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Theorem gsumply1subr 22186

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 22185	. . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐵 ∈ (SubRing‘𝑆))
8		subrgsubg 20522	. . . 4 ⊢ (𝐵 ∈ (SubRing‘𝑆) → 𝐵 ∈ (SubGrp‘𝑆))
9		subgsubm 19090	. . . 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 2737	. . 3 ⊢ (𝑆 ↾_s 𝐵) = (𝑆 ↾_s 𝐵)
13	1, 10, 11, 12	gsumsubm 18772	. 2 ⊢ (𝜑 → (𝑆 Σ_g 𝐹) = ((𝑆 ↾_s 𝐵) Σ_g 𝐹))
14	11, 1	fexd 7183	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
15		ovexd 7403	. . 3 ⊢ (𝜑 → (𝑆 ↾_s 𝐵) ∈ V)
16	5	fvexi 6856	. . . 4 ⊢ 𝑈 ∈ V
17	16	a1i 11	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
18		eqid 2737	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
19	6	oveq2i 7379	. . . . 5 ⊢ (𝑆 ↾_s 𝐵) = (𝑆 ↾_s (Base‘𝑈))
20	3, 4, 5, 18, 2, 19	ressply1bas 22181	. . . 4 ⊢ (𝜑 → (Base‘𝑈) = (Base‘(𝑆 ↾_s 𝐵)))
21	20	eqcomd 2743	. . 3 ⊢ (𝜑 → (Base‘(𝑆 ↾_s 𝐵)) = (Base‘𝑈))
22	12	subrgring 20519	. . . 4 ⊢ (𝐵 ∈ (SubRing‘𝑆) → (𝑆 ↾_s 𝐵) ∈ Ring)
23		ringmgm 20191	. . . 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 22181	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆 ↾_s 𝐵)))
27	26	eqcomd 2743	. . . . . . . . 9 ⊢ (𝜑 → (Base‘(𝑆 ↾_s 𝐵)) = 𝐵)
28	27	eleq2d 2823	. . . . . . . 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 2823	. . . . . . . 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 22182	. . . . 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 2743	. . 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 6674	. . 3 ⊢ (𝜑 → Fun 𝐹)
40	11	frnd 6678	. . . 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 18618	. 2 ⊢ (𝜑 → ((𝑆 ↾_s 𝐵) Σ_g 𝐹) = (𝑈 Σ_g 𝐹))
43	13, 42	eqtrd 2772	1 ⊢ (𝜑 → (𝑆 Σ_g 𝐹) = (𝑈 Σ_g 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3442 ran crn 5633 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 Basecbs 17148 ↾_s cress 17169 +_gcplusg 17189 Σ_g cgsu 17372 Mgmcmgm 18575 SubMndcsubmnd 18719 SubGrpcsubg 19062 Ringcrg 20180 SubRingcsubrg 20514 Poly₁cpl1 22129

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-ofr 7633 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-sup 9357 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-fzo 13583 df-seq 13937 df-hash 14266 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-hom 17213 df-cco 17214 df-0g 17373 df-gsum 17374 df-prds 17379 df-pws 17381 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-submnd 18721 df-grp 18878 df-minusg 18879 df-mulg 19010 df-subg 19065 df-ghm 19154 df-cntz 19258 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-subrng 20491 df-subrg 20515 df-psr 21877 df-mpl 21879 df-opsr 21881 df-psr1 22132 df-ply1 22134

This theorem is referenced by: (None)

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