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Theorem psgndiflemA 21535
Description: Lemma 2 for psgndif 21536. (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 6891 . . . . . . . . . . . 12 (𝑀 = π‘Š β†’ (β™―β€˜π‘€) = (β™―β€˜π‘Š))
21eqeq1d 2727 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ ((β™―β€˜π‘€) = (β™―β€˜π‘Ÿ) ↔ (β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ)))
31oveq2d 7431 . . . . . . . . . . . 12 (𝑀 = π‘Š β†’ (0..^(β™―β€˜π‘€)) = (0..^(β™―β€˜π‘Š)))
4 fveq1 6890 . . . . . . . . . . . . . . . 16 (𝑀 = π‘Š β†’ (π‘€β€˜π‘–) = (π‘Šβ€˜π‘–))
54fveq1d 6893 . . . . . . . . . . . . . . 15 (𝑀 = π‘Š β†’ ((π‘€β€˜π‘–)β€˜π‘›) = ((π‘Šβ€˜π‘–)β€˜π‘›))
65eqeq1d 2727 . . . . . . . . . . . . . 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 3168 . . . . . . . . . . . . 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 3330 . . . . . . . . . . 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 3169 . . . . . . . . 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 3599 . . . . . . . 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 19443 . . . . . . . 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 725 . . . 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 405 . . 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 7423 . . . . . . . . 9 ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) β†’ (-1↑(β™―β€˜π‘Š)) = (-1↑(β™―β€˜π‘Ÿ)))
2221adantr 479 . . . . . . . 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 729 . . . . . . 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 773 . . . . . . . . 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 725 . . . . . . . 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 773 . . . . . . . 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 479 . . . . . . . 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 775 . . . . . . . . . 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 480 . . . . . . . . . . 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 725 . . . . . . . . . 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 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†’ (β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ))
3534adantr 479 . . . . . . . . . 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 776 . . . . . . . . . . 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 479 . . . . . . . . . 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 21534 . . . . . . . . . . . 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 416 . . . . . . . . . . 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 2731 . . . . . . . . . 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 2734 . . . . . . . . . . . 12 (SymGrpβ€˜π‘) = 𝑍
4645oveq1i 7425 . . . . . . . . . . 11 ((SymGrpβ€˜π‘) Ξ£g π‘ˆ) = (𝑍 Ξ£g π‘ˆ)
4744, 46eqtrdi 2781 . . . . . . . . . 10 (𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ) β†’ 𝑄 = (𝑍 Ξ£g π‘ˆ))
4847adantl 480 . . . . . . . . 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 2765 . . . . . . . 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 19456 . . . . . . 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 2765 . . . . . 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 411 . . . . 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 411 . . . 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 3137 . . 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 411 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051  βˆƒwrex 3060  {crab 3419   βˆ– cdif 3937  {csn 4624  ran crn 5673   β†Ύ cres 5674  β€˜cfv 6542  (class class class)co 7415  Fincfn 8960  0cc0 11136  1c1 11137  -cneg 11473  ..^cfzo 13657  β†‘cexp 14056  β™―chash 14319  Word cword 14494  Basecbs 17177   Ξ£g cgsu 17419  SymGrpcsymg 19323  pmTrspcpmtr 19398
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 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-ot 4633  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  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-se 5628  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 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-tpos 8228  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-2o 8484  df-er 8721  df-map 8843  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-div 11900  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12501  df-xnn0 12573  df-z 12587  df-uz 12851  df-rp 13005  df-fz 13515  df-fzo 13658  df-seq 13997  df-exp 14057  df-hash 14320  df-word 14495  df-lsw 14543  df-concat 14551  df-s1 14576  df-substr 14621  df-pfx 14651  df-splice 14730  df-reverse 14739  df-s2 14829  df-struct 17113  df-sets 17130  df-slot 17148  df-ndx 17160  df-base 17178  df-ress 17207  df-plusg 17243  df-tset 17249  df-0g 17420  df-gsum 17421  df-mre 17563  df-mrc 17564  df-acs 17566  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-mhm 18737  df-submnd 18738  df-efmnd 18823  df-grp 18895  df-minusg 18896  df-subg 19080  df-ghm 19170  df-gim 19215  df-oppg 19299  df-symg 19324  df-pmtr 19399  df-psgn 19448
This theorem is referenced by:  psgndif  21536
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