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Mirrors > Home > ILE Home > Th. List > srgen1zr | GIF version |
Description: The only semiring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 25-Jan-2020.) |
Ref | Expression |
---|---|
srg1zr.b | ⊢ 𝐵 = (Base‘𝑅) |
srg1zr.p | ⊢ + = (+g‘𝑅) |
srg1zr.t | ⊢ ∗ = (.r‘𝑅) |
srgen1zr.p | ⊢ 𝑍 = (0g‘𝑅) |
Ref | Expression |
---|---|
srgen1zr | ⊢ ((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1o ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srg1zr.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
2 | srgen1zr.p | . . . 4 ⊢ 𝑍 = (0g‘𝑅) | |
3 | 1, 2 | srg0cl 13165 | . . 3 ⊢ (𝑅 ∈ SRing → 𝑍 ∈ 𝐵) |
4 | 3 | 3ad2ant1 1018 | . 2 ⊢ ((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → 𝑍 ∈ 𝐵) |
5 | en1eqsnbi 6950 | . . . 4 ⊢ (𝑍 ∈ 𝐵 → (𝐵 ≈ 1o ↔ 𝐵 = {𝑍})) | |
6 | 5 | adantl 277 | . . 3 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 ≈ 1o ↔ 𝐵 = {𝑍})) |
7 | srg1zr.p | . . . 4 ⊢ + = (+g‘𝑅) | |
8 | srg1zr.t | . . . 4 ⊢ ∗ = (.r‘𝑅) | |
9 | 1, 7, 8 | srg1zr 13175 | . . 3 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))) |
10 | 6, 9 | bitrd 188 | . 2 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 ≈ 1o ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))) |
11 | 4, 10 | mpdan 421 | 1 ⊢ ((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1o ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 {csn 3594 ⟨cop 3597 class class class wbr 4005 × cxp 4626 Fn wfn 5213 ‘cfv 5218 1oc1o 6412 ≈ cen 6740 Basecbs 12464 +gcplusg 12538 .rcmulr 12539 0gc0g 12710 SRingcsrg 13151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-addcom 7913 ax-addass 7915 ax-i2m1 7918 ax-0lt1 7919 ax-0id 7921 ax-rnegex 7922 ax-pre-ltirr 7925 ax-pre-ltadd 7929 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-iord 4368 df-on 4370 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-1o 6419 df-er 6537 df-en 6743 df-fin 6745 df-pnf 7996 df-mnf 7997 df-ltxr 7999 df-inn 8922 df-2 8980 df-3 8981 df-ndx 12467 df-slot 12468 df-base 12470 df-sets 12471 df-plusg 12551 df-mulr 12552 df-0g 12712 df-plusf 12779 df-mgm 12780 df-sgrp 12813 df-mnd 12823 df-cmn 13095 df-mgp 13136 df-srg 13152 |
This theorem is referenced by: (None) |
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