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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 4366 | . . . 4 ⊢ ¬ ∃𝑝 ∈ ∅ ((𝑂‘𝑝)‘𝑥) = 0 | |
2 | eqid 2735 | . . . . . 6 ⊢ (Monic1p‘𝑈) = (Monic1p‘𝑈) | |
3 | eqid 2735 | . . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
4 | elirng.s | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) | |
5 | irngval.u | . . . . . . . 8 ⊢ 𝑈 = (𝑅 ↾s 𝑆) | |
6 | 5 | subrgring 20591 | . . . . . . 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 20264 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring) |
11 | 8 | fveq2i 6910 | . . . . . . . 8 ⊢ (♯‘𝐵) = (♯‘(Base‘𝑅)) |
12 | 0ringirng.1 | . . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑅 ∈ NzRing) | |
13 | 0ringnnzr 20542 | . . . . . . . . . 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 584 | . . . . . . . 8 ⊢ (𝜑 → (♯‘(Base‘𝑅)) = 1) |
16 | 11, 15 | eqtrid 2787 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐵) = 1) |
17 | 8 | subrgss 20589 | . . . . . . . . 9 ⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ 𝐵) |
18 | 5, 8 | ressbas2 17283 | . . . . . . . . 9 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝑈)) |
19 | 4, 17, 18 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 = (Base‘𝑈)) |
20 | 19, 4 | eqeltrrd 2840 | . . . . . . 7 ⊢ (𝜑 → (Base‘𝑈) ∈ (SubRing‘𝑅)) |
21 | 8, 10, 16, 20 | 0ringsubrg 33238 | . . . . . 6 ⊢ (𝜑 → (♯‘(Base‘𝑈)) = 1) |
22 | 2, 3, 7, 21 | 0ringmon1p 33563 | . . . . 5 ⊢ (𝜑 → (Monic1p‘𝑈) = ∅) |
23 | 22 | rexeqdv 3325 | . . . 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 33701 | . . . 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 816 | . 2 ⊢ (𝜑 → ¬ 𝑥 ∈ (𝑅 IntgRing 𝑆)) |
30 | 29 | eq0rdv 4413 | 1 ⊢ (𝜑 → (𝑅 IntgRing 𝑆) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ⊆ wss 3963 ∅c0 4339 ‘cfv 6563 (class class class)co 7431 1c1 11154 ♯chash 14366 Basecbs 17245 ↾s cress 17274 0gc0g 17486 Ringcrg 20251 CRingccrg 20252 NzRingcnzr 20529 SubRingcsubrg 20586 evalSub1 ces1 22333 Monic1pcmn1 26180 IntgRing cirng 33698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-srg 20205 df-ring 20253 df-cring 20254 df-rhm 20489 df-nzr 20530 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lsp 20988 df-cnfld 21383 df-assa 21891 df-asp 21892 df-ascl 21893 df-psr 21947 df-mvr 21948 df-mpl 21949 df-opsr 21951 df-evls 22116 df-psr1 22197 df-ply1 22199 df-coe1 22200 df-evls1 22335 df-mdeg 26109 df-deg1 26110 df-mon1 26185 df-irng 33699 |
This theorem is referenced by: (None) |
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