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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 20187 | . . . . . . 7 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd) | |
| 3 | 2 | 3ad2ant1 1134 | . . . . . 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 18754 | . . . . 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 1193 | . . . 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 18668 | . . . 4 ⊢ ((𝑅 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {〈〈𝑍, 𝑍〉, 𝑍〉})) |
| 12 | 6, 7, 8, 11 | syl3anc 1373 | . . 3 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ + = {〈〈𝑍, 𝑍〉, 𝑍〉})) |
| 13 | simpl1 1192 | . . . . . 6 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → 𝑅 ∈ SRing) | |
| 14 | eqid 2737 | . . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
| 15 | 14 | srgmgp 20188 | . . . . . 6 ⊢ (𝑅 ∈ SRing → (mulGrp‘𝑅) ∈ Mnd) |
| 16 | mndmgm 18754 | . . . . . 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 20141 | . . . . . . . . 9 ⊢ ∗ = (+g‘(mulGrp‘𝑅)) |
| 20 | 19 | fneq1i 6665 | . . . . . . . 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 1136 | . . . . . 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 20142 | . . . . . 6 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
| 25 | eqid 2737 | . . . . . 6 ⊢ (+g‘(mulGrp‘𝑅)) = (+g‘(mulGrp‘𝑅)) | |
| 26 | 24, 25 | mgmb1mgm1 18668 | . . . . 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 1373 | . . . 4 ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ (+g‘(mulGrp‘𝑅)) = {〈〈𝑍, 𝑍〉, 𝑍〉})) |
| 28 | 19 | eqcomi 2746 | . . . . . 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 2739 | . . . 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 1087 = wceq 1540 ∈ wcel 2108 {csn 4626 〈cop 4632 × cxp 5683 Fn wfn 6556 ‘cfv 6561 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 Mgmcmgm 18651 Mndcmnd 18747 mulGrpcmgp 20137 SRingcsrg 20183 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-plusf 18652 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-cmn 19800 df-mgp 20138 df-srg 20184 |
| This theorem is referenced by: srgen1zr 20213 ring1zr 20777 |
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