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Theorem gsumply1subr 22174

Description: Evaluate a group sum in a polynomial ring over a subring. (Contributed by AV, 22-Sep-2019.) (Proof shortened by AV, 31-Jan-2020.)

**Hypotheses**
Ref	Expression
subrgply1.s	⊢ 𝑆 = (Poly₁‘𝑅)
subrgply1.h	⊢ 𝐻 = (𝑅 ↾_s 𝑇)
subrgply1.u	⊢ 𝑈 = (Poly₁‘𝐻)
subrgply1.b	⊢ 𝐵 = (Base‘𝑈)
gsumply1subr.s	⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))
gsumply1subr.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gsumply1subr.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

**Assertion**
Ref	Expression
gsumply1subr	⊢ (𝜑 → (𝑆 Σ_g 𝐹) = (𝑈 Σ_g 𝐹))

Proof of Theorem gsumply1subr Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumply1subr.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		gsumply1subr.s	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅))
3		subrgply1.s	. . . . 5 ⊢ 𝑆 = (Poly₁‘𝑅)
4		subrgply1.h	. . . . 5 ⊢ 𝐻 = (𝑅 ↾_s 𝑇)
5		subrgply1.u	. . . . 5 ⊢ 𝑈 = (Poly₁‘𝐻)
6		subrgply1.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑈)
7	3, 4, 5, 6	subrgply1 22173	. . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐵 ∈ (SubRing‘𝑆))
8		subrgsubg 20542	. . . 4 ⊢ (𝐵 ∈ (SubRing‘𝑆) → 𝐵 ∈ (SubGrp‘𝑆))
9		subgsubm 19136	. . . 4 ⊢ (𝐵 ∈ (SubGrp‘𝑆) → 𝐵 ∈ (SubMnd‘𝑆))
10	2, 7, 8, 9	4syl 19	. . 3 ⊢ (𝜑 → 𝐵 ∈ (SubMnd‘𝑆))
11		gsumply1subr.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
12		eqid 2736	. . 3 ⊢ (𝑆 ↾_s 𝐵) = (𝑆 ↾_s 𝐵)
13	1, 10, 11, 12	gsumsubm 18818	. 2 ⊢ (𝜑 → (𝑆 Σ_g 𝐹) = ((𝑆 ↾_s 𝐵) Σ_g 𝐹))
14	11, 1	fexd 7224	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
15		ovexd 7445	. . 3 ⊢ (𝜑 → (𝑆 ↾_s 𝐵) ∈ V)
16	5	fvexi 6895	. . . 4 ⊢ 𝑈 ∈ V
17	16	a1i 11	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
18		eqid 2736	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
19	6	oveq2i 7421	. . . . 5 ⊢ (𝑆 ↾_s 𝐵) = (𝑆 ↾_s (Base‘𝑈))
20	3, 4, 5, 18, 2, 19	ressply1bas 22169	. . . 4 ⊢ (𝜑 → (Base‘𝑈) = (Base‘(𝑆 ↾_s 𝐵)))
21	20	eqcomd 2742	. . 3 ⊢ (𝜑 → (Base‘(𝑆 ↾_s 𝐵)) = (Base‘𝑈))
22	12	subrgring 20539	. . . 4 ⊢ (𝐵 ∈ (SubRing‘𝑆) → (𝑆 ↾_s 𝐵) ∈ Ring)
23		ringmgm 20209	. . . 4 ⊢ ((𝑆 ↾_s 𝐵) ∈ Ring → (𝑆 ↾_s 𝐵) ∈ Mgm)
24	2, 7, 22, 23	4syl 19	. . 3 ⊢ (𝜑 → (𝑆 ↾_s 𝐵) ∈ Mgm)
25		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾_s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾_s 𝐵)))) → 𝜑)
26	3, 4, 5, 6, 2, 12	ressply1bas 22169	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆 ↾_s 𝐵)))
27	26	eqcomd 2742	. . . . . . . . 9 ⊢ (𝜑 → (Base‘(𝑆 ↾_s 𝐵)) = 𝐵)
28	27	eleq2d 2821	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (Base‘(𝑆 ↾_s 𝐵)) ↔ 𝑠 ∈ 𝐵))
29	28	biimpcd 249	. . . . . . 7 ⊢ (𝑠 ∈ (Base‘(𝑆 ↾_s 𝐵)) → (𝜑 → 𝑠 ∈ 𝐵))
30	29	adantr 480	. . . . . 6 ⊢ ((𝑠 ∈ (Base‘(𝑆 ↾_s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾_s 𝐵))) → (𝜑 → 𝑠 ∈ 𝐵))
31	30	impcom 407	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾_s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾_s 𝐵)))) → 𝑠 ∈ 𝐵)
32	27	eleq2d 2821	. . . . . . . 8 ⊢ (𝜑 → (𝑡 ∈ (Base‘(𝑆 ↾_s 𝐵)) ↔ 𝑡 ∈ 𝐵))
33	32	biimpcd 249	. . . . . . 7 ⊢ (𝑡 ∈ (Base‘(𝑆 ↾_s 𝐵)) → (𝜑 → 𝑡 ∈ 𝐵))
34	33	adantl 481	. . . . . 6 ⊢ ((𝑠 ∈ (Base‘(𝑆 ↾_s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾_s 𝐵))) → (𝜑 → 𝑡 ∈ 𝐵))
35	34	impcom 407	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾_s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾_s 𝐵)))) → 𝑡 ∈ 𝐵)
36	3, 4, 5, 6, 2, 12	ressply1add 22170	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑡 ∈ 𝐵)) → (𝑠(+_g‘𝑈)𝑡) = (𝑠(+_g‘(𝑆 ↾_s 𝐵))𝑡))
37	25, 31, 35, 36	syl12anc 836	. . . 4 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾_s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾_s 𝐵)))) → (𝑠(+_g‘𝑈)𝑡) = (𝑠(+_g‘(𝑆 ↾_s 𝐵))𝑡))
38	37	eqcomd 2742	. . 3 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾_s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾_s 𝐵)))) → (𝑠(+_g‘(𝑆 ↾_s 𝐵))𝑡) = (𝑠(+_g‘𝑈)𝑡))
39	11	ffund 6715	. . 3 ⊢ (𝜑 → Fun 𝐹)
40	11	frnd 6719	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
41	40, 26	sseqtrd 4000	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘(𝑆 ↾_s 𝐵)))
42	14, 15, 17, 21, 24, 38, 39, 41	gsummgmpropd 18664	. 2 ⊢ (𝜑 → ((𝑆 ↾_s 𝐵) Σ_g 𝐹) = (𝑈 Σ_g 𝐹))
43	13, 42	eqtrd 2771	1 ⊢ (𝜑 → (𝑆 Σ_g 𝐹) = (𝑈 Σ_g 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3464 ran crn 5660 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 Basecbs 17233 ↾_s cress 17256 +_gcplusg 17276 Σ_g cgsu 17459 Mgmcmgm 18621 SubMndcsubmnd 18765 SubGrpcsubg 19108 Ringcrg 20198 SubRingcsubrg 20534 Poly₁cpl1 22117

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-ofr 7677 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-sup 9459 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-fz 13530 df-fzo 13677 df-seq 14025 df-hash 14354 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 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 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-mhm 18766 df-submnd 18767 df-grp 18924 df-minusg 18925 df-mulg 19056 df-subg 19111 df-ghm 19201 df-cntz 19305 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-subrng 20511 df-subrg 20535 df-psr 21874 df-mpl 21876 df-opsr 21878 df-psr1 22120 df-ply1 22122

This theorem is referenced by: (None)

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