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Theorem psgndiflemA 21494
Description: Lemma 2 for psgndif 21495. (Contributed by AV, 31-Jan-2019.)
Hypotheses
Ref Expression
psgnfix.p 𝑃 = (Baseβ€˜(SymGrpβ€˜π‘))
psgnfix.t 𝑇 = ran (pmTrspβ€˜(𝑁 βˆ– {𝐾}))
psgnfix.s 𝑆 = (SymGrpβ€˜(𝑁 βˆ– {𝐾}))
psgnfix.z 𝑍 = (SymGrpβ€˜π‘)
psgnfix.r 𝑅 = ran (pmTrspβ€˜π‘)
Assertion
Ref Expression
psgndiflemA (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) β†’ ((π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅) β†’ (𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ) β†’ (-1↑(β™―β€˜π‘Š)) = (-1↑(β™―β€˜π‘ˆ)))))
Distinct variable groups:   𝐾,π‘ž   𝑃,π‘ž   𝑄,π‘ž
Allowed substitution hints:   𝑅(π‘ž)   𝑆(π‘ž)   𝑇(π‘ž)   π‘ˆ(π‘ž)   𝑁(π‘ž)   π‘Š(π‘ž)   𝑍(π‘ž)

Proof of Theorem psgndiflemA
Dummy variables 𝑀 𝑖 𝑛 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6885 . . . . . . . . . . . 12 (𝑀 = π‘Š β†’ (β™―β€˜π‘€) = (β™―β€˜π‘Š))
21eqeq1d 2728 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ ((β™―β€˜π‘€) = (β™―β€˜π‘Ÿ) ↔ (β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ)))
31oveq2d 7421 . . . . . . . . . . . 12 (𝑀 = π‘Š β†’ (0..^(β™―β€˜π‘€)) = (0..^(β™―β€˜π‘Š)))
4 fveq1 6884 . . . . . . . . . . . . . . . 16 (𝑀 = π‘Š β†’ (π‘€β€˜π‘–) = (π‘Šβ€˜π‘–))
54fveq1d 6887 . . . . . . . . . . . . . . 15 (𝑀 = π‘Š β†’ ((π‘€β€˜π‘–)β€˜π‘›) = ((π‘Šβ€˜π‘–)β€˜π‘›))
65eqeq1d 2728 . . . . . . . . . . . . . 14 (𝑀 = π‘Š β†’ (((π‘€β€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›) ↔ ((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))
76ralbidv 3171 . . . . . . . . . . . . 13 (𝑀 = π‘Š β†’ (βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘€β€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›) ↔ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))
87anbi2d 628 . . . . . . . . . . . 12 (𝑀 = π‘Š β†’ ((((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘€β€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)) ↔ (((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›))))
93, 8raleqbidv 3336 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜π‘€))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘€β€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)) ↔ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›))))
102, 9anbi12d 630 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (((β™―β€˜π‘€) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘€))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘€β€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›))) ↔ ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))))
1110rexbidv 3172 . . . . . . . . 9 (𝑀 = π‘Š β†’ (βˆƒπ‘Ÿ ∈ Word 𝑅((β™―β€˜π‘€) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘€))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘€β€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›))) ↔ βˆƒπ‘Ÿ ∈ Word 𝑅((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))))
1211rspccv 3603 . . . . . . . 8 (βˆ€π‘€ ∈ Word π‘‡βˆƒπ‘Ÿ ∈ Word 𝑅((β™―β€˜π‘€) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘€))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘€β€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›))) β†’ (π‘Š ∈ Word 𝑇 β†’ βˆƒπ‘Ÿ ∈ Word 𝑅((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))))
13 psgnfix.t . . . . . . . . 9 𝑇 = ran (pmTrspβ€˜(𝑁 βˆ– {𝐾}))
14 psgnfix.r . . . . . . . . 9 𝑅 = ran (pmTrspβ€˜π‘)
1513, 14pmtrdifwrdel2 19406 . . . . . . . 8 (𝐾 ∈ 𝑁 β†’ βˆ€π‘€ ∈ Word π‘‡βˆƒπ‘Ÿ ∈ Word 𝑅((β™―β€˜π‘€) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘€))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘€β€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›))))
1612, 15syl11 33 . . . . . . 7 (π‘Š ∈ Word 𝑇 β†’ (𝐾 ∈ 𝑁 β†’ βˆƒπ‘Ÿ ∈ Word 𝑅((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))))
17163ad2ant1 1130 . . . . . 6 ((π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅) β†’ (𝐾 ∈ 𝑁 β†’ βˆƒπ‘Ÿ ∈ Word 𝑅((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))))
