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Mirrors > Home > MPE Home > Th. List > symg2hash | Structured version Visualization version GIF version |
Description: The symmetric group on a (proper) pair has cardinality 2. (Contributed by AV, 9-Dec-2018.) |
Ref | Expression |
---|---|
symg1bas.1 | ⊢ 𝐺 = (SymGrp‘𝐴) |
symg1bas.2 | ⊢ 𝐵 = (Base‘𝐺) |
symg2bas.0 | ⊢ 𝐴 = {𝐼, 𝐽} |
Ref | Expression |
---|---|
symg2hash | ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ∈ 𝑊 ∧ 𝐼 ≠ 𝐽) → (♯‘𝐵) = 2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | symg2bas.0 | . . . 4 ⊢ 𝐴 = {𝐼, 𝐽} | |
2 | prfi 9050 | . . . 4 ⊢ {𝐼, 𝐽} ∈ Fin | |
3 | 1, 2 | eqeltri 2836 | . . 3 ⊢ 𝐴 ∈ Fin |
4 | symg1bas.1 | . . . 4 ⊢ 𝐺 = (SymGrp‘𝐴) | |
5 | symg1bas.2 | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
6 | 4, 5 | symghash 18966 | . . 3 ⊢ (𝐴 ∈ Fin → (♯‘𝐵) = (!‘(♯‘𝐴))) |
7 | 3, 6 | ax-mp 5 | . 2 ⊢ (♯‘𝐵) = (!‘(♯‘𝐴)) |
8 | 1 | fveq2i 6771 | . . . . 5 ⊢ (♯‘𝐴) = (♯‘{𝐼, 𝐽}) |
9 | elex 3448 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V) | |
10 | elex 3448 | . . . . . . 7 ⊢ (𝐽 ∈ 𝑊 → 𝐽 ∈ V) | |
11 | id 22 | . . . . . . 7 ⊢ (𝐼 ≠ 𝐽 → 𝐼 ≠ 𝐽) | |
12 | 9, 10, 11 | 3anim123i 1149 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ∈ 𝑊 ∧ 𝐼 ≠ 𝐽) → (𝐼 ∈ V ∧ 𝐽 ∈ V ∧ 𝐼 ≠ 𝐽)) |
13 | hashprb 14093 | . . . . . 6 ⊢ ((𝐼 ∈ V ∧ 𝐽 ∈ V ∧ 𝐼 ≠ 𝐽) ↔ (♯‘{𝐼, 𝐽}) = 2) | |
14 | 12, 13 | sylib 217 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ∈ 𝑊 ∧ 𝐼 ≠ 𝐽) → (♯‘{𝐼, 𝐽}) = 2) |
15 | 8, 14 | eqtrid 2791 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ∈ 𝑊 ∧ 𝐼 ≠ 𝐽) → (♯‘𝐴) = 2) |
16 | 15 | fveq2d 6772 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ∈ 𝑊 ∧ 𝐼 ≠ 𝐽) → (!‘(♯‘𝐴)) = (!‘2)) |
17 | fac2 13974 | . . 3 ⊢ (!‘2) = 2 | |
18 | 16, 17 | eqtrdi 2795 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ∈ 𝑊 ∧ 𝐼 ≠ 𝐽) → (!‘(♯‘𝐴)) = 2) |
19 | 7, 18 | eqtrid 2791 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ∈ 𝑊 ∧ 𝐼 ≠ 𝐽) → (♯‘𝐵) = 2) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ≠ wne 2944 Vcvv 3430 {cpr 4568 ‘cfv 6430 Fincfn 8707 2c2 12011 !cfa 13968 ♯chash 14025 Basecbs 16893 SymGrpcsymg 18955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-oadd 8285 df-er 8472 df-map 8591 df-pm 8592 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-dju 9643 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-xnn0 12289 df-z 12303 df-uz 12565 df-fz 13222 df-seq 13703 df-fac 13969 df-bc 13998 df-hash 14026 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-tset 16962 df-efmnd 18489 df-symg 18956 |
This theorem is referenced by: symg2bas 18981 |
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