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Mirrors > Home > MPE Home > Th. List > mnd1 | Structured version Visualization version GIF version |
Description: The (smallest) structure representing a trivial monoid consists of one element. (Contributed by AV, 28-Apr-2019.) (Proof shortened by AV, 11-Feb-2020.) |
Ref | Expression |
---|---|
mnd1.m | ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} |
Ref | Expression |
---|---|
mnd1 | ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnd1.m | . . 3 ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} | |
2 | 1 | sgrp1 18755 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Smgrp) |
3 | df-ov 7434 | . . . . 5 ⊢ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) | |
4 | opex 5475 | . . . . . 6 ⊢ 〈𝐼, 𝐼〉 ∈ V | |
5 | fvsng 7200 | . . . . . 6 ⊢ ((〈𝐼, 𝐼〉 ∈ V ∧ 𝐼 ∈ 𝑉) → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) | |
6 | 4, 5 | mpan 690 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) |
7 | 3, 6 | eqtrid 2787 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼) |
8 | oveq2 7439 | . . . . . . 7 ⊢ (𝑦 = 𝐼 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
9 | id 22 | . . . . . . 7 ⊢ (𝑦 = 𝐼 → 𝑦 = 𝐼) | |
10 | 8, 9 | eqeq12d 2751 | . . . . . 6 ⊢ (𝑦 = 𝐼 → ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼)) |
11 | oveq1 7438 | . . . . . . 7 ⊢ (𝑦 = 𝐼 → (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
12 | 11, 9 | eqeq12d 2751 | . . . . . 6 ⊢ (𝑦 = 𝐼 → ((𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑦 ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼)) |
13 | 10, 12 | anbi12d 632 | . . . . 5 ⊢ (𝑦 = 𝐼 → (((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑦) ↔ ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼 ∧ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼))) |
14 | 13 | ralsng 4680 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (∀𝑦 ∈ {𝐼} ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑦) ↔ ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼 ∧ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼))) |
15 | 7, 7, 14 | mpbir2and 713 | . . 3 ⊢ (𝐼 ∈ 𝑉 → ∀𝑦 ∈ {𝐼} ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑦)) |
16 | oveq1 7438 | . . . . . 6 ⊢ (𝑥 = 𝐼 → (𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦)) | |
17 | 16 | eqeq1d 2737 | . . . . 5 ⊢ (𝑥 = 𝐼 → ((𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦)) |
18 | 17 | ovanraleqv 7455 | . . . 4 ⊢ (𝑥 = 𝐼 → (∀𝑦 ∈ {𝐼} ((𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝑥) = 𝑦) ↔ ∀𝑦 ∈ {𝐼} ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑦))) |
19 | 18 | rexsng 4681 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (∃𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} ((𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝑥) = 𝑦) ↔ ∀𝑦 ∈ {𝐼} ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑦))) |
20 | 15, 19 | mpbird 257 | . 2 ⊢ (𝐼 ∈ 𝑉 → ∃𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} ((𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝑥) = 𝑦)) |
21 | snex 5442 | . . . 4 ⊢ {𝐼} ∈ V | |
22 | 1 | grpbase 17332 | . . . 4 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀)) |
23 | 21, 22 | ax-mp 5 | . . 3 ⊢ {𝐼} = (Base‘𝑀) |
24 | snex 5442 | . . . 4 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V | |
25 | 1 | grpplusg 17334 | . . . 4 ⊢ ({〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V → {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀)) |
26 | 24, 25 | ax-mp 5 | . . 3 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀) |
27 | 23, 26 | ismnddef 18762 | . 2 ⊢ (𝑀 ∈ Mnd ↔ (𝑀 ∈ Smgrp ∧ ∃𝑥 ∈ {𝐼}∀𝑦 ∈ {𝐼} ((𝑥{〈〈𝐼, 𝐼〉, 𝐼〉}𝑦) = 𝑦 ∧ (𝑦{〈〈𝐼, 𝐼〉, 𝐼〉}𝑥) = 𝑦))) |
28 | 2, 20, 27 | sylanbrc 583 | 1 ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 Vcvv 3478 {csn 4631 {cpr 4633 〈cop 4637 ‘cfv 6563 (class class class)co 7431 ndxcnx 17227 Basecbs 17245 +gcplusg 17298 Smgrpcsgrp 18744 Mndcmnd 18760 |
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-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 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-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mgm 18666 df-sgrp 18745 df-mnd 18761 |
This theorem is referenced by: grp1 19078 ring1 20324 |
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