1817com12 32 . . . . 5 (𝐾 ∈ 𝑁 β†’ ((π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅) β†’ βˆƒπ‘Ÿ ∈ Word 𝑅((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))))
1918ad2antlr 724 . . . 4 (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) β†’ ((π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅) β†’ βˆƒπ‘Ÿ ∈ Word 𝑅((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))))
2019imp 406 . . 3 ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅)) β†’ βˆƒπ‘Ÿ ∈ Word 𝑅((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›))))
21 oveq2 7413 . . . . . . . . 9 ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) β†’ (-1↑(β™―β€˜π‘Š)) = (-1↑(β™―β€˜π‘Ÿ)))
2221adantr 480 . . . . . . . 8 (((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›))) β†’ (-1↑(β™―β€˜π‘Š)) = (-1↑(β™―β€˜π‘Ÿ)))
2322ad3antlr 728 . . . . . . 7 ((((π‘Ÿ ∈ Word 𝑅 ∧ ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ)) β†’ (-1↑(β™―β€˜π‘Š)) = (-1↑(β™―β€˜π‘Ÿ)))
24 psgnfix.z . . . . . . . 8 𝑍 = (SymGrpβ€˜π‘)
25 simplll 772 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅)) β†’ 𝑁 ∈ Fin)
2625ad2antlr 724 . . . . . . . 8 ((((π‘Ÿ ∈ Word 𝑅 ∧ ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ)) β†’ 𝑁 ∈ Fin)
27 simplll 772 . . . . . . . 8 ((((π‘Ÿ ∈ Word 𝑅 ∧ ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ)) β†’ π‘Ÿ ∈ Word 𝑅)
28 simprr3 1220 . . . . . . . . 9 (((π‘Ÿ ∈ Word 𝑅 ∧ ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅))) β†’ π‘ˆ ∈ Word 𝑅)
2928adantr 480 . . . . . . . 8 ((((π‘Ÿ ∈ Word 𝑅 ∧ ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ)) β†’ π‘ˆ ∈ Word 𝑅)
30 simplrl 774 . . . . . . . . . 10 ((((π‘Ÿ ∈ Word 𝑅 ∧ ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ)) β†’ ((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}))
31 3simpa 1145 . . . . . . . . . . . 12 ((π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅) β†’ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š)))
3231adantl 481 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅)) β†’ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š)))
3332ad2antlr 724 . . . . . . . . . 10 ((((π‘Ÿ ∈ Word 𝑅 ∧ ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ)) β†’ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š)))
34 simplrl 774 . . . . . . . . . . 11 (((π‘Ÿ ∈ Word 𝑅 ∧ ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅))) β†’ (β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ))
3534adantr 480 . . . . . . . . . 10 ((((π‘Ÿ ∈ Word 𝑅 ∧ ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ)) β†’ (β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ))
36 simplrr 775 . . . . . . . . . . 11 (((π‘Ÿ ∈ Word 𝑅 ∧ ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅))) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))
3736adantr 480 . . . . . . . . . 10 ((((π‘Ÿ ∈ Word 𝑅 ∧ ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ)) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))
38 psgnfix.p . . . . . . . . . . . . 13 𝑃 = (Baseβ€˜(SymGrpβ€˜π‘))
39 psgnfix.s . . . . . . . . . . . . 13 𝑆 = (SymGrpβ€˜(𝑁 βˆ– {𝐾}))
4038, 13, 39, 24, 14psgndiflemB 21493 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) β†’ ((π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š)) β†’ ((π‘Ÿ ∈ Word 𝑅 ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›))) β†’ 𝑄 = (𝑍 Ξ£g π‘Ÿ))))
4140imp31 417 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š))) ∧ (π‘Ÿ ∈ Word 𝑅 ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) β†’ 𝑄 = (𝑍 Ξ£g π‘Ÿ))
4241eqcomd 2732 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š))) ∧ (π‘Ÿ ∈ Word 𝑅 ∧ (β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) β†’ (𝑍 Ξ£g π‘Ÿ) = 𝑄)
4330, 33, 27, 35, 37, 42syl23anc 1374 . . . . . . . . 9 ((((π‘Ÿ ∈ Word 𝑅 ∧ ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ)) β†’ (𝑍 Ξ£g π‘Ÿ) = 𝑄)
