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Theorem psrmulvalfi 15188

Description: The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)

**Hypotheses**
Ref	Expression
psrmulr.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
psrmulr.b	⊢ 𝐵 = (Base‘𝑆)
psrmulr.m	⊢ · = (._r‘𝑅)
psrmulr.t	⊢ ∙ = (._r‘𝑆)
psrmulr.d	⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
psrmulfval.i	⊢ (𝜑 → 𝐹 ∈ 𝐵)
psrmulfval.r	⊢ (𝜑 → 𝐺 ∈ 𝐵)
psrmulval.r	⊢ (𝜑 → 𝑋 ∈ 𝐷)
psrmulvalfi.fi	⊢ (𝜑 → 𝐼 ∈ Fin)
psrmulvalfi.ring	⊢ (𝜑 → 𝑅 ∈ Ring)

**Assertion**
Ref	Expression
psrmulvalfi	⊢ (𝜑 → ((𝐹 ∙ 𝐺)‘𝑋) = (𝑅 Σ_g (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘_𝑓 − 𝑘))))))

Distinct variable groups: · ,𝑘 𝐵,𝑘 𝐷,𝑘,𝑦 𝑘,𝐹 𝑘,𝐺 ℎ,𝐼,𝑘,𝑦 𝑅,𝑘 𝑘,𝑋,𝑦 𝜑,𝑘 ℎ,𝑋 Allowed substitution hints: 𝜑(𝑦, ℎ) 𝐵(𝑦, ℎ) 𝐷(ℎ) 𝑅(𝑦, ℎ) 𝑆(𝑦, ℎ, 𝑘) ∙ (𝑦, ℎ, 𝑘) · (𝑦, ℎ) 𝐹(𝑦, ℎ) 𝐺(𝑦, ℎ)

