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Theorem gsumply1subr 22147

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 22146	. . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐵 ∈ (SubRing‘𝑆))
8		subrgsubg 20494	. . . 4 ⊢ (𝐵 ∈ (SubRing‘𝑆) → 𝐵 ∈ (SubGrp‘𝑆))
9		subgsubm 19063	. . . 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 2733	. . 3 ⊢ (𝑆 ↾_s 𝐵) = (𝑆 ↾_s 𝐵)
13	1, 10, 11, 12	gsumsubm 18745	. 2 ⊢ (𝜑 → (𝑆 Σ_g 𝐹) = ((𝑆 ↾_s 𝐵) Σ_g 𝐹))
14	11, 1	fexd 7167	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
15		ovexd 7387	. . 3 ⊢ (𝜑 → (𝑆 ↾_s 𝐵) ∈ V)
16	5	fvexi 6842	. . . 4 ⊢ 𝑈 ∈ V
17	16	a1i 11	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
18		eqid 2733	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
19	6	oveq2i 7363	. . . . 5 ⊢ (𝑆 ↾_s 𝐵) = (𝑆 ↾_s (Base‘𝑈))
20	3, 4, 5, 18, 2, 19	ressply1bas 22142	. . . 4 ⊢ (𝜑 → (Base‘𝑈) = (Base‘(𝑆 ↾_s 𝐵)))
21	20	eqcomd 2739	. . 3 ⊢ (𝜑 → (Base‘(𝑆 ↾_s 𝐵)) = (Base‘𝑈))
22	12	subrgring 20491	. . . 4 ⊢ (𝐵 ∈ (SubRing‘𝑆) → (𝑆 ↾_s 𝐵) ∈ Ring)
23		ringmgm 20164	. . . 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 22142	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆 ↾_s 𝐵)))
27	26	eqcomd 2739	. . . . . . . . 9 ⊢ (𝜑 → (Base‘(𝑆 ↾_s 𝐵)) = 𝐵)
28	27	eleq2d 2819	. . . . . . . 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 2819	. . . . . . . 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 22143	. . . . 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 2739	. . 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 6660	. . 3 ⊢ (𝜑 → Fun 𝐹)
40	11	frnd 6664	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
41	40, 26	sseqtrd 3967	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘(𝑆 ↾_s 𝐵)))
42	14, 15, 17, 21, 24, 38, 39, 41	gsummgmpropd 18591	. 2 ⊢ (𝜑 → ((𝑆 ↾_s 𝐵) Σ_g 𝐹) = (𝑈 Σ_g 𝐹))
43	13, 42	eqtrd 2768	1 ⊢ (𝜑 → (𝑆 Σ_g 𝐹) = (𝑈 Σ_g 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 Vcvv 3437 ran crn 5620 ⟶wf 6482 ‘cfv 6486 (class class class)co 7352 Basecbs 17122 ↾_s cress 17143 +_gcplusg 17163 Σ_g cgsu 17346 Mgmcmgm 18548 SubMndcsubmnd 18692 SubGrpcsubg 19035 Ringcrg 20153 SubRingcsubrg 20486 Poly₁cpl1 22090

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 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090

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 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-ofr 7617 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-pm 8759 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9253 df-sup 9333 df-oi 9403 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-dec 12595 df-uz 12739 df-fz 13410 df-fzo 13557 df-seq 13911 df-hash 14240 df-struct 17060 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-mulr 17177 df-sca 17179 df-vsca 17180 df-ip 17181 df-tset 17182 df-ple 17183 df-ds 17185 df-hom 17187 df-cco 17188 df-0g 17347 df-gsum 17348 df-prds 17353 df-pws 17355 df-mre 17490 df-mrc 17491 df-acs 17493 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-mhm 18693 df-submnd 18694 df-grp 18851 df-minusg 18852 df-mulg 18983 df-subg 19038 df-ghm 19127 df-cntz 19231 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-subrng 20463 df-subrg 20487 df-psr 21848 df-mpl 21850 df-opsr 21852 df-psr1 22093 df-ply1 22095

This theorem is referenced by: (None)

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