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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmaplem2N | Structured version Visualization version GIF version |
Description: Lemma to convert a frequently-used union condition. TODO: see if this can be applied to other hdmap* theorems. (Contributed by NM, 17-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hdmaplem1.v | β’ π = (Baseβπ) |
hdmaplem1.n | β’ π = (LSpanβπ) |
hdmaplem1.w | β’ (π β π β LMod) |
hdmaplem1.z | β’ (π β π β π) |
hdmaplem1.j | β’ (π β Β¬ π β ((πβ{π}) βͺ (πβ{π}))) |
hdmaplem1.y | β’ (π β π β π) |
Ref | Expression |
---|---|
hdmaplem2N | β’ (π β (πβ{π}) β (πβ{π})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmaplem1.v | . 2 β’ π = (Baseβπ) | |
2 | hdmaplem1.n | . 2 β’ π = (LSpanβπ) | |
3 | hdmaplem1.w | . 2 β’ (π β π β LMod) | |
4 | hdmaplem1.z | . 2 β’ (π β π β π) | |
5 | hdmaplem1.y | . 2 β’ (π β π β π) | |
6 | hdmaplem1.j | . . 3 β’ (π β Β¬ π β ((πβ{π}) βͺ (πβ{π}))) | |
7 | elun2 4171 | . . 3 β’ (π β (πβ{π}) β π β ((πβ{π}) βͺ (πβ{π}))) | |
8 | 6, 7 | nsyl 140 | . 2 β’ (π β Β¬ π β (πβ{π})) |
9 | 1, 2, 3, 4, 5, 8 | lspsnne2 21008 | 1 β’ (π β (πβ{π}) β (πβ{π})) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 = wceq 1533 β wcel 2098 β wne 2930 βͺ cun 3938 {csn 4624 βcfv 6542 Basecbs 17177 LModclmod 20745 LSpanclspn 20857 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
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-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-plusg 17243 df-0g 17420 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-grp 18895 df-minusg 18896 df-sbg 18897 df-mgp 20077 df-ur 20124 df-ring 20177 df-lmod 20747 df-lss 20818 df-lsp 20858 |
This theorem is referenced by: (None) |
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