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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 π. (Contributed by Thierry Arnoux, 9-Sep-2018.) |
| Ref | Expression |
|---|---|
| signsw.p | ⒠⨣ = (π β {-1, 0, 1}, π β {-1, 0, 1} β¦ if(π = 0, π, π)) |
| signsw.w | β’ π = {β¨(Baseβndx), {-1, 0, 1}β©, β¨(+gβndx), ⨣ β©} |
| Ref | Expression |
|---|---|
| signsw0g | β’ 0 = (0gβπ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | c0ex 11212 | . . . . 5 β’ 0 β V | |
| 2 | 1 | tpid2 4774 | . . . 4 β’ 0 β {-1, 0, 1} |
| 3 | signsw.p | . . . . 5 ⒠⨣ = (π β {-1, 0, 1}, π β {-1, 0, 1} β¦ if(π = 0, π, π)) | |
| 4 | 3 | signsw0glem 33850 | . . . 4 β’ βπ’ β {-1, 0, 1} ((0 ⨣ π’) = π’ β§ (π’ ⨣ 0) = π’) |
| 5 | 2, 4 | pm3.2i 471 | . . 3 β’ (0 β {-1, 0, 1} β§ βπ’ β {-1, 0, 1} ((0 ⨣ π’) = π’ β§ (π’ ⨣ 0) = π’)) |
| 6 | signsw.w | . . . . . 6 β’ π = {β¨(Baseβndx), {-1, 0, 1}β©, β¨(+gβndx), ⨣ β©} | |
| 7 | 3, 6 | signswbase 33851 | . . . . 5 β’ {-1, 0, 1} = (Baseβπ) |
| 8 | eqid 2732 | . . . . 5 β’ (0gβπ) = (0gβπ) | |
| 9 | 3, 6 | signswplusg 33852 | . . . . 5 ⒠⨣ = (+gβπ) |
| 10 | oveq1 7418 | . . . . . . . . . 10 β’ (π = 0 β (π ⨣ π’) = (0 ⨣ π’)) | |
| 11 | 10 | eqeq1d 2734 | . . . . . . . . 9 β’ (π = 0 β ((π ⨣ π’) = π’ β (0 ⨣ π’) = π’)) |
| 12 | 11 | ovanraleqv 7435 | . . . . . . . 8 β’ (π = 0 β (βπ’ β {-1, 0, 1} ((π ⨣ π’) = π’ β§ (π’ ⨣ π) = π’) β βπ’ β {-1, 0, 1} ((0 ⨣ π’) = π’ β§ (π’ ⨣ 0) = π’))) |
| 13 | 12 | rspcev 3612 | . . . . . . 7 β’ ((0 β {-1, 0, 1} β§ βπ’ β {-1, 0, 1} ((0 ⨣ π’) = π’ β§ (π’ ⨣ 0) = π’)) β βπ β {-1, 0, 1}βπ’ β {-1, 0, 1} ((π ⨣ π’) = π’ β§ (π’ ⨣ π) = π’)) |
| 14 | 2, 4, 13 | mp2an 690 | . . . . . 6 β’ βπ β {-1, 0, 1}βπ’ β {-1, 0, 1} ((π ⨣ π’) = π’ β§ (π’ ⨣ π) = π’) |
| 15 | 14 | a1i 11 | . . . . 5 β’ (β€ β βπ β {-1, 0, 1}βπ’ β {-1, 0, 1} ((π ⨣ π’) = π’ β§ (π’ ⨣ π) = π’)) |
| 16 | 7, 8, 9, 15 | ismgmid 18590 | . . . 4 β’ (β€ β ((0 β {-1, 0, 1} β§ βπ’ β {-1, 0, 1} ((0 ⨣ π’) = π’ β§ (π’ ⨣ 0) = π’)) β (0gβπ) = 0)) |
| 17 | 16 | mptru 1548 | . . 3 β’ ((0 β {-1, 0, 1} β§ βπ’ β {-1, 0, 1} ((0 ⨣ π’) = π’ β§ (π’ ⨣ 0) = π’)) β (0gβπ) = 0) |
| 18 | 5, 17 | mpbi 229 | . 2 β’ (0gβπ) = 0 |
| 19 | 18 | eqcomi 2741 | 1 β’ 0 = (0gβπ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wb 205 β§ wa 396 = wceq 1541 β€wtru 1542 β wcel 2106 βwral 3061 βwrex 3070 ifcif 4528 {cpr 4630 {ctp 4632 β¨cop 4634 βcfv 6543 (class class class)co 7411 β cmpo 7413 0cc0 11112 1c1 11113 -cneg 11449 ndxcnx 17130 Basecbs 17148 +gcplusg 17201 0gc0g 17389 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-struct 17084 df-slot 17119 df-ndx 17131 df-base 17149 df-plusg 17214 df-0g 17391 |
| This theorem is referenced by: (None) |
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