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Mirrors > Home > MPE Home > Th. List > rngen1zr | Structured version Visualization version GIF version |
Description: The only (unital) ring with one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 25-Jan-2020.) |
Ref | Expression |
---|---|
ring1zr.b | ⊢ 𝐵 = (Base‘𝑅) |
ring1zr.p | ⊢ + = (+g‘𝑅) |
ring1zr.t | ⊢ ∗ = (.r‘𝑅) |
Ref | Expression |
---|---|
rngen1zr | ⊢ (((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 ≈ 1o ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en1eqsnbi 9309 | . . 3 ⊢ (𝑍 ∈ 𝐵 → (𝐵 ≈ 1o ↔ 𝐵 = {𝑍})) | |
2 | 1 | adantl 480 | . 2 ⊢ (((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 ≈ 1o ↔ 𝐵 = {𝑍})) |
3 | ring1zr.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
4 | ring1zr.p | . . 3 ⊢ + = (+g‘𝑅) | |
5 | ring1zr.t | . . 3 ⊢ ∗ = (.r‘𝑅) | |
6 | 3, 4, 5 | ring1zr 20678 | . 2 ⊢ (((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))) |
7 | 2, 6 | bitrd 278 | 1 ⊢ (((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 ≈ 1o ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {csn 4632 ⟨cop 4638 class class class wbr 5152 × cxp 5680 Fn wfn 6548 ‘cfv 6553 1oc1o 8488 ≈ cen 8969 Basecbs 17189 +gcplusg 17242 .rcmulr 17243 Ringcrg 20187 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-plusg 17255 df-0g 17432 df-plusf 18608 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-grp 18907 df-minusg 18908 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-srg 20141 df-ring 20189 |
This theorem is referenced by: ringen1zr 20680 |
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