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Theorem gsumply1subr 22275

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 22274	. . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐵 ∈ (SubRing‘𝑆))
8		subrgsubg 20606	. . . 4 ⊢ (𝐵 ∈ (SubRing‘𝑆) → 𝐵 ∈ (SubGrp‘𝑆))
9		subgsubm 19173	. . . 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 2761	. . 3 ⊢ (𝑆 ↾_s 𝐵) = (𝑆 ↾_s 𝐵)
13	1, 10, 11, 12	gsumsubm 18852	. 2 ⊢ (𝜑 → (𝑆 Σ_g 𝐹) = ((𝑆 ↾_s 𝐵) Σ_g 𝐹))
14	11, 1	fexd 7207	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
15		ovexd 7427	. . 3 ⊢ (𝜑 → (𝑆 ↾_s 𝐵) ∈ V)
16	5	fvexi 6877	. . . 4 ⊢ 𝑈 ∈ V
17	16	a1i 11	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
18		eqid 2761	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
19	6	oveq2i 7403	. . . . 5 ⊢ (𝑆 ↾_s 𝐵) = (𝑆 ↾_s (Base‘𝑈))
20	3, 4, 5, 18, 2, 19	ressply1bas 22270	. . . 4 ⊢ (𝜑 → (Base‘𝑈) = (Base‘(𝑆 ↾_s 𝐵)))
21	20	eqcomd 2767	. . 3 ⊢ (𝜑 → (Base‘(𝑆 ↾_s 𝐵)) = (Base‘𝑈))
22	12	subrgring 20603	. . . 4 ⊢ (𝐵 ∈ (SubRing‘𝑆) → (𝑆 ↾_s 𝐵) ∈ Ring)
23		ringmgm 20273	. . . 4 ⊢ ((𝑆 ↾_s 𝐵) ∈ Ring → (𝑆 ↾_s 𝐵) ∈ Mgm)
24	2, 7, 22, 23	4syl 19	. . 3 ⊢ (𝜑 → (𝑆 ↾_s 𝐵) ∈ Mgm)
25		simpl 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾_s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾_s 𝐵)))) → 𝜑)
26	3, 4, 5, 6, 2, 12	ressply1bas 22270	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆 ↾_s 𝐵)))
27	26	eqcomd 2767	. . . . . . . . 9 ⊢ (𝜑 → (Base‘(𝑆 ↾_s 𝐵)) = 𝐵)
28	27	eleq2d 2847	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (Base‘(𝑆 ↾_s 𝐵)) ↔ 𝑠 ∈ 𝐵))
29	28	biimpcd 251	. . . . . . 7 ⊢ (𝑠 ∈ (Base‘(𝑆 ↾_s 𝐵)) → (𝜑 → 𝑠 ∈ 𝐵))
30	29	adantr 484	. . . . . 6 ⊢ ((𝑠 ∈ (Base‘(𝑆 ↾_s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾_s 𝐵))) → (𝜑 → 𝑠 ∈ 𝐵))
31	30	impcom 411	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾_s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾_s 𝐵)))) → 𝑠 ∈ 𝐵)
32	27	eleq2d 2847	. . . . . . . 8 ⊢ (𝜑 → (𝑡 ∈ (Base‘(𝑆 ↾_s 𝐵)) ↔ 𝑡 ∈ 𝐵))
33	32	biimpcd 251	. . . . . . 7 ⊢ (𝑡 ∈ (Base‘(𝑆 ↾_s 𝐵)) → (𝜑 → 𝑡 ∈ 𝐵))
34	33	adantl 485	. . . . . 6 ⊢ ((𝑠 ∈ (Base‘(𝑆 ↾_s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾_s 𝐵))) → (𝜑 → 𝑡 ∈ 𝐵))
35	34	impcom 411	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾_s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾_s 𝐵)))) → 𝑡 ∈ 𝐵)
36	3, 4, 5, 6, 2, 12	ressply1add 22271	. . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑡 ∈ 𝐵)) → (𝑠(+_g‘𝑈)𝑡) = (𝑠(+_g‘(𝑆 ↾_s 𝐵))𝑡))
37	25, 31, 35, 36	syl12anc 847	. . . 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 2767	. . 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 6692	. . 3 ⊢ (𝜑 → Fun 𝐹)
40	11	frnd 6696	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
41	40, 26	sseqtrd 3972	. . 3 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘(𝑆 ↾_s 𝐵)))
42	14, 15, 17, 21, 24, 38, 39, 41	gsummgmpropd 18698	. 2 ⊢ (𝜑 → ((𝑆 ↾_s 𝐵) Σ_g 𝐹) = (𝑈 Σ_g 𝐹))
43	13, 42	eqtrd 2796	1 ⊢ (𝜑 → (𝑆 Σ_g 𝐹) = (𝑈 Σ_g 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 Vcvv 3453 ran crn 5646 ⟶wf 6513 ‘cfv 6517 (class class class)co 7392 Basecbs 17228 ↾_s cress 17249 +_gcplusg 17269 Σ_g cgsu 17452 Mgmcmgm 18655 SubMndcsubmnd 18799 SubGrpcsubg 19145 Ringcrg 20262 SubRingcsubrg 20598 Poly₁cpl1 22219

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 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147

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 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-ofr 7657 df-om 7843 df-1st 7966 df-2nd 7967 df-supp 8136 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8673 df-map 8805 df-pm 8806 df-ixp 8876 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-fsupp 9305 df-sup 9385 df-oi 9455 df-card 9894 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12479 df-z 12566 df-dec 12686 df-uz 12837 df-fz 13510 df-fzo 13657 df-seq 14012 df-hash 14341 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-hom 17293 df-cco 17294 df-0g 17453 df-gsum 17454 df-prds 17459 df-pws 17461 df-mre 17597 df-mrc 17598 df-acs 17600 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-mhm 18800 df-submnd 18801 df-grp 18961 df-minusg 18962 df-mulg 19093 df-subg 19148 df-ghm 19237 df-cntz 19340 df-cmn 19805 df-abl 19806 df-mgp 20170 df-rng 20182 df-ur 20211 df-ring 20264 df-subrng 20575 df-subrg 20599 df-psr 21941 df-mpl 21943 df-opsr 21945 df-psr1 22222 df-ply1 22224

This theorem is referenced by: (None)

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