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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rloc0g | Structured version Visualization version GIF version | ||
| Description: The zero of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.) | 
| Ref | Expression | 
|---|---|
| rloc0g.1 | ⊢ 0 = (0g‘𝑅) | 
| rloc0g.2 | ⊢ 1 = (1r‘𝑅) | 
| rloc0g.3 | ⊢ 𝐿 = (𝑅 RLocal 𝑆) | 
| rloc0g.4 | ⊢ ∼ = (𝑅 ~RL 𝑆) | 
| rloc0g.5 | ⊢ (𝜑 → 𝑅 ∈ CRing) | 
| rloc0g.6 | ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) | 
| rloc0g.o | ⊢ 𝑂 = [〈 0 , 1 〉] ∼ | 
| Ref | Expression | 
|---|---|
| rloc0g | ⊢ (𝜑 → 𝑂 = (0g‘𝐿)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rloc0g.o | . 2 ⊢ 𝑂 = [〈 0 , 1 〉] ∼ | |
| 2 | eqid 2737 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 3 | eqid 2737 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
| 4 | eqid 2737 | . . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
| 5 | rloc0g.3 | . . . . 5 ⊢ 𝐿 = (𝑅 RLocal 𝑆) | |
| 6 | rloc0g.4 | . . . . 5 ⊢ ∼ = (𝑅 ~RL 𝑆) | |
| 7 | rloc0g.5 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
| 8 | rloc0g.6 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) | |
| 9 | 2, 3, 4, 5, 6, 7, 8 | rloccring 33274 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ CRing) | 
| 10 | 9 | crnggrpd 20244 | . . 3 ⊢ (𝜑 → 𝐿 ∈ Grp) | 
| 11 | 7 | crnggrpd 20244 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Grp) | 
| 12 | rloc0g.1 | . . . . . . . 8 ⊢ 0 = (0g‘𝑅) | |
| 13 | 2, 12 | grpidcl 18983 | . . . . . . 7 ⊢ (𝑅 ∈ Grp → 0 ∈ (Base‘𝑅)) | 
| 14 | 11, 13 | syl 17 | . . . . . 6 ⊢ (𝜑 → 0 ∈ (Base‘𝑅)) | 
| 15 | eqid 2737 | . . . . . . . . 9 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
| 16 | rloc0g.2 | . . . . . . . . 9 ⊢ 1 = (1r‘𝑅) | |
| 17 | 15, 16 | ringidval 20180 | . . . . . . . 8 ⊢ 1 = (0g‘(mulGrp‘𝑅)) | 
| 18 | 17 | subm0cl 18824 | . . . . . . 7 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 1 ∈ 𝑆) | 
| 19 | 8, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 1 ∈ 𝑆) | 
| 20 | 14, 19 | opelxpd 5724 | . . . . 5 ⊢ (𝜑 → 〈 0 , 1 〉 ∈ ((Base‘𝑅) × 𝑆)) | 
| 21 | 6 | ovexi 7465 | . . . . . 6 ⊢ ∼ ∈ V | 
| 22 | 21 | ecelqsi 8813 | . . . . 5 ⊢ (〈 0 , 1 〉 ∈ ((Base‘𝑅) × 𝑆) → [〈 0 , 1 〉] ∼ ∈ (((Base‘𝑅) × 𝑆) / ∼ )) | 
| 23 | 20, 22 | syl 17 | . . . 4 ⊢ (𝜑 → [〈 0 , 1 〉] ∼ ∈ (((Base‘𝑅) × 𝑆) / ∼ )) | 
| 24 | eqid 2737 | . . . . 5 ⊢ (-g‘𝑅) = (-g‘𝑅) | |
| 25 | eqid 2737 | . . . . 5 ⊢ ((Base‘𝑅) × 𝑆) = ((Base‘𝑅) × 𝑆) | |
| 26 | 15, 2 | mgpbas 20142 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅)) | 
| 27 | 26 | submss 18822 | . . . . . 6 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 𝑆 ⊆ (Base‘𝑅)) | 
| 28 | 8, 27 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑅)) | 
