elirng - Mathbox for Thierry Arnoux

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Theorem elirng 33360

Description: Property for an element 𝑋 of a field 𝑅 to be integral over a subring 𝑆. (Contributed by Thierry Arnoux, 28-Jan-2025.)

**Hypotheses**
Ref	Expression
irngval.o	⊢ 𝑂 = (𝑅 evalSub₁ 𝑆)
irngval.u	⊢ 𝑈 = (𝑅 ↾_s 𝑆)
irngval.b	⊢ 𝐵 = (Base‘𝑅)
irngval.0	⊢ 0 = (0_g‘𝑅)
elirng.r	⊢ (𝜑 → 𝑅 ∈ CRing)
elirng.s	⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))

**Assertion**
Ref	Expression
elirng	⊢ (𝜑 → (𝑋 ∈ (𝑅 IntgRing 𝑆) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑓 ∈ (Monic_1p‘𝑈)((𝑂‘𝑓)‘𝑋) = 0 )))

Distinct variable groups: 𝐵,𝑓 𝑓,𝑂 𝑅,𝑓 𝑆,𝑓 𝑈,𝑓 𝑓,𝑋 𝜑,𝑓 Allowed substitution hint: 0 (𝑓)

**Proof of Theorem elirng**
Step	Hyp	Ref	Expression
1		irngval.o	. . . . . 6 ⊢ 𝑂 = (𝑅 evalSub₁ 𝑆)
2		irngval.u	. . . . . 6 ⊢ 𝑈 = (𝑅 ↾_s 𝑆)
3		irngval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
4		irngval.0	. . . . . 6 ⊢ 0 = (0_g‘𝑅)
5		elirng.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ CRing)
6	5	crngringd 20185	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring)
7		elirng.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))
8	3	subrgss 20510	. . . . . . 7 ⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ 𝐵)
9	7, 8	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
10	1, 2, 3, 4, 6, 9	irngval 33359	. . . . 5 ⊢ (𝜑 → (𝑅 IntgRing 𝑆) = ∪ 𝑓 ∈ (Monic_1p‘𝑈)(^◡(𝑂‘𝑓) “ { 0 }))
11	10	eleq2d 2815	. . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝑅 IntgRing 𝑆) ↔ 𝑋 ∈ ∪ 𝑓 ∈ (Monic_1p‘𝑈)(^◡(𝑂‘𝑓) “ { 0 })))
12		eliun 5000	. . . 4 ⊢ (𝑋 ∈ ∪ 𝑓 ∈ (Monic_1p‘𝑈)(^◡(𝑂‘𝑓) “ { 0 }) ↔ ∃𝑓 ∈ (Monic_1p‘𝑈)𝑋 ∈ (^◡(𝑂‘𝑓) “ { 0 }))
13	11, 12	bitrdi 287	. . 3 ⊢ (𝜑 → (𝑋 ∈ (𝑅 IntgRing 𝑆) ↔ ∃𝑓 ∈ (Monic_1p‘𝑈)𝑋 ∈ (^◡(𝑂‘𝑓) “ { 0 })))
14		eqid 2728	. . . . . 6 ⊢ (𝑅 ↑_s 𝐵) = (𝑅 ↑_s 𝐵)
15		eqid 2728	. . . . . 6 ⊢ (Base‘(𝑅 ↑_s 𝐵)) = (Base‘(𝑅 ↑_s 𝐵))
16	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (Monic_1p‘𝑈)) → 𝑅 ∈ Ring)
17	3	fvexi 6911	. . . . . . 7 ⊢ 𝐵 ∈ V
18	17	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (Monic_1p‘𝑈)) → 𝐵 ∈ V)
19		eqid 2728	. . . . . . . . . . 11 ⊢ (Poly₁‘𝑈) = (Poly₁‘𝑈)
20	1, 3, 14, 2, 19	evls1rhm 22240	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑅)) → 𝑂 ∈ ((Poly₁‘𝑈) RingHom (𝑅 ↑_s 𝐵)))
21	5, 7, 20	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → 𝑂 ∈ ((Poly₁‘𝑈) RingHom (𝑅 ↑_s 𝐵)))
22		eqid 2728	. . . . . . . . . 10 ⊢ (Base‘(Poly₁‘𝑈)) = (Base‘(Poly₁‘𝑈))
23	22, 15	rhmf 20423	. . . . . . . . 9 ⊢ (𝑂 ∈ ((Poly₁‘𝑈) RingHom (𝑅 ↑_s 𝐵)) → 𝑂:(Base‘(Poly₁‘𝑈))⟶(Base‘(𝑅 ↑_s 𝐵)))
24	21, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑂:(Base‘(Poly₁‘𝑈))⟶(Base‘(𝑅 ↑_s 𝐵)))
25	24	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Monic_1p‘𝑈)) → 𝑂:(Base‘(Poly₁‘𝑈))⟶(Base‘(𝑅 ↑_s 𝐵)))
26		eqid 2728	. . . . . . . . 9 ⊢ (Monic_1p‘𝑈) = (Monic_1p‘𝑈)
27	19, 22, 26	mon1pcl 26079	. . . . . . . 8 ⊢ (𝑓 ∈ (Monic_1p‘𝑈) → 𝑓 ∈ (Base‘(Poly₁‘𝑈)))
