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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 5433 | . . . 4 ⊢ {𝐼} ∈ V | |
2 | mnd1.m | . . . . 5 ⊢ 𝑀 = {〈(Base‘ndx), {𝐼}〉, 〈(+g‘ndx), {〈〈𝐼, 𝐼〉, 𝐼〉}〉} | |
3 | 2 | grpbase 17267 | . . . 4 ⊢ ({𝐼} ∈ V → {𝐼} = (Base‘𝑀)) |
4 | 1, 3 | ax-mp 5 | . . 3 ⊢ {𝐼} = (Base‘𝑀) |
5 | eqid 2728 | . . 3 ⊢ (0g‘𝑀) = (0g‘𝑀) | |
6 | snex 5433 | . . . 4 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V | |
7 | 2 | grpplusg 17269 | . . . 4 ⊢ ({〈〈𝐼, 𝐼〉, 𝐼〉} ∈ V → {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀)) |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ {〈〈𝐼, 𝐼〉, 𝐼〉} = (+g‘𝑀) |
9 | snidg 4663 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ {𝐼}) | |
10 | velsn 4645 | . . . . 5 ⊢ (𝑎 ∈ {𝐼} ↔ 𝑎 = 𝐼) | |
11 | df-ov 7423 | . . . . . . 7 ⊢ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) | |
12 | opex 5466 | . . . . . . . 8 ⊢ 〈𝐼, 𝐼〉 ∈ V | |
13 | fvsng 7189 | . . . . . . . 8 ⊢ ((〈𝐼, 𝐼〉 ∈ V ∧ 𝐼 ∈ 𝑉) → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) | |
14 | 12, 13 | mpan 689 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → ({〈〈𝐼, 𝐼〉, 𝐼〉}‘〈𝐼, 𝐼〉) = 𝐼) |
15 | 11, 14 | eqtrid 2780 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼) |
16 | oveq2 7428 | . . . . . . 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 2744 | . . . . . 6 ⊢ (𝑎 = 𝐼 → ((𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑎) = 𝑎 ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼)) |
19 | 15, 18 | syl5ibrcom 246 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑎 = 𝐼 → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑎) = 𝑎)) |
20 | 10, 19 | biimtrid 241 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑎 ∈ {𝐼} → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑎) = 𝑎)) |
21 | 20 | imp 406 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑎 ∈ {𝐼}) → (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝑎) = 𝑎) |
22 | oveq1 7427 | . . . . . . 7 ⊢ (𝑎 = 𝐼 → (𝑎{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼)) | |
23 | 22, 17 | eqeq12d 2744 | . . . . . 6 ⊢ (𝑎 = 𝐼 → ((𝑎{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑎 ↔ (𝐼{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝐼)) |
24 | 15, 23 | syl5ibrcom 246 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑎 = 𝐼 → (𝑎{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑎)) |
25 | 10, 24 | biimtrid 241 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑎 ∈ {𝐼} → (𝑎{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑎)) |
26 | 25 | imp 406 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑎 ∈ {𝐼}) → (𝑎{〈〈𝐼, 𝐼〉, 𝐼〉}𝐼) = 𝑎) |
27 | 4, 5, 8, 9, 21, 26 | ismgmid2 18628 | . 2 ⊢ (𝐼 ∈ 𝑉 → 𝐼 = (0g‘𝑀)) |
28 | 27 | eqcomd 2734 | 1 ⊢ (𝐼 ∈ 𝑉 → (0g‘𝑀) = 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3471 {csn 4629 {cpr 4631 〈cop 4635 ‘cfv 6548 (class class class)co 7420 ndxcnx 17162 Basecbs 17180 +gcplusg 17233 0gc0g 17421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-struct 17116 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-0g 17423 |
This theorem is referenced by: grp1 19003 |
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