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Theorem gsumply1subr 22251

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 22250	. . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐵 ∈ (SubRing‘𝑆))
8		subrgsubg 20594	. . . 4 ⊢ (𝐵 ∈ (SubRing‘𝑆) → 𝐵 ∈ (SubGrp‘𝑆))
9		subgsubm 19179	. . . 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 2735	. . 3 ⊢ (𝑆 ↾_s 𝐵) = (𝑆 ↾_s 𝐵)
13	1, 10, 11, 12	gsumsubm 18861	. 2 ⊢ (𝜑 → (𝑆 Σ_g 𝐹) = ((𝑆 ↾_s 𝐵) Σ_g 𝐹))
14	11, 1	fexd 7247	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
15		ovexd 7466	. . 3 ⊢ (𝜑 → (𝑆 ↾_s 𝐵) ∈ V)
16	5	fvexi 6921	. . . 4 ⊢ 𝑈 ∈ V
17	16	a1i 11	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
18		eqid 2735	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
19	6	oveq2i 7442	. . . . 5 ⊢ (𝑆 ↾_s 𝐵) = (𝑆 ↾_s (Base‘𝑈))
20	3, 4, 5, 18, 2, 19	ressply1bas 22246	. . . 4 ⊢ (𝜑 → (Base‘𝑈) = (Base‘(𝑆 ↾_s 𝐵)))
21	20	eqcomd 2741	. . 3 ⊢ (𝜑 → (Base‘(𝑆 ↾_s 𝐵)) = (Base‘𝑈))
22	12	subrgring 20591	. . . 4 ⊢ (𝐵 ∈ (SubRing‘𝑆) → (𝑆 ↾_s 𝐵) ∈ Ring)
23		ringmgm 20262	. . . 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 22246	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆 ↾_s 𝐵)))
27	26	eqcomd 2741	. . . . . . . . 9 ⊢ (𝜑 → (Base‘(𝑆 ↾_s 𝐵)) = 𝐵)
28	27	eleq2d 2825	. . . . . . . 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 2825	. . . . . . . 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 22247	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑡 ∈ 𝐵)) → (𝑠(+_g‘𝑈)𝑡) = (𝑠(+_g‘(𝑆 ↾_s 𝐵))𝑡))
37	25, 31, 35, 36	syl12anc 837	. . . 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 2741	. . 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 6741	. . 3 ⊢ (𝜑 → Fun 𝐹)
40	11	frnd 6745	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
41	40, 26	sseqtrd 4036	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘(𝑆 ↾_s 𝐵)))
42	14, 15, 17, 21, 24, 38, 39, 41	gsummgmpropd 18707	. 2 ⊢ (𝜑 → ((𝑆 ↾_s 𝐵) Σ_g 𝐹) = (𝑈 Σ_g 𝐹))
43	13, 42	eqtrd 2775	1 ⊢ (𝜑 → (𝑆 Σ_g 𝐹) = (𝑈 Σ_g 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ran crn 5690 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾_s cress 17274 +_gcplusg 17298 Σ_g cgsu 17487 Mgmcmgm 18664 SubMndcsubmnd 18808 SubGrpcsubg 19151 Ringcrg 20251 SubRingcsubrg 20586 Poly₁cpl1 22194

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-subrng 20563 df-subrg 20587 df-psr 21947 df-mpl 21949 df-opsr 21951 df-psr1 22197 df-ply1 22199

This theorem is referenced by: (None)

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