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| Mirrors > Home > MPE Home > Th. List > psgnprfval1 | Structured version Visualization version GIF version | ||
| Description: The permutation sign of the identity for a pair. (Contributed by AV, 11-Dec-2018.) | 
| Ref | Expression | 
|---|---|
| psgnprfval.0 | ⊢ 𝐷 = {1, 2} | 
| psgnprfval.g | ⊢ 𝐺 = (SymGrp‘𝐷) | 
| psgnprfval.b | ⊢ 𝐵 = (Base‘𝐺) | 
| psgnprfval.t | ⊢ 𝑇 = ran (pmTrsp‘𝐷) | 
| psgnprfval.n | ⊢ 𝑁 = (pmSgn‘𝐷) | 
| Ref | Expression | 
|---|---|
| psgnprfval1 | ⊢ (𝑁‘{〈1, 1〉, 〈2, 2〉}) = 1 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | psgnprfval.0 | . . . . . . 7 ⊢ 𝐷 = {1, 2} | |
| 2 | prex 5437 | . . . . . . 7 ⊢ {1, 2} ∈ V | |
| 3 | 1, 2 | eqeltri 2837 | . . . . . 6 ⊢ 𝐷 ∈ V | 
| 4 | psgnprfval.g | . . . . . . 7 ⊢ 𝐺 = (SymGrp‘𝐷) | |
| 5 | 4 | symgid 19419 | . . . . . 6 ⊢ (𝐷 ∈ V → ( I ↾ 𝐷) = (0g‘𝐺)) | 
| 6 | 3, 5 | ax-mp 5 | . . . . 5 ⊢ ( I ↾ 𝐷) = (0g‘𝐺) | 
| 7 | 6 | gsum0 18697 | . . . 4 ⊢ (𝐺 Σg ∅) = ( I ↾ 𝐷) | 
| 8 | reseq2 5992 | . . . . . 6 ⊢ (𝐷 = {1, 2} → ( I ↾ 𝐷) = ( I ↾ {1, 2})) | |
| 9 | 1ex 11257 | . . . . . . 7 ⊢ 1 ∈ V | |
| 10 | 2nn 12339 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
| 11 | residpr 7163 | . . . . . . 7 ⊢ ((1 ∈ V ∧ 2 ∈ ℕ) → ( I ↾ {1, 2}) = {〈1, 1〉, 〈2, 2〉}) | |
| 12 | 9, 10, 11 | mp2an 692 | . . . . . 6 ⊢ ( I ↾ {1, 2}) = {〈1, 1〉, 〈2, 2〉} | 
| 13 | 8, 12 | eqtrdi 2793 | . . . . 5 ⊢ (𝐷 = {1, 2} → ( I ↾ 𝐷) = {〈1, 1〉, 〈2, 2〉}) | 
| 14 | 1, 13 | ax-mp 5 | . . . 4 ⊢ ( I ↾ 𝐷) = {〈1, 1〉, 〈2, 2〉} | 
| 15 | 7, 14 | eqtr2i 2766 | . . 3 ⊢ {〈1, 1〉, 〈2, 2〉} = (𝐺 Σg ∅) | 
| 16 | 15 | fveq2i 6909 | . 2 ⊢ (𝑁‘{〈1, 1〉, 〈2, 2〉}) = (𝑁‘(𝐺 Σg ∅)) | 
| 17 | wrd0 14577 | . . 3 ⊢ ∅ ∈ Word 𝑇 | |
| 18 | psgnprfval.t | . . . 4 ⊢ 𝑇 = ran (pmTrsp‘𝐷) | |
| 19 | psgnprfval.n | . . . 4 ⊢ 𝑁 = (pmSgn‘𝐷) | |
| 20 | 4, 18, 19 | psgnvalii 19527 | . . 3 ⊢ ((𝐷 ∈ V ∧ ∅ ∈ Word 𝑇) → (𝑁‘(𝐺 Σg ∅)) = (-1↑(♯‘∅))) | 
| 21 | 3, 17, 20 | mp2an 692 | . 2 ⊢ (𝑁‘(𝐺 Σg ∅)) = (-1↑(♯‘∅)) | 
| 22 | hash0 14406 | . . . 4 ⊢ (♯‘∅) = 0 | |
| 23 | 22 | oveq2i 7442 | . . 3 ⊢ (-1↑(♯‘∅)) = (-1↑0) | 
| 24 | neg1cn 12380 | . . . 4 ⊢ -1 ∈ ℂ | |
| 25 | exp0 14106 | . . . 4 ⊢ (-1 ∈ ℂ → (-1↑0) = 1) | |
| 26 | 24, 25 | ax-mp 5 | . . 3 ⊢ (-1↑0) = 1 | 
| 27 | 23, 26 | eqtri 2765 | . 2 ⊢ (-1↑(♯‘∅)) = 1 | 
| 28 | 16, 21, 27 | 3eqtri 2769 | 1 ⊢ (𝑁‘{〈1, 1〉, 〈2, 2〉}) = 1 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 {cpr 4628 〈cop 4632 I cid 5577 ran crn 5686 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 -cneg 11493 ℕcn 12266 2c2 12321 ↑cexp 14102 ♯chash 14369 Word cword 14552 Basecbs 17247 0gc0g 17484 Σg cgsu 17485 SymGrpcsymg 19386 pmTrspcpmtr 19459 pmSgncpsgn 19507 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-xor 1512 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-word 14553 df-lsw 14601 df-concat 14609 df-s1 14634 df-substr 14679 df-pfx 14709 df-splice 14788 df-reverse 14797 df-s2 14887 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-tset 17316 df-0g 17486 df-gsum 17487 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-efmnd 18882 df-grp 18954 df-minusg 18955 df-subg 19141 df-ghm 19231 df-gim 19277 df-oppg 19364 df-symg 19387 df-pmtr 19460 df-psgn 19509 | 
| This theorem is referenced by: m2detleiblem1 22630 m2detleiblem5 22631 | 
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