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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 | pm4.24 563 | . 2 ⊢ (𝐵 = {𝑍} ↔ (𝐵 = {𝑍} ∧ 𝐵 = {𝑍})) | |
2 | srgmnd 20208 | . . . . . . 7 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd) | |
3 | 2 | 3ad2ant1 1132 | . . . . . 6 ⊢ ((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → 𝑅 ∈ Mnd) |
4 | 3 | adantr 480 | . . . . 5 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → 𝑅 ∈ Mnd) |
5 | mndmgm 18767 | . . . . 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 484 | . . . 4 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ 𝐵) | |
8 | simpl2 1191 | . . . 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 18681 | . . . 4 ⊢ ((𝑅 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {〈〈𝑍, 𝑍〉, 𝑍〉})) |
12 | 6, 7, 8, 11 | syl3anc 1370 | . . 3 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ + = {〈〈𝑍, 𝑍〉, 𝑍〉})) |
13 | simpl1 1190 | . . . . . 6 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → 𝑅 ∈ SRing) | |
14 | eqid 2735 | . . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
15 | 14 | srgmgp 20209 | . . . . . 6 ⊢ (𝑅 ∈ SRing → (mulGrp‘𝑅) ∈ Mnd) |
16 | mndmgm 18767 | . . . . . 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 20156 | . . . . . . . . 9 ⊢ ∗ = (+g‘(mulGrp‘𝑅)) |
20 | 19 | fneq1i 6666 | . . . . . . . 8 ⊢ ( ∗ Fn (𝐵 × 𝐵) ↔ (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵)) |
21 | 20 | biimpi 216 | . . . . . . 7 ⊢ ( ∗ Fn (𝐵 × 𝐵) → (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵)) |
22 | 21 | 3ad2ant3 1134 | . . . . . 6 ⊢ ((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵)) |
23 | 22 | adantr 480 | . . . . 5 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵)) |
24 | 14, 9 | mgpbas 20158 | . . . . . 6 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
25 | eqid 2735 | . . . . . 6 ⊢ (+g‘(mulGrp‘𝑅)) = (+g‘(mulGrp‘𝑅)) | |
26 | 24, 25 | mgmb1mgm1 18681 | . . . . 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 1370 | . . . 4 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ (+g‘(mulGrp‘𝑅)) = {〈〈𝑍, 𝑍〉, 𝑍〉})) |
28 | 19 | eqcomi 2744 | . . . . . 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 2737 | . . . 4 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → ((+g‘(mulGrp‘𝑅)) = {〈〈𝑍, 𝑍〉, 𝑍〉} ↔ ∗ = {〈〈𝑍, 𝑍〉, 𝑍〉})) |
31 | 27, 30 | bitrd 279 | . . 3 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ∗ = {〈〈𝑍, 𝑍〉, 𝑍〉})) |
32 | 12, 31 | anbi12d 632 | . 2 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → ((𝐵 = {𝑍} ∧ 𝐵 = {𝑍}) ↔ ( + = {〈〈𝑍, 𝑍〉, 𝑍〉} ∧ ∗ = {〈〈𝑍, 𝑍〉, 𝑍〉}))) |
33 | 1, 32 | bitrid 283 | 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 {csn 4631 〈cop 4637 × cxp 5687 Fn wfn 6558 ‘cfv 6563 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 Mgmcmgm 18664 Mndcmnd 18760 mulGrpcmgp 20152 SRingcsrg 20204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-plusf 18665 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-cmn 19815 df-mgp 20153 df-srg 20205 |
This theorem is referenced by: srgen1zr 20234 ring1zr 20794 |
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