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Mirrors > Home > MPE Home > Th. List > rngisomring | Structured version Visualization version GIF version |
Description: If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then both rings are unital. (Contributed by AV, 27-Feb-2025.) |
Ref | Expression |
---|---|
rngisomring | ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝑆 ∈ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1135 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝑆 ∈ Rng) | |
2 | eqid 2727 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | eqid 2727 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
4 | 2, 3 | rngisomfv1 20386 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (𝐹‘(1r‘𝑅)) ∈ (Base‘𝑆)) |
5 | 4 | 3adant2 1129 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (𝐹‘(1r‘𝑅)) ∈ (Base‘𝑆)) |
6 | oveq1 7421 | . . . . . . 7 ⊢ (𝑖 = (𝐹‘(1r‘𝑅)) → (𝑖(.r‘𝑆)𝑥) = ((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥)) | |
7 | 6 | eqeq1d 2729 | . . . . . 6 ⊢ (𝑖 = (𝐹‘(1r‘𝑅)) → ((𝑖(.r‘𝑆)𝑥) = 𝑥 ↔ ((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥)) |
8 | oveq2 7422 | . . . . . . 7 ⊢ (𝑖 = (𝐹‘(1r‘𝑅)) → (𝑥(.r‘𝑆)𝑖) = (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅)))) | |
9 | 8 | eqeq1d 2729 | . . . . . 6 ⊢ (𝑖 = (𝐹‘(1r‘𝑅)) → ((𝑥(.r‘𝑆)𝑖) = 𝑥 ↔ (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥)) |
10 | 7, 9 | anbi12d 630 | . . . . 5 ⊢ (𝑖 = (𝐹‘(1r‘𝑅)) → (((𝑖(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)𝑖) = 𝑥) ↔ (((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥))) |
11 | 10 | ralbidv 3172 | . . . 4 ⊢ (𝑖 = (𝐹‘(1r‘𝑅)) → (∀𝑥 ∈ (Base‘𝑆)((𝑖(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)𝑖) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝑆)(((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥))) |
12 | 11 | adantl 481 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑖 = (𝐹‘(1r‘𝑅))) → (∀𝑥 ∈ (Base‘𝑆)((𝑖(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)𝑖) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝑆)(((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥))) |
13 | eqid 2727 | . . . 4 ⊢ (.r‘𝑆) = (.r‘𝑆) | |
14 | 2, 3, 13 | rngisom1 20387 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → ∀𝑥 ∈ (Base‘𝑆)(((𝐹‘(1r‘𝑅))(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)(𝐹‘(1r‘𝑅))) = 𝑥)) |
15 | 5, 12, 14 | rspcedvd 3609 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → ∃𝑖 ∈ (Base‘𝑆)∀𝑥 ∈ (Base‘𝑆)((𝑖(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)𝑖) = 𝑥)) |
16 | 3, 13 | isringrng 20205 | . 2 ⊢ (𝑆 ∈ Ring ↔ (𝑆 ∈ Rng ∧ ∃𝑖 ∈ (Base‘𝑆)∀𝑥 ∈ (Base‘𝑆)((𝑖(.r‘𝑆)𝑥) = 𝑥 ∧ (𝑥(.r‘𝑆)𝑖) = 𝑥))) |
17 | 1, 15, 16 | sylanbrc 582 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝑆 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 ‘cfv 6542 (class class class)co 7414 Basecbs 17165 .rcmulr 17219 Rngcrng 20076 1rcur 20105 Ringcrg 20157 RngIso crngim 20356 |
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 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
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-op 4631 df-uni 4904 df-iun 4993 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-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-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-plusg 17231 df-0g 17408 df-mgm 18585 df-mgmhm 18637 df-sgrp 18664 df-mnd 18680 df-grp 18878 df-minusg 18879 df-ghm 19152 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-ring 20159 df-rnghm 20357 df-rngim 20358 |
This theorem is referenced by: rngringbdlem2 21179 |
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