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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > erngplus2-rN | Structured version Visualization version GIF version |
Description: Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
erngset.h-r | β’ π» = (LHypβπΎ) |
erngset.t-r | β’ π = ((LTrnβπΎ)βπ) |
erngset.e-r | β’ πΈ = ((TEndoβπΎ)βπ) |
erngset.d-r | β’ π· = ((EDRingRβπΎ)βπ) |
erng.p-r | β’ + = (+gβπ·) |
Ref | Expression |
---|---|
erngplus2-rN | β’ (((πΎ β HL β§ π β π») β§ (π β πΈ β§ π β πΈ β§ πΉ β π)) β ((π + π)βπΉ) = ((πβπΉ) β (πβπΉ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erngset.h-r | . . . 4 β’ π» = (LHypβπΎ) | |
2 | erngset.t-r | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
3 | erngset.e-r | . . . 4 β’ πΈ = ((TEndoβπΎ)βπ) | |
4 | erngset.d-r | . . . 4 β’ π· = ((EDRingRβπΎ)βπ) | |
5 | erng.p-r | . . . 4 β’ + = (+gβπ·) | |
6 | 1, 2, 3, 4, 5 | erngplus-rN 40336 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β πΈ β§ π β πΈ)) β (π + π) = (π β π β¦ ((πβπ) β (πβπ)))) |
7 | 6 | 3adantr3 1168 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β πΈ β§ π β πΈ β§ πΉ β π)) β (π + π) = (π β π β¦ ((πβπ) β (πβπ)))) |
8 | fveq2 6890 | . . . 4 β’ (π = πΉ β (πβπ) = (πβπΉ)) | |
9 | fveq2 6890 | . . . 4 β’ (π = πΉ β (πβπ) = (πβπΉ)) | |
10 | 8, 9 | coeq12d 5862 | . . 3 β’ (π = πΉ β ((πβπ) β (πβπ)) = ((πβπΉ) β (πβπΉ))) |
11 | 10 | adantl 480 | . 2 β’ ((((πΎ β HL β§ π β π») β§ (π β πΈ β§ π β πΈ β§ πΉ β π)) β§ π = πΉ) β ((πβπ) β (πβπ)) = ((πβπΉ) β (πβπΉ))) |
12 | simpr3 1193 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β πΈ β§ π β πΈ β§ πΉ β π)) β πΉ β π) | |
13 | fvex 6903 | . . . 4 β’ (πβπΉ) β V | |
14 | fvex 6903 | . . . 4 β’ (πβπΉ) β V | |
15 | 13, 14 | coex 7932 | . . 3 β’ ((πβπΉ) β (πβπΉ)) β V |
16 | 15 | a1i 11 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β πΈ β§ π β πΈ β§ πΉ β π)) β ((πβπΉ) β (πβπΉ)) β V) |
17 | 7, 11, 12, 16 | fvmptd 7005 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β πΈ β§ π β πΈ β§ πΉ β π)) β ((π + π)βπΉ) = ((πβπΉ) β (πβπΉ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 Vcvv 3463 β¦ cmpt 5227 β ccom 5677 βcfv 6543 (class class class)co 7413 +gcplusg 17227 HLchlt 38874 LHypclh 39509 LTrncltrn 39626 TEndoctendo 40277 EDRingRcedring-rN 40279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-struct 17110 df-slot 17145 df-ndx 17157 df-base 17175 df-plusg 17240 df-mulr 17241 df-edring-rN 40281 |
This theorem is referenced by: (None) |
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