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Mirrors > Home > MPE Home > Th. List > mnd1id | Structured version Visualization version GIF version |
Description: The singleton element of a trivial monoid is its identity element. (Contributed by AV, 23-Jan-2020.) |
Ref | Expression |
---|---|
mnd1.m | ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} |
Ref | Expression |
---|---|
mnd1id | ⊢ (𝐼 ∈ 𝑉 → (0g‘𝑀) = 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5322 | . . . 4 ⊢ {𝐼} ∈ V | |
2 | mnd1.m | . . . . 5 ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} | |
3 | 2 | grpbase 16598 | . . . 4 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀)) |
4 | 1, 3 | ax-mp 5 | . . 3 ⊢ {𝐼} = (Base‘𝑀) |
5 | eqid 2818 | . . 3 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
6 | snex 5322 | . . . 4 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V | |
7 | 2 | grpplusg 16599 | . . . 4 ⊢ ({〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V → {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀)) |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀) |
9 | snidg 4589 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ {𝐼}) | |
10 | velsn 4573 | . . . . 5 ⊢ (𝑎 ∈ {𝐼} ↔ 𝑎 = 𝐼) | |
11 | df-ov 7148 | . . . . . . 7 ⊢ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) | |
12 | opex 5347 | . . . . . . . 8 ⊢ 〈𝐼, 𝐼〉 ∈ V | |
13 | fvsng 6934 | . . . . . . . 8 ⊢ ((〈𝐼, 𝐼〉 ∈ V ∧ 𝐼 ∈ 𝑉) → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) | |
14 | 12, 13 | mpan 686 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) |
15 | 11, 14 | syl5eq 2865 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼) |
16 | oveq2 7153 | . . . . . . 7 ⊢ (𝑎 = 𝐼 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑎) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
17 | id 22 | . . . . . . 7 ⊢ (𝑎 = 𝐼 → 𝑎 = 𝐼) | |
18 | 16, 17 | eqeq12d 2834 | . . . . . 6 ⊢ (𝑎 = 𝐼 → ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑎) = 𝑎 ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼)) |
19 | 15, 18 | syl5ibrcom 248 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑎 = 𝐼 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑎) = 𝑎)) |
20 | 10, 19 | syl5bi 243 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑎 ∈ {𝐼} → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑎) = 𝑎)) |
21 | 20 | imp 407 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑎 ∈ {𝐼}) → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑎) = 𝑎) |
22 | oveq1 7152 | . . . . . . 7 ⊢ (𝑎 = 𝐼 → (𝑎{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
23 | 22, 17 | eqeq12d 2834 | . . . . . 6 ⊢ (𝑎 = 𝐼 → ((𝑎{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑎 ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼)) |
24 | 15, 23 | syl5ibrcom 248 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑎 = 𝐼 → (𝑎{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑎)) |
25 | 10, 24 | syl5bi 243 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑎 ∈ {𝐼} → (𝑎{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑎)) |
26 | 25 | imp 407 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑎 ∈ {𝐼}) → (𝑎{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑎) |
27 | 4, 5, 8, 9, 21, 26 | ismgmid2 17866 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝐼 = (0g‘𝑀)) |
28 | 27 | eqcomd 2824 | 1 ⊢ (𝐼 ∈ 𝑉 → (0g‘𝑀) = 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 {cpr 4559 〈cop 4563 ‘cfv 6348 (class class class)co 7145 ndxcnx 16468 Basecbs 16471 +gcplusg 16553 0gc0g 16701 |
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-rmo 3143 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-int 4868 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-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 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-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-plusg 16566 df-0g 16703 |
This theorem is referenced by: grp1 18144 |
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