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Mirrors > Home > MPE Home > Th. List > znle2 | Structured version Visualization version GIF version |
Description: The ordering of the β€/nβ€ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
Ref | Expression |
---|---|
znle2.y | β’ π = (β€/nβ€βπ) |
znle2.f | β’ πΉ = ((β€RHomβπ) βΎ π) |
znle2.w | β’ π = if(π = 0, β€, (0..^π)) |
znle2.l | β’ β€ = (leβπ) |
Ref | Expression |
---|---|
znle2 | β’ (π β β0 β β€ = ((πΉ β β€ ) β β‘πΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 β’ (RSpanββ€ring) = (RSpanββ€ring) | |
2 | eqid 2733 | . . 3 β’ (β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π}))) = (β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π}))) | |
3 | znle2.y | . . 3 β’ π = (β€/nβ€βπ) | |
4 | eqid 2733 | . . 3 β’ ((β€RHomβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))) βΎ π) = ((β€RHomβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))) βΎ π) | |
5 | znle2.w | . . 3 β’ π = if(π = 0, β€, (0..^π)) | |
6 | znle2.l | . . 3 β’ β€ = (leβπ) | |
7 | 1, 2, 3, 4, 5, 6 | znle 20962 | . 2 β’ (π β β0 β β€ = ((((β€RHomβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))) βΎ π) β β€ ) β β‘((β€RHomβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))) βΎ π))) |
8 | 1, 2, 3 | znzrh 20972 | . . . . . 6 β’ (π β β0 β (β€RHomβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))) = (β€RHomβπ)) |
9 | 8 | reseq1d 5940 | . . . . 5 β’ (π β β0 β ((β€RHomβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))) βΎ π) = ((β€RHomβπ) βΎ π)) |
10 | znle2.f | . . . . 5 β’ πΉ = ((β€RHomβπ) βΎ π) | |
11 | 9, 10 | eqtr4di 2791 | . . . 4 β’ (π β β0 β ((β€RHomβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))) βΎ π) = πΉ) |
12 | 11 | coeq1d 5821 | . . 3 β’ (π β β0 β (((β€RHomβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))) βΎ π) β β€ ) = (πΉ β β€ )) |
13 | 11 | cnveqd 5835 | . . 3 β’ (π β β0 β β‘((β€RHomβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))) βΎ π) = β‘πΉ) |
14 | 12, 13 | coeq12d 5824 | . 2 β’ (π β β0 β ((((β€RHomβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))) βΎ π) β β€ ) β β‘((β€RHomβ(β€ring /s (β€ring ~QG ((RSpanββ€ring)β{π})))) βΎ π)) = ((πΉ β β€ ) β β‘πΉ)) |
15 | 7, 14 | eqtrd 2773 | 1 β’ (π β β0 β β€ = ((πΉ β β€ ) β β‘πΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 ifcif 4490 {csn 4590 β‘ccnv 5636 βΎ cres 5639 β ccom 5641 βcfv 6500 (class class class)co 7361 0cc0 11059 β€ cle 11198 β0cn0 12421 β€cz 12507 ..^cfzo 13576 lecple 17148 /s cqus 17395 ~QG cqg 18932 RSpancrsp 20677 β€ringczring 20892 β€RHomczrh 20923 β€/nβ€czn 20926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-addf 11138 ax-mulf 11139 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-fz 13434 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-mhm 18609 df-grp 18759 df-minusg 18760 df-subg 18933 df-ghm 19014 df-cmn 19572 df-mgp 19905 df-ur 19922 df-ring 19974 df-cring 19975 df-rnghom 20156 df-subrg 20262 df-cnfld 20820 df-zring 20893 df-zrh 20927 df-zn 20930 |
This theorem is referenced by: znleval 20984 |
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