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Mirrors > Home > MPE Home > Th. List > cgr3swap13 | Structured version Visualization version GIF version |
Description: Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 3-Oct-2020.) |
Ref | Expression |
---|---|
tgcgrxfr.p | β’ π = (BaseβπΊ) |
tgcgrxfr.m | β’ β = (distβπΊ) |
tgcgrxfr.i | β’ πΌ = (ItvβπΊ) |
tgcgrxfr.r | β’ βΌ = (cgrGβπΊ) |
tgcgrxfr.g | β’ (π β πΊ β TarskiG) |
tgbtwnxfr.a | β’ (π β π΄ β π) |
tgbtwnxfr.b | β’ (π β π΅ β π) |
tgbtwnxfr.c | β’ (π β πΆ β π) |
tgbtwnxfr.d | β’ (π β π· β π) |
tgbtwnxfr.e | β’ (π β πΈ β π) |
tgbtwnxfr.f | β’ (π β πΉ β π) |
tgbtwnxfr.2 | β’ (π β β¨βπ΄π΅πΆββ© βΌ β¨βπ·πΈπΉββ©) |
Ref | Expression |
---|---|
cgr3swap13 | β’ (π β β¨βπΆπ΅π΄ββ© βΌ β¨βπΉπΈπ·ββ©) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgcgrxfr.p | . 2 β’ π = (BaseβπΊ) | |
2 | tgcgrxfr.m | . 2 β’ β = (distβπΊ) | |
3 | tgcgrxfr.i | . 2 β’ πΌ = (ItvβπΊ) | |
4 | tgcgrxfr.r | . 2 β’ βΌ = (cgrGβπΊ) | |
5 | tgcgrxfr.g | . 2 β’ (π β πΊ β TarskiG) | |
6 | tgbtwnxfr.b | . 2 β’ (π β π΅ β π) | |
7 | tgbtwnxfr.c | . 2 β’ (π β πΆ β π) | |
8 | tgbtwnxfr.a | . 2 β’ (π β π΄ β π) | |
9 | tgbtwnxfr.e | . 2 β’ (π β πΈ β π) | |
10 | tgbtwnxfr.f | . 2 β’ (π β πΉ β π) | |
11 | tgbtwnxfr.d | . 2 β’ (π β π· β π) | |
12 | tgbtwnxfr.2 | . . . 4 β’ (π β β¨βπ΄π΅πΆββ© βΌ β¨βπ·πΈπΉββ©) | |
13 | 1, 2, 3, 4, 5, 8, 6, 7, 11, 9, 10, 12 | cgr3swap12 28371 | . . 3 β’ (π β β¨βπ΅π΄πΆββ© βΌ β¨βπΈπ·πΉββ©) |
14 | 1, 2, 3, 4, 5, 6, 8, 7, 9, 11, 10, 13 | cgr3swap23 28372 | . 2 β’ (π β β¨βπ΅πΆπ΄ββ© βΌ β¨βπΈπΉπ·ββ©) |
15 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14 | cgr3swap12 28371 | 1 β’ (π β β¨βπΆπ΅π΄ββ© βΌ β¨βπΉπΈπ·ββ©) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 class class class wbr 5143 βcfv 6543 β¨βcs3 14825 Basecbs 17179 distcds 17241 TarskiGcstrkg 28275 Itvcitv 28281 cgrGccgrg 28358 |
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 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 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 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-pm 8846 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-hash 14322 df-word 14497 df-concat 14553 df-s1 14578 df-s2 14831 df-s3 14832 df-trkgc 28296 df-trkgcb 28298 df-trkg 28301 df-cgrg 28359 |
This theorem is referenced by: (None) |
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