28	27	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Monic_1p‘𝑈)) → 𝑓 ∈ (Base‘(Poly₁‘𝑈)))
29	25, 28	ffvelcdmd 7095	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (Monic_1p‘𝑈)) → (𝑂‘𝑓) ∈ (Base‘(𝑅 ↑_s 𝐵)))
30	14, 3, 15, 16, 18, 29	pwselbas 17470	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (Monic_1p‘𝑈)) → (𝑂‘𝑓):𝐵⟶𝐵)
31		ffn 6722	. . . . 5 ⊢ ((𝑂‘𝑓):𝐵⟶𝐵 → (𝑂‘𝑓) Fn 𝐵)
32		fniniseg 7069	. . . . 5 ⊢ ((𝑂‘𝑓) Fn 𝐵 → (𝑋 ∈ (^◡(𝑂‘𝑓) “ { 0 }) ↔ (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑓)‘𝑋) = 0 )))
33	30, 31, 32	3syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (Monic_1p‘𝑈)) → (𝑋 ∈ (^◡(𝑂‘𝑓) “ { 0 }) ↔ (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑓)‘𝑋) = 0 )))
34	33	rexbidva 3173	. . 3 ⊢ (𝜑 → (∃𝑓 ∈ (Monic_1p‘𝑈)𝑋 ∈ (^◡(𝑂‘𝑓) “ { 0 }) ↔ ∃𝑓 ∈ (Monic_1p‘𝑈)(𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑓)‘𝑋) = 0 )))
35	13, 34	bitrd 279	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝑅 IntgRing 𝑆) ↔ ∃𝑓 ∈ (Monic_1p‘𝑈)(𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑓)‘𝑋) = 0 )))
36		r19.42v 3187	. 2 ⊢ (∃𝑓 ∈ (Monic_1p‘𝑈)(𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑓)‘𝑋) = 0 ) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑓 ∈ (Monic_1p‘𝑈)((𝑂‘𝑓)‘𝑋) = 0 ))
37	35, 36	bitrdi 287	1 ⊢ (𝜑 → (𝑋 ∈ (𝑅 IntgRing 𝑆) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑓 ∈ (Monic_1p‘𝑈)((𝑂‘𝑓)‘𝑋) = 0 )))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃wrex 3067 Vcvv 3471 ⊆ wss 3947 {csn 4629 ∪ ciun 4996 ^◡ccnv 5677 “ cima 5681 Fn wfn 6543 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 Basecbs 17179 ↾_s cress 17208 0_gc0g 17420 ↑_s cpws 17427 Ringcrg 20172 CRingccrg 20173 RingHom crh 20407 SubRingcsubrg 20505 Poly₁cpl1 22095 evalSub₁ ces1 22231 Monic_1pcmn1 26060 IntgRing cirng 33357

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-ofr 7686 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-pm 8847 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9386 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-fzo 13660 df-seq 13999 df-hash 14322 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-hom 17256 df-cco 17257 df-0g 17422 df-gsum 17423 df-prds 17428 df-pws 17430 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18892 df-minusg 18893 df-sbg 18894 df-mulg 19023 df-subg 19077 df-ghm 19167 df-cntz 19267 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-srg 20126 df-ring 20174 df-cring 20175 df-rhm 20410 df-subrng 20482 df-subrg 20507 df-lmod 20744 df-lss 20815 df-lsp 20855 df-assa 21786 df-asp 21787 df-ascl 21788 df-psr 21841 df-mvr 21842 df-mpl 21843 df-opsr 21845 df-evls 22017 df-psr1 22098 df-ply1 22100 df-evls1 22233 df-mon1 26065 df-irng 33358

This theorem is referenced by: irngss 33361 irngssv 33362 0ringirng 33363 irngnzply1lem 33364 irngnzply1 33365 irredminply 33384

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