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Mirrors > Home > MPE Home > Th. List > srg1zr | Structured version Visualization version GIF version |
Description: The only semiring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) |
Ref | Expression |
---|---|
srg1zr.b | ⊢ 𝐵 = (Base‘𝑅) |
srg1zr.p | ⊢ + = (+g‘𝑅) |
srg1zr.t | ⊢ ∗ = (.r‘𝑅) |
Ref | Expression |
---|---|
srg1zr | ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {〈〈𝑍, 𝑍〉, 𝑍〉} ∧ ∗ = {〈〈𝑍, 𝑍〉, 𝑍〉}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm4.24 564 | . 2 ⊢ (𝐵 = {𝑍} ↔ (𝐵 = {𝑍} ∧ 𝐵 = {𝑍})) | |
2 | srgmnd 19188 | . . . . . . 7 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd) | |
3 | 2 | 3ad2ant1 1125 | . . . . . 6 ⊢ ((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → 𝑅 ∈ Mnd) |
4 | 3 | adantr 481 | . . . . 5 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → 𝑅 ∈ Mnd) |
5 | mndmgm 17906 | . . . . 5 ⊢ (𝑅 ∈ Mnd → 𝑅 ∈ Mgm) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → 𝑅 ∈ Mgm) |
7 | simpr 485 | . . . 4 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ 𝐵) | |
8 | simpl2 1184 | . . . 4 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → + Fn (𝐵 × 𝐵)) | |
9 | srg1zr.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
10 | srg1zr.p | . . . . 5 ⊢ + = (+g‘𝑅) | |
11 | 9, 10 | mgmb1mgm1 17853 | . . . 4 ⊢ ((𝑅 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {〈〈𝑍, 𝑍〉, 𝑍〉})) |
12 | 6, 7, 8, 11 | syl3anc 1363 | . . 3 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ + = {〈〈𝑍, 𝑍〉, 𝑍〉})) |
13 | simpl1 1183 | . . . . . 6 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → 𝑅 ∈ SRing) | |
14 | eqid 2818 | . . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
15 | 14 | srgmgp 19189 | . . . . . 6 ⊢ (𝑅 ∈ SRing → (mulGrp‘𝑅) ∈ Mnd) |
16 | mndmgm 17906 | . . . . . 6 ⊢ ((mulGrp‘𝑅) ∈ Mnd → (mulGrp‘𝑅) ∈ Mgm) | |
17 | 13, 15, 16 | 3syl 18 | . . . . 5 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (mulGrp‘𝑅) ∈ Mgm) |
18 | srg1zr.t | . . . . . . . . . 10 ⊢ ∗ = (.r‘𝑅) | |
19 | 14, 18 | mgpplusg 19172 | . . . . . . . . 9 ⊢ ∗ = (+g‘(mulGrp‘𝑅)) |
20 | 19 | fneq1i 6443 | . . . . . . . 8 ⊢ ( ∗ Fn (𝐵 × 𝐵) ↔ (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵)) |
21 | 20 | biimpi 217 | . . . . . . 7 ⊢ ( ∗ Fn (𝐵 × 𝐵) → (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵)) |
22 | 21 | 3ad2ant3 1127 | . . . . . 6 ⊢ ((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵)) |
23 | 22 | adantr 481 | . . . . 5 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵)) |
24 | 14, 9 | mgpbas 19174 | . . . . . 6 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
25 | eqid 2818 | . . . . . 6 ⊢ (+g‘(mulGrp‘𝑅)) = (+g‘(mulGrp‘𝑅)) | |
26 | 24, 25 | mgmb1mgm1 17853 | . . . . 5 ⊢ (((mulGrp‘𝑅) ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ (+g‘(mulGrp‘𝑅)) = {〈〈𝑍, 𝑍〉, 𝑍〉})) |
27 | 17, 7, 23, 26 | syl3anc 1363 | . . . 4 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ (+g‘(mulGrp‘𝑅)) = {〈〈𝑍, 𝑍〉, 𝑍〉})) |
28 | 19 | eqcomi 2827 | . . . . . 6 ⊢ (+g‘(mulGrp‘𝑅)) = ∗ |
29 | 28 | a1i 11 | . . . . 5 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (+g‘(mulGrp‘𝑅)) = ∗ ) |
30 | 29 | eqeq1d 2820 | . . . 4 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → ((+g‘(mulGrp‘𝑅)) = {〈〈𝑍, 𝑍〉, 𝑍〉} ↔ ∗ = {〈〈𝑍, 𝑍〉, 𝑍〉})) |
31 | 27, 30 | bitrd 280 | . . 3 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ∗ = {〈〈𝑍, 𝑍〉, 𝑍〉})) |
32 | 12, 31 | anbi12d 630 | . 2 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → ((𝐵 = {𝑍} ∧ 𝐵 = {𝑍}) ↔ ( + = {〈〈𝑍, 𝑍〉, 𝑍〉} ∧ ∗ = {〈〈𝑍, 𝑍〉, 𝑍〉}))) |
33 | 1, 32 | syl5bb 284 | 1 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {〈〈𝑍, 𝑍〉, 𝑍〉} ∧ ∗ = {〈〈𝑍, 𝑍〉, 𝑍〉}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 {csn 4557 〈cop 4563 × cxp 5546 Fn wfn 6343 ‘cfv 6348 Basecbs 16471 +gcplusg 16553 .rcmulr 16554 Mgmcmgm 17838 Mndcmnd 17899 mulGrpcmgp 19168 SRingcsrg 19184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-plusf 17839 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-cmn 18837 df-mgp 19169 df-srg 19185 |
This theorem is referenced by: srgen1zr 19209 ring1zr 19976 |
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