Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > signsw0g | Structured version Visualization version GIF version |
Description: The neutral element of 𝑊. (Contributed by Thierry Arnoux, 9-Sep-2018.) |
Ref | Expression |
---|---|
signsw.p | ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏)) |
signsw.w | ⊢ 𝑊 = {〈(Base‘ndx), {-1, 0, 1}〉, 〈(+g‘ndx), ⨣ 〉} |
Ref | Expression |
---|---|
signsw0g | ⊢ 0 = (0g‘𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 10970 | . . . . 5 ⊢ 0 ∈ V | |
2 | 1 | tpid2 4712 | . . . 4 ⊢ 0 ∈ {-1, 0, 1} |
3 | signsw.p | . . . . 5 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏)) | |
4 | 3 | signsw0glem 32528 | . . . 4 ⊢ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢) |
5 | 2, 4 | pm3.2i 471 | . . 3 ⊢ (0 ∈ {-1, 0, 1} ∧ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)) |
6 | signsw.w | . . . . . 6 ⊢ 𝑊 = {〈(Base‘ndx), {-1, 0, 1}〉, 〈(+g‘ndx), ⨣ 〉} | |
7 | 3, 6 | signswbase 32529 | . . . . 5 ⊢ {-1, 0, 1} = (Base‘𝑊) |
8 | eqid 2740 | . . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
9 | 3, 6 | signswplusg 32530 | . . . . 5 ⊢ ⨣ = (+g‘𝑊) |
10 | oveq1 7278 | . . . . . . . . . 10 ⊢ (𝑒 = 0 → (𝑒 ⨣ 𝑢) = (0 ⨣ 𝑢)) | |
11 | 10 | eqeq1d 2742 | . . . . . . . . 9 ⊢ (𝑒 = 0 → ((𝑒 ⨣ 𝑢) = 𝑢 ↔ (0 ⨣ 𝑢) = 𝑢)) |
12 | 11 | ovanraleqv 7295 | . . . . . . . 8 ⊢ (𝑒 = 0 → (∀𝑢 ∈ {-1, 0, 1} ((𝑒 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 𝑒) = 𝑢) ↔ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢))) |
13 | 12 | rspcev 3561 | . . . . . . 7 ⊢ ((0 ∈ {-1, 0, 1} ∧ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)) → ∃𝑒 ∈ {-1, 0, 1}∀𝑢 ∈ {-1, 0, 1} ((𝑒 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 𝑒) = 𝑢)) |
14 | 2, 4, 13 | mp2an 689 | . . . . . 6 ⊢ ∃𝑒 ∈ {-1, 0, 1}∀𝑢 ∈ {-1, 0, 1} ((𝑒 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 𝑒) = 𝑢) |
15 | 14 | a1i 11 | . . . . 5 ⊢ (⊤ → ∃𝑒 ∈ {-1, 0, 1}∀𝑢 ∈ {-1, 0, 1} ((𝑒 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 𝑒) = 𝑢)) |
16 | 7, 8, 9, 15 | ismgmid 18347 | . . . 4 ⊢ (⊤ → ((0 ∈ {-1, 0, 1} ∧ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)) ↔ (0g‘𝑊) = 0)) |
17 | 16 | mptru 1549 | . . 3 ⊢ ((0 ∈ {-1, 0, 1} ∧ ∀𝑢 ∈ {-1, 0, 1} ((0 ⨣ 𝑢) = 𝑢 ∧ (𝑢 ⨣ 0) = 𝑢)) ↔ (0g‘𝑊) = 0) |
18 | 5, 17 | mpbi 229 | . 2 ⊢ (0g‘𝑊) = 0 |
19 | 18 | eqcomi 2749 | 1 ⊢ 0 = (0g‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1542 ⊤wtru 1543 ∈ wcel 2110 ∀wral 3066 ∃wrex 3067 ifcif 4465 {cpr 4569 {ctp 4571 〈cop 4573 ‘cfv 6432 (class class class)co 7271 ∈ cmpo 7273 0cc0 10872 1c1 10873 -cneg 11206 ndxcnx 16892 Basecbs 16910 +gcplusg 16960 0gc0g 17148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-struct 16846 df-slot 16881 df-ndx 16893 df-base 16911 df-plusg 16973 df-0g 17150 |
This theorem is referenced by: (None) |
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