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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0ringirng | Structured version Visualization version GIF version |
Description: A zero ring 𝑅 has no integral elements. (Contributed by Thierry Arnoux, 5-Feb-2025.) |
Ref | Expression |
---|---|
irngval.o | ⊢ 𝑂 = (𝑅 evalSub1 𝑆) |
irngval.u | ⊢ 𝑈 = (𝑅 ↾s 𝑆) |
irngval.b | ⊢ 𝐵 = (Base‘𝑅) |
irngval.0 | ⊢ 0 = (0g‘𝑅) |
elirng.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
elirng.s | ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) |
0ringirng.1 | ⊢ (𝜑 → ¬ 𝑅 ∈ NzRing) |
Ref | Expression |
---|---|
0ringirng | ⊢ (𝜑 → (𝑅 IntgRing 𝑆) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rex0 4353 | . . . 4 ⊢ ¬ ∃𝑝 ∈ ∅ ((𝑂‘𝑝)‘𝑥) = 0 | |
2 | eqid 2727 | . . . . . 6 ⊢ (Monic1p‘𝑈) = (Monic1p‘𝑈) | |
3 | eqid 2727 | . . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
4 | elirng.s | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) | |
5 | irngval.u | . . . . . . . 8 ⊢ 𝑈 = (𝑅 ↾s 𝑆) | |
6 | 5 | subrgring 20502 | . . . . . . 7 ⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑈 ∈ Ring) |
7 | 4, 6 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Ring) |
8 | irngval.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
9 | elirng.r | . . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
10 | 9 | crngringd 20177 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring) |
11 | 8 | fveq2i 6894 | . . . . . . . 8 ⊢ (♯‘𝐵) = (♯‘(Base‘𝑅)) |
12 | 0ringirng.1 | . . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑅 ∈ NzRing) | |
13 | 0ringnnzr 20451 | . . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → ((♯‘(Base‘𝑅)) = 1 ↔ ¬ 𝑅 ∈ NzRing)) | |
14 | 13 | biimpar 477 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ ¬ 𝑅 ∈ NzRing) → (♯‘(Base‘𝑅)) = 1) |
15 | 10, 12, 14 | syl2anc 583 | . . . . . . . 8 ⊢ (𝜑 → (♯‘(Base‘𝑅)) = 1) |
16 | 11, 15 | eqtrid 2779 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐵) = 1) |
17 | 8 | subrgss 20500 | . . . . . . . . 9 ⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ 𝐵) |
18 | 5, 8 | ressbas2 17209 | . . . . . . . . 9 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝑈)) |
19 | 4, 17, 18 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 = (Base‘𝑈)) |
20 | 19, 4 | eqeltrrd 2829 | . . . . . . 7 ⊢ (𝜑 → (Base‘𝑈) ∈ (SubRing‘𝑅)) |
21 | 8, 10, 16, 20 | 0ringsubrg 32917 | . . . . . 6 ⊢ (𝜑 → (♯‘(Base‘𝑈)) = 1) |
22 | 2, 3, 7, 21 | 0ringmon1p 33168 | . . . . 5 ⊢ (𝜑 → (Monic1p‘𝑈) = ∅) |
23 | 22 | rexeqdv 3321 | . . . 4 ⊢ (𝜑 → (∃𝑝 ∈ (Monic1p‘𝑈)((𝑂‘𝑝)‘𝑥) = 0 ↔ ∃𝑝 ∈ ∅ ((𝑂‘𝑝)‘𝑥) = 0 )) |
24 | 1, 23 | mtbiri 327 | . . 3 ⊢ (𝜑 → ¬ ∃𝑝 ∈ (Monic1p‘𝑈)((𝑂‘𝑝)‘𝑥) = 0 ) |
25 | irngval.o | . . . . 5 ⊢ 𝑂 = (𝑅 evalSub1 𝑆) | |
26 | irngval.0 | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
27 | 25, 5, 8, 26, 9, 4 | elirng 33296 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑅 IntgRing 𝑆) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑝 ∈ (Monic1p‘𝑈)((𝑂‘𝑝)‘𝑥) = 0 ))) |
28 | 27 | simplbda 499 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑅 IntgRing 𝑆)) → ∃𝑝 ∈ (Monic1p‘𝑈)((𝑂‘𝑝)‘𝑥) = 0 ) |
29 | 24, 28 | mtand 815 | . 2 ⊢ (𝜑 → ¬ 𝑥 ∈ (𝑅 IntgRing 𝑆)) |
30 | 29 | eq0rdv 4400 | 1 ⊢ (𝜑 → (𝑅 IntgRing 𝑆) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1534 ∈ wcel 2099 ∃wrex 3065 ⊆ wss 3944 ∅c0 4318 ‘cfv 6542 (class class class)co 7414 1c1 11131 ♯chash 14313 Basecbs 17171 ↾s cress 17200 0gc0g 17412 Ringcrg 20164 CRingccrg 20165 NzRingcnzr 20440 SubRingcsubrg 20495 evalSub1 ces1 22219 Monic1pcmn1 26048 IntgRing cirng 33293 |
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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-addf 11209 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-map 8838 df-pm 8839 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-sup 9457 df-oi 9525 df-dju 9916 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-xnn0 12567 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-fzo 13652 df-seq 13991 df-hash 14314 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-starv 17239 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-hom 17248 df-cco 17249 df-0g 17414 df-gsum 17415 df-prds 17420 df-pws 17422 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-submnd 18732 df-grp 18884 df-minusg 18885 df-sbg 18886 df-mulg 19015 df-subg 19069 df-ghm 19159 df-cntz 19259 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-srg 20118 df-ring 20166 df-cring 20167 df-rhm 20400 df-nzr 20441 df-subrng 20472 df-subrg 20497 df-lmod 20734 df-lss 20805 df-lsp 20845 df-cnfld 21267 df-assa 21774 df-asp 21775 df-ascl 21776 df-psr 21829 df-mvr 21830 df-mpl 21831 df-opsr 21833 df-evls 22005 df-psr1 22086 df-ply1 22088 df-coe1 22089 df-evls1 22221 df-mdeg 25975 df-deg1 25976 df-mon1 26053 df-irng 33294 |
This theorem is referenced by: (None) |
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