Proof of Theorem psrmulvalfi Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4135	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑦 ∘_𝑟 ≤ 𝑥 ↔ 𝑦 ∘_𝑟 ≤ 𝑋))
2	1	rabbidv 2810	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑥} = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋})
3		fvoveq1 6111	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐺‘(𝑥 ∘_𝑓 − 𝑘)) = (𝐺‘(𝑋 ∘_𝑓 − 𝑘)))
4	3	oveq2d 6104	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑘) · (𝐺‘(𝑥 ∘_𝑓 − 𝑘))) = ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘_𝑓 − 𝑘))))
5	2, 4	mpteq12dv 4214	. . 3 ⊢ (𝑥 = 𝑋 → (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑥} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑥 ∘_𝑓 − 𝑘)))) = (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘_𝑓 − 𝑘)))))
6	5	oveq2d 6104	. 2 ⊢ (𝑥 = 𝑋 → (𝑅 Σ_g (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑥} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑥 ∘_𝑓 − 𝑘))))) = (𝑅 Σ_g (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘_𝑓 − 𝑘))))))
7		psrmulr.s	. . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
8		psrmulr.b	. . 3 ⊢ 𝐵 = (Base‘𝑆)
9		psrmulr.m	. . 3 ⊢ · = (._r‘𝑅)
10		psrmulr.t	. . 3 ⊢ ∙ = (._r‘𝑆)
11		psrmulr.d	. . 3 ⊢ 𝐷 = {ℎ ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
12		psrmulfval.i	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
13		psrmulfval.r	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
14	7, 8, 9, 10, 11, 12, 13	psrmulfval 15187	. 2 ⊢ (𝜑 → (𝐹 ∙ 𝐺) = (𝑥 ∈ 𝐷 ↦ (𝑅 Σ_g (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑥} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑥 ∘_𝑓 − 𝑘)))))))
15		psrmulval.r	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
16		eqid 2238	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
17		eqid 2238	. . 3 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
18		psrmulvalfi.ring	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
19	18	ringcmnd 14447	. . 3 ⊢ (𝜑 → 𝑅 ∈ CMnd)
20		psrmulvalfi.fi	. . . 4 ⊢ (𝜑 → 𝐼 ∈ Fin)
21	11	psrbaglefifi 15172	. . . 4 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝐼 ∈ Fin) → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋} ∈ Fin)
22	15, 20, 21	syl2anc 415	. . 3 ⊢ (𝜑 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋} ∈ Fin)
23	18	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋}) → 𝑅 ∈ Ring)
24	7, 16, 20, 8, 12	psrelbasfi 15180	. . . . . . 7 ⊢ (𝜑 → 𝐹:(ℕ₀ ↑_𝑚 𝐼)⟶(Base‘𝑅))
25	24	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋}) → 𝐹:(ℕ₀ ↑_𝑚 𝐼)⟶(Base‘𝑅))
26		elrabi 2979	. . . . . . . 8 ⊢ (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋} → 𝑘 ∈ 𝐷)
27	26	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋}) → 𝑘 ∈ 𝐷)
28	11	psrbagfi 15168	. . . . . . . . 9 ⊢ (𝐼 ∈ Fin → 𝐷 = (ℕ₀ ↑_𝑚 𝐼))
29	20, 28	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐷 = (ℕ₀ ↑_𝑚 𝐼))
30	29	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋}) → 𝐷 = (ℕ₀ ↑_𝑚 𝐼))
31	27, 30	eleqtrd 2317	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋}) → 𝑘 ∈ (ℕ₀ ↑_𝑚 𝐼))
32	25, 31	ffvelcdmd 5847	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋}) → (𝐹‘𝑘) ∈ (Base‘𝑅))
33	7, 16, 20, 8, 13	psrelbasfi 15180	. . . . . . 7 ⊢ (𝜑 → 𝐺:(ℕ₀ ↑_𝑚 𝐼)⟶(Base‘𝑅))
34	33	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋}) → 𝐺:(ℕ₀ ↑_𝑚 𝐼)⟶(Base‘𝑅))
35		eqid 2238	. . . . . . . . . 10 ⊢ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋} = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋}
36	11, 35	psrbagconcl 15173	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋}) → (𝑋 ∘_𝑓 − 𝑘) ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋})
37	15, 36	sylan 283	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋}) → (𝑋 ∘_𝑓 − 𝑘) ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋})
38		elrabi 2979	. . . . . . . 8 ⊢ ((𝑋 ∘_𝑓 − 𝑘) ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋} → (𝑋 ∘_𝑓 − 𝑘) ∈ 𝐷)
39	37, 38	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋}) → (𝑋 ∘_𝑓 − 𝑘) ∈ 𝐷)
40	39, 30	eleqtrd 2317	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋}) → (𝑋 ∘_𝑓 − 𝑘) ∈ (ℕ₀ ↑_𝑚 𝐼))
41	34, 40	ffvelcdmd 5847	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋}) → (𝐺‘(𝑋 ∘_𝑓 − 𝑘)) ∈ (Base‘𝑅))
42	16, 9, 23, 32, 41	ringcld 14429	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋}) → ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘_𝑓 − 𝑘))) ∈ (Base‘𝑅))
43	42	fmpttd 5866	. . 3 ⊢ (𝜑 → (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘_𝑓 − 𝑘)))):{𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋}⟶(Base‘𝑅))
44	16, 17, 19, 22, 43	gsumclfi 14249	. 2 ⊢ (𝜑 → (𝑅 Σ_g (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘_𝑓 − 𝑘))))) ∈ (Base‘𝑅))
45	6, 14, 15, 44	fvmptd4 5803	1 ⊢ (𝜑 → ((𝐹 ∙ 𝐺)‘𝑋) = (𝑅 Σ_g (𝑘 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑋} ↦ ((𝐹‘𝑘) · (𝐺‘(𝑋 ∘_𝑓 − 𝑘))))))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 = wceq 1402 ∈ wcel 2209 {crab 2532 class class class wbr 4131 ↦ cmpt 4193 ^◡ccnv 4775 “ cima 4779 ⟶wf 5376 ‘cfv 5380 (class class class)co 6088 ∘_𝑓 cof 6303 ∘_𝑟 cofr 6304 ↑_𝑚 cmap 6926 Fincfn 7026 ≤ cle 8367 − cmin 8504 ℕcn 9313 ℕ₀cn0 9574 Basecbs 13410 ._rcmulr 13491 0_gc0g 13669 Σ_g cgsu 14240 Ringcrg 14407 mPwSer cmps 15154

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303 ax-arch 8304 ax-caucvg 8305

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-tp 3717 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-of 6305 df-ofr 6306 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-frec 6666 df-1o 6691 df-oadd 6695 df-er 6811 df-map 6928 df-ixp 6985 df-en 7027 df-dom 7028 df-fin 7029 df-sup 7330 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-5 9375 df-6 9376 df-7 9377 df-8 9378 df-9 9379 df-n0 9575 df-z 9656 df-uz 9938 df-rp 10072 df-fz 10429 df-fzo 10567 df-seqfrec 10906 df-exp 10997 df-ihash 11237 df-cj 11629 df-re 11630 df-im 11631 df-rsqrt 11786 df-abs 11787 df-struct 13412 df-ndx 13413 df-slot 13414 df-base 13416 df-sets 13417 df-plusg 13503 df-mulr 13504 df-sca 13506 df-vsca 13507 df-tset 13509 df-rest 13654 df-topn 13655 df-0g 13671 df-gzsum 13672 df-topgen 13673 df-pt 13674 df-mgm 13735 df-sgrp 13776 df-mnd 13789 df-grp 13867 df-minusg 13868 df-mulg 13982 df-cmn 14179 df-abl 14180 df-gsumfi 14241 df-mgp 14325 df-ur 14370 df-ring 14409 df-psr 15156

This theorem is used by: psrlidmfi 15199 psrridmfi 15200 psrass1fi 15201

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