘𝑅) ∈ 𝑇)
19	5, 17, 18	3syl 18	. . . . 5 ⊢ (𝜑 → (0_g‘𝑅) ∈ 𝑇)
20	16, 19	eqeltrrd 2834	. . . 4 ⊢ (𝜑 → (0_g‘𝐻) ∈ 𝑇)
21	20	fvresd 6911	. . 3 ⊢ (𝜑 → (((algSc‘𝑆) ↾ 𝑇)‘(0_g‘𝐻)) = ((algSc‘𝑆)‘(0_g‘𝐻)))
22	8, 13, 21	3eqtr3d 2780	. 2 ⊢ (𝜑 → (0_g‘𝑈) = ((algSc‘𝑆)‘(0_g‘𝐻)))
23	16	fveq2d 6895	. 2 ⊢ (𝜑 → ((algSc‘𝑆)‘(0_g‘𝑅)) = ((algSc‘𝑆)‘(0_g‘𝐻)))
24		ressply10g.6	. . 3 ⊢ 𝑍 = (0_g‘𝑆)
25		subrgrcl 20323	. . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
26	5, 25	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
27	1, 2, 14, 24, 26	ply1ascl0 32647	. 2 ⊢ (𝜑 → ((algSc‘𝑆)‘(0_g‘𝑅)) = 𝑍)
28	22, 23, 27	3eqtr2rd 2779	1 ⊢ (𝜑 → 𝑍 = (0_g‘𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ↾ cres 5678 ‘cfv 6543 (class class class)co 7408 Basecbs 17143 ↾_s cress 17172 0_gc0g 17384 SubGrpcsubg 18999 Ringcrg 20055 SubRingcsubrg 20314 algSccascl 21406 Poly₁cpl1 21700

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-ofr 7670 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 df-fzo 13627 df-seq 13966 df-hash 14290 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mhm 18670 df-submnd 18671 df-grp 18821 df-minusg 18822 df-sbg 18823 df-mulg 18950 df-subg 19002 df-ghm 19089 df-cntz 19180 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-ring 20057 df-subrg 20316 df-lmod 20472 df-lss 20542 df-ascl 21409 df-psr 21461 df-mpl 21463 df-opsr 21465 df-psr1 21703 df-ply1 21705

This theorem is referenced by: ressply1mon1p 32652 ressply1invg 32653 irngnzply1lem 32749 irngnzply1 32750 irngnminplynz 32766 algextdeglem1 32767

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