44 id 22 . . . . . . . . . . 11 (𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ) β†’ 𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ))
4524eqcomi 2735 . . . . . . . . . . . 12 (SymGrpβ€˜π‘) = 𝑍
4645oveq1i 7415 . . . . . . . . . . 11 ((SymGrpβ€˜π‘) Ξ£g π‘ˆ) = (𝑍 Ξ£g π‘ˆ)
4744, 46eqtrdi 2782 . . . . . . . . . 10 (𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ) β†’ 𝑄 = (𝑍 Ξ£g π‘ˆ))
4847adantl 481 . . . . . . . . 9 ((((π‘Ÿ ∈ Word 𝑅 ∧ ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ)) β†’ 𝑄 = (𝑍 Ξ£g π‘ˆ))
4943, 48eqtrd 2766 . . . . . . . 8 ((((π‘Ÿ ∈ Word 𝑅 ∧ ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ)) β†’ (𝑍 Ξ£g π‘Ÿ) = (𝑍 Ξ£g π‘ˆ))
5024, 14, 26, 27, 29, 49psgnuni 19419 . . . . . . 7 ((((π‘Ÿ ∈ Word 𝑅 ∧ ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ)) β†’ (-1↑(β™―β€˜π‘Ÿ)) = (-1↑(β™―β€˜π‘ˆ)))
5123, 50eqtrd 2766 . . . . . 6 ((((π‘Ÿ ∈ Word 𝑅 ∧ ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ)) β†’ (-1↑(β™―β€˜π‘Š)) = (-1↑(β™―β€˜π‘ˆ)))
5251ex 412 . . . . 5 (((π‘Ÿ ∈ Word 𝑅 ∧ ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅))) β†’ (𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ) β†’ (-1↑(β™―β€˜π‘Š)) = (-1↑(β™―β€˜π‘ˆ))))
5352ex 412 . . . 4 ((π‘Ÿ ∈ Word 𝑅 ∧ ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›)))) β†’ ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅)) β†’ (𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ) β†’ (-1↑(β™―β€˜π‘Š)) = (-1↑(β™―β€˜π‘ˆ)))))
5453rexlimiva 3141 . . 3 (βˆƒπ‘Ÿ ∈ Word 𝑅((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π‘Š))(((π‘Ÿβ€˜π‘–)β€˜πΎ) = 𝐾 ∧ βˆ€π‘› ∈ (𝑁 βˆ– {𝐾})((π‘Šβ€˜π‘–)β€˜π‘›) = ((π‘Ÿβ€˜π‘–)β€˜π‘›))) β†’ ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅)) β†’ (𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ) β†’ (-1↑(β™―β€˜π‘Š)) = (-1↑(β™―β€˜π‘ˆ)))))
5520, 54mpcom 38 . 2 ((((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) ∧ (π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅)) β†’ (𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ) β†’ (-1↑(β™―β€˜π‘Š)) = (-1↑(β™―β€˜π‘ˆ))))
5655ex 412 1 (((𝑁 ∈ Fin ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ {π‘ž ∈ 𝑃 ∣ (π‘žβ€˜πΎ) = 𝐾}) β†’ ((π‘Š ∈ Word 𝑇 ∧ (𝑄 β†Ύ (𝑁 βˆ– {𝐾})) = (𝑆 Ξ£g π‘Š) ∧ π‘ˆ ∈ Word 𝑅) β†’ (𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ) β†’ (-1↑(β™―β€˜π‘Š)) = (-1↑(β™―β€˜π‘ˆ)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βˆƒwrex 3064  {crab 3426   βˆ– cdif 3940  {csn 4623  ran crn 5670   β†Ύ cres 5671  β€˜cfv 6537  (class class class)co 7405  Fincfn 8941  0cc0 11112  1c1 11113  -cneg 11449  ..^cfzo 13633  β†‘cexp 14032  β™―chash 14295  Word cword 14470  Basecbs 17153   Ξ£g cgsu 17395  SymGrpcsymg 19286  pmTrspcpmtr 19361
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  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 396  df-or 845  df-3or 1085  df-3an 1086  df-xor 1505  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-ot 4632  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-tpos 8212  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-2o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  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-div 11876  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-xnn0 12549  df-z 12563  df-uz 12827  df-rp 12981  df-fz 13491  df-fzo 13634  df-seq 13973  df-exp 14033  df-hash 14296  df-word 14471  df-lsw 14519  df-concat 14527  df-s1 14552  df-substr 14597  df-pfx 14627  df-splice 14706  df-reverse 14715  df-s2 14805  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-tset 17225  df-0g 17396  df-gsum 17397  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-mhm 18713  df-submnd 18714  df-efmnd 18794  df-grp 18866  df-minusg 18867  df-subg 19050  df-ghm 19139  df-gim 19184  df-oppg 19262  df-symg 19287  df-pmtr 19362  df-psgn 19411
This theorem is referenced by:  psgndif  21495
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