| 29 | 2, 12, 3, 24, 25, 5, 6, 7, 28 | rlocbas 33271 | . . . 4 ⊢ (𝜑 → (((Base‘𝑅) × 𝑆) / ∼ ) = (Base‘𝐿)) | 
| 30 | 23, 29 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → [〈 0 , 1 〉] ∼ ∈ (Base‘𝐿)) | 
| 31 | eqid 2737 | . . . . 5 ⊢ (+g‘𝐿) = (+g‘𝐿) | |
| 32 | 2, 3, 4, 5, 6, 7, 8, 14, 14, 19, 19, 31 | rlocaddval 33272 | . . . 4 ⊢ (𝜑 → ([〈 0 , 1 〉] ∼ (+g‘𝐿)[〈 0 , 1 〉] ∼ ) = [〈(( 0 (.r‘𝑅) 1 )(+g‘𝑅)( 0 (.r‘𝑅) 1 )), ( 1 (.r‘𝑅) 1 )〉] ∼ ) | 
| 33 | 7 | crngringd 20243 | . . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring) | 
| 34 | 2, 3, 16, 33, 14 | ringridmd 20270 | . . . . . . . 8 ⊢ (𝜑 → ( 0 (.r‘𝑅) 1 ) = 0 ) | 
| 35 | 34, 34 | oveq12d 7449 | . . . . . . 7 ⊢ (𝜑 → (( 0 (.r‘𝑅) 1 )(+g‘𝑅)( 0 (.r‘𝑅) 1 )) = ( 0 (+g‘𝑅) 0 )) | 
| 36 | 2, 4, 12, 11, 14 | grplidd 18987 | . . . . . . 7 ⊢ (𝜑 → ( 0 (+g‘𝑅) 0 ) = 0 ) | 
| 37 | 35, 36 | eqtrd 2777 | . . . . . 6 ⊢ (𝜑 → (( 0 (.r‘𝑅) 1 )(+g‘𝑅)( 0 (.r‘𝑅) 1 )) = 0 ) | 
| 38 | 28, 19 | sseldd 3984 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ (Base‘𝑅)) | 
| 39 | 2, 3, 16, 33, 38 | ringlidmd 20269 | . . . . . 6 ⊢ (𝜑 → ( 1 (.r‘𝑅) 1 ) = 1 ) | 
| 40 | 37, 39 | opeq12d 4881 | . . . . 5 ⊢ (𝜑 → 〈(( 0 (.r‘𝑅) 1 )(+g‘𝑅)( 0 (.r‘𝑅) 1 )), ( 1 (.r‘𝑅) 1 )〉 = 〈 0 , 1 〉) | 
| 41 | 40 | eceq1d 8785 | . . . 4 ⊢ (𝜑 → [〈(( 0 (.r‘𝑅) 1 )(+g‘𝑅)( 0 (.r‘𝑅) 1 )), ( 1 (.r‘𝑅) 1 )〉] ∼ = [〈 0 , 1 〉] ∼ ) | 
| 42 | 32, 41 | eqtrd 2777 | . . 3 ⊢ (𝜑 → ([〈 0 , 1 〉] ∼ (+g‘𝐿)[〈 0 , 1 〉] ∼ ) = [〈 0 , 1 〉] ∼ ) | 
| 43 | eqid 2737 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
| 44 | eqid 2737 | . . . . 5 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
| 45 | 43, 31, 44 | isgrpid2 18994 | . . . 4 ⊢ (𝐿 ∈ Grp → (([〈 0 , 1 〉] ∼ ∈ (Base‘𝐿) ∧ ([〈 0 , 1 〉] ∼ (+g‘𝐿)[〈 0 , 1 〉] ∼ ) = [〈 0 , 1 〉] ∼ ) ↔ (0g‘𝐿) = [〈 0 , 1 〉] ∼ )) | 
| 46 | 45 | biimpa 476 | . . 3 ⊢ ((𝐿 ∈ Grp ∧ ([〈 0 , 1 〉] ∼ ∈ (Base‘𝐿) ∧ ([〈 0 , 1 〉] ∼ (+g‘𝐿)[〈 0 , 1 〉] ∼ ) = [〈 0 , 1 〉] ∼ )) → (0g‘𝐿) = [〈 0 , 1 〉] ∼ ) | 
| 47 | 10, 30, 42, 46 | syl12anc 837 | . 2 ⊢ (𝜑 → (0g‘𝐿) = [〈 0 , 1 〉] ∼ ) | 
| 48 | 1, 47 | eqtr4id 2796 | 1 ⊢ (𝜑 → 𝑂 = (0g‘𝐿)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 〈cop 4632 × cxp 5683 ‘cfv 6561 (class class class)co 7431 [cec 8743 / cqs 8744 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 0gc0g 17484 SubMndcsubmnd 18795 Grpcgrp 18951 -gcsg 18953 mulGrpcmgp 20137 1rcur 20178 CRingccrg 20231 ~RL cerl 33257 RLocal crloc 33258 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-ec 8747 df-qs 8751 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-0g 17486 df-imas 17553 df-qus 17554 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-erl 33259 df-rloc 33260 | 
| This theorem is referenced by: fracfld 33310 | 
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