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Theorem rhmpsrfilem2 15185

Description: Lemma for rhmpsr et al. (Contributed by SN, 8-Feb-2025.)

**Hypotheses**
Ref	Expression
rhmpsrlem1.d	⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
rhmpsrlem1.r	⊢ (𝜑 → 𝑅 ∈ Ring)
rhmpsrlem2.i	⊢ (𝜑 → 𝐼 ∈ Fin)
rhmpsrlem1.x	⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
rhmpsrlem1.y	⊢ (𝜑 → 𝑌:𝐷⟶(Base‘𝑅))

**Assertion**
Ref	Expression
rhmpsrfilem2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑅 Σ_g (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑥)(._r‘𝑅)(𝑌‘(𝑘 ∘_𝑓 − 𝑥))))) ∈ (Base‘𝑅))

Distinct variable groups: 𝑥,𝐷,𝑦 𝑓,𝐼,𝑦 𝜑,𝑥 𝑓,𝑘,𝑦,𝑥 𝑥,𝑅 𝑥,𝑓 Allowed substitution hints: 𝜑(𝑦, 𝑓, 𝑘) 𝐷(𝑓, 𝑘) 𝑅(𝑦, 𝑓, 𝑘) 𝐼(𝑥, 𝑘) 𝑋(𝑥, 𝑦, 𝑓, 𝑘) 𝑌(𝑥, 𝑦, 𝑓, 𝑘)

**Proof of Theorem rhmpsrfilem2**
Step	Hyp	Ref	Expression
1		eqid 2238	. 2 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2238	. 2 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
3		rhmpsrlem1.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
4	3	ringcmnd 14447	. . 3 ⊢ (𝜑 → 𝑅 ∈ CMnd)
5	4	adantr 276	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑅 ∈ CMnd)
6		simpr 110	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑘 ∈ 𝐷)
7		rhmpsrlem2.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ Fin)
8	7	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝐼 ∈ Fin)
9		rhmpsrlem1.d	. . . 4 ⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_𝑚 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
10	9	psrbaglefifi 15172	. . 3 ⊢ ((𝑘 ∈ 𝐷 ∧ 𝐼 ∈ Fin) → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘} ∈ Fin)
11	6, 8, 10	syl2anc 415	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘} ∈ Fin)
12		eqid 2238	. . . 4 ⊢ (._r‘𝑅) = (._r‘𝑅)
13	3	ad2antrr 492	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘}) → 𝑅 ∈ Ring)
14		rhmpsrlem1.x	. . . . . 6 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
15	14	ad2antrr 492	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘}) → 𝑋:𝐷⟶(Base‘𝑅))
16		elrabi 2979	. . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘} → 𝑥 ∈ 𝐷)
17	16	adantl 277	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘}) → 𝑥 ∈ 𝐷)
18	15, 17	ffvelcdmd 5847	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘}) → (𝑋‘𝑥) ∈ (Base‘𝑅))
19		rhmpsrlem1.y	. . . . . 6 ⊢ (𝜑 → 𝑌:𝐷⟶(Base‘𝑅))
20	19	ad2antrr 492	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘}) → 𝑌:𝐷⟶(Base‘𝑅))
21	6	adantr 276	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘}) → 𝑘 ∈ 𝐷)
22	9	psrbagf 15163	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → 𝑥:𝐼⟶ℕ₀)
23	17, 22	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘}) → 𝑥:𝐼⟶ℕ₀)
24		breq1 4134	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦 ∘_𝑟 ≤ 𝑘 ↔ 𝑥 ∘_𝑟 ≤ 𝑘))
25	24	elrab 2982	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘} ↔ (𝑥 ∈ 𝐷 ∧ 𝑥 ∘_𝑟 ≤ 𝑘))
26	25	simprbi 275	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘} → 𝑥 ∘_𝑟 ≤ 𝑘)
27	26	adantl 277	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘}) → 𝑥 ∘_𝑟 ≤ 𝑘)
28	9	psrbagcon 15171	. . . . . . 7 ⊢ ((𝑘 ∈ 𝐷 ∧ 𝑥:𝐼⟶ℕ₀ ∧ 𝑥 ∘_𝑟 ≤ 𝑘) → ((𝑘 ∘_𝑓 − 𝑥) ∈ 𝐷 ∧ (𝑘 ∘_𝑓 − 𝑥) ∘_𝑟 ≤ 𝑘))
29	21, 23, 27, 28	syl3anc 1278	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘}) → ((𝑘 ∘_𝑓 − 𝑥) ∈ 𝐷 ∧ (𝑘 ∘_𝑓 − 𝑥) ∘_𝑟 ≤ 𝑘))
30	29	simpld 112	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘}) → (𝑘 ∘_𝑓 − 𝑥) ∈ 𝐷)
31	20, 30	ffvelcdmd 5847	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘}) → (𝑌‘(𝑘 ∘_𝑓 − 𝑥)) ∈ (Base‘𝑅))
32	1, 12, 13, 18, 31	ringcld 14429	. . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘}) → ((𝑋‘𝑥)(._r‘𝑅)(𝑌‘(𝑘 ∘_𝑓 − 𝑥))) ∈ (Base‘𝑅))
33	32	fmpttd 5866	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑥)(._r‘𝑅)(𝑌‘(𝑘 ∘_𝑓 − 𝑥)))):{𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘}⟶(Base‘𝑅))
34	1, 2, 5, 11, 33	gsumclfi 14249	1 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → (𝑅 Σ_g (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘} ↦ ((𝑋‘𝑥)(._r‘𝑅)(𝑌‘(𝑘 ∘_𝑓 − 𝑥))))) ∈ (Base‘𝑅))

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 CMndccmn 14177 Σ_g cgsu 14240 Ringcrg 14407

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-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-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-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-ndx 13413 df-slot 13414 df-base 13416 df-sets 13417 df-plusg 13503 df-mulr 13504 df-0g 13671 df-gzsum 13672 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

This theorem is used by: psrmulclfilem 15189 psrdifi 15202 psrdirfi 15203

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