rhmpsrfilem2 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > rhmpsrfilem2		GIF version

Theorem rhmpsrfilem2 15125

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 14392	. . 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 15115	. . 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 5844	. . . 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 15106	. . . . . . . 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 4133	. . . . . . . . . 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 15114	. . . . . . 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 5844	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘}) → (𝑌‘(𝑘 ∘_𝑓 − 𝑥)) ∈ (Base‘𝑅))
32	1, 12, 13, 18, 31	ringcld 14374	. . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐷) ∧ 𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘_𝑟 ≤ 𝑘}) → ((𝑋‘𝑥)(._r‘𝑅)(𝑌‘(𝑘 ∘_𝑓 − 𝑥))) ∈ (Base‘𝑅))
33	32	fmpttd 5863	. 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 14211	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 4130 ↦ cmpt 4192 ^◡ccnv 4773 “ cima 4777 ⟶wf 5373 ‘cfv 5377 (class class class)co 6085 ∘_𝑓 cof 6300 ∘_𝑟 cofr 6301 ↑_𝑚 cmap 6922 Fincfn 7022 ≤ cle 8362 − cmin 8499 ℕcn 9307 ℕ₀cn0 9568 Basecbs 13404 ._rcmulr 13484 0_gc0g 13662 CMndccmn 14139 Σ_g cgsu 14202 Ringcrg 14352

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 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8271 ax-resscn 8272 ax-1cn 8273 ax-1re 8274 ax-icn 8275 ax-addcl 8276 ax-addrcl 8277 ax-mulcl 8278 ax-mulrcl 8279 ax-addcom 8280 ax-mulcom 8281 ax-addass 8282 ax-mulass 8283 ax-distr 8284 ax-i2m1 8285 ax-0lt1 8286 ax-1rid 8287 ax-0id 8288 ax-rnegex 8289 ax-precex 8290 ax-cnre 8291 ax-pre-ltirr 8292 ax-pre-ltwlin 8293 ax-pre-lttrn 8294 ax-pre-apti 8295 ax-pre-ltadd 8296 ax-pre-mulgt0 8297 ax-pre-mulext 8298 ax-arch 8299 ax-caucvg 8300

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 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-of 6302 df-ofr 6303 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-oadd 6691 df-er 6807 df-map 6924 df-en 7023 df-dom 7024 df-fin 7025 df-sup 7325 df-pnf 8363 df-mnf 8364 df-xr 8365 df-ltxr 8366 df-le 8367 df-sub 8501 df-neg 8502 df-reap 8906 df-ap 8913 df-div 9006 df-inn 9308 df-2 9366 df-3 9367 df-4 9368 df-n0 9569 df-z 9650 df-uz 9932 df-rp 10066 df-fz 10423 df-fzo 10561 df-seqfrec 10900 df-exp 10991 df-ihash 11231 df-cj 11623 df-re 11624 df-im 11625 df-rsqrt 11780 df-abs 11781 df-ndx 13407 df-slot 13408 df-base 13410 df-sets 13411 df-plusg 13496 df-mulr 13497 df-0g 13664 df-gzsum 13665 df-mgm 13728 df-sgrp 13769 df-mnd 13782 df-grp 13860 df-minusg 13861 df-mulg 13975 df-cmn 14141 df-abl 14142 df-gsumfi 14203 df-mgp 14270 df-ur 14315 df-ring 14354

This theorem is used by: psrmulclfilem 15129

W3C validator