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Mirrors > Home > MPE Home > Th. List > efmnd2hash | Structured version Visualization version GIF version |
Description: The monoid of endofunctions on a (proper) pair has cardinality 4. (Contributed by AV, 18-Feb-2024.) |
Ref | Expression |
---|---|
efmnd1bas.1 | β’ πΊ = (EndoFMndβπ΄) |
efmnd1bas.2 | β’ π΅ = (BaseβπΊ) |
efmnd2bas.0 | β’ π΄ = {πΌ, π½} |
Ref | Expression |
---|---|
efmnd2hash | β’ ((πΌ β π β§ π½ β π β§ πΌ β π½) β (β―βπ΅) = 4) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efmnd2bas.0 | . . . 4 β’ π΄ = {πΌ, π½} | |
2 | prfi 9324 | . . . 4 β’ {πΌ, π½} β Fin | |
3 | 1, 2 | eqeltri 2827 | . . 3 β’ π΄ β Fin |
4 | efmnd1bas.1 | . . . 4 β’ πΊ = (EndoFMndβπ΄) | |
5 | efmnd1bas.2 | . . . 4 β’ π΅ = (BaseβπΊ) | |
6 | 4, 5 | efmndhash 18793 | . . 3 β’ (π΄ β Fin β (β―βπ΅) = ((β―βπ΄)β(β―βπ΄))) |
7 | 3, 6 | ax-mp 5 | . 2 β’ (β―βπ΅) = ((β―βπ΄)β(β―βπ΄)) |
8 | 1 | fveq2i 6893 | . . . . 5 β’ (β―βπ΄) = (β―β{πΌ, π½}) |
9 | elex 3491 | . . . . . . 7 β’ (πΌ β π β πΌ β V) | |
10 | elex 3491 | . . . . . . 7 β’ (π½ β π β π½ β V) | |
11 | id 22 | . . . . . . 7 β’ (πΌ β π½ β πΌ β π½) | |
12 | 9, 10, 11 | 3anim123i 1149 | . . . . . 6 β’ ((πΌ β π β§ π½ β π β§ πΌ β π½) β (πΌ β V β§ π½ β V β§ πΌ β π½)) |
13 | hashprb 14361 | . . . . . 6 β’ ((πΌ β V β§ π½ β V β§ πΌ β π½) β (β―β{πΌ, π½}) = 2) | |
14 | 12, 13 | sylib 217 | . . . . 5 β’ ((πΌ β π β§ π½ β π β§ πΌ β π½) β (β―β{πΌ, π½}) = 2) |
15 | 8, 14 | eqtrid 2782 | . . . 4 β’ ((πΌ β π β§ π½ β π β§ πΌ β π½) β (β―βπ΄) = 2) |
16 | 15, 15 | oveq12d 7429 | . . 3 β’ ((πΌ β π β§ π½ β π β§ πΌ β π½) β ((β―βπ΄)β(β―βπ΄)) = (2β2)) |
17 | sq2 14165 | . . 3 β’ (2β2) = 4 | |
18 | 16, 17 | eqtrdi 2786 | . 2 β’ ((πΌ β π β§ π½ β π β§ πΌ β π½) β ((β―βπ΄)β(β―βπ΄)) = 4) |
19 | 7, 18 | eqtrid 2782 | 1 β’ ((πΌ β π β§ π½ β π β§ πΌ β π½) β (β―βπ΅) = 4) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1085 = wceq 1539 β wcel 2104 β wne 2938 Vcvv 3472 {cpr 4629 βcfv 6542 (class class class)co 7411 Fincfn 8941 2c2 12271 4c4 12273 βcexp 14031 β―chash 14294 Basecbs 17148 EndoFMndcefmnd 18785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 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 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 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 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-oadd 8472 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 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-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-seq 13971 df-exp 14032 df-hash 14295 df-struct 17084 df-slot 17119 df-ndx 17131 df-base 17149 df-plusg 17214 df-tset 17220 df-efmnd 18786 |
This theorem is referenced by: (None) |
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