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Theorem psgndiflemA 21028
Description: Lemma 2 for psgndif 21029. (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 6846 . . . . . . . . . . . 12 (𝑀 = π‘Š β†’ (β™―β€˜π‘€) = (β™―β€˜π‘Š))
21eqeq1d 2735 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ ((β™―β€˜π‘€) = (β™―β€˜π‘Ÿ) ↔ (β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ)))
31oveq2d 7377 . . . . . . . . . . . 12 (𝑀 = π‘Š β†’ (0..^(β™―β€˜π‘€)) = (0..^(β™―β€˜π‘Š)))
4 fveq1 6845 . . . . . . . . . . . . . . . 16 (𝑀 = π‘Š β†’ (π‘€β€˜π‘–) = (π‘Šβ€˜π‘–))
54fveq1d 6848 . . . . . . . . . . . . . . 15 (𝑀 = π‘Š β†’ ((π‘€β€˜π‘–)β€˜π‘›) = ((π‘Šβ€˜π‘–)β€˜π‘›))
65eqeq1d 2735 . . . . . . . . . . . . . 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 630 . . . . . . . . . . . 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 3318 . . . . . . . . . . 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 632 . . . . . . . . . 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 3580 . . . . . . . 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 19276 . . . . . . . 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 1134 . . . . . 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 726 . . . 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 408 . . 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 7369 . . . . . . . . 9 ((β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ) β†’ (-1↑(β™―β€˜π‘Š)) = (-1↑(β™―β€˜π‘Ÿ)))
2221adantr 482 . . . . . . . 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 730 . . . . . . 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 774 . . . . . . . . 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 726 . . . . . . . 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 774 . . . . . . . 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 1224 . . . . . . . . 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 482 . . . . . . . 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 776 . . . . . . . . . 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 1149 . . . . . . . . . . . 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 483 . . . . . . . . . . 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 726 . . . . . . . . . 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 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†’ (β™―β€˜π‘Š) = (β™―β€˜π‘Ÿ))
3534adantr 482 . . . . . . . . . 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 777 . . . . . . . . . . 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 482 . . . . . . . . . 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 21027 . . . . . . . . . . . 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 419 . . . . . . . . . . 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 2739 . . . . . . . . . 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 1378 . . . . . . . . 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 2742 . . . . . . . . . . . 12 (SymGrpβ€˜π‘) = 𝑍
4645oveq1i 7371 . . . . . . . . . . 11 ((SymGrpβ€˜π‘) Ξ£g π‘ˆ) = (𝑍 Ξ£g π‘ˆ)
4744, 46eqtrdi 2789 . . . . . . . . . 10 (𝑄 = ((SymGrpβ€˜π‘) Ξ£g π‘ˆ) β†’ 𝑄 = (𝑍 Ξ£g π‘ˆ))
4847adantl 483 . . . . . . . . 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 2773 . . . . . . . 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 19289 . . . . . . 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 2773 . . . . . 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 414 . . . . 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 414 . . . 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 414 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  {crab 3406   βˆ– cdif 3911  {csn 4590  ran crn 5638   β†Ύ cres 5639  β€˜cfv 6500  (class class class)co 7361  Fincfn 8889  0cc0 11059  1c1 11060  -cneg 11394  ..^cfzo 13576  β†‘cexp 13976  β™―chash 14239  Word cword 14411  Basecbs 17091   Ξ£g cgsu 17330  SymGrpcsymg 19156  pmTrspcpmtr 19231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-xor 1511  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-ot 4599  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-tpos 8161  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-xnn0 12494  df-z 12508  df-uz 12772  df-rp 12924  df-fz 13434  df-fzo 13577  df-seq 13916  df-exp 13977  df-hash 14240  df-word 14412  df-lsw 14460  df-concat 14468  df-s1 14493  df-substr 14538  df-pfx 14568  df-splice 14647  df-reverse 14656  df-s2 14746  df-struct 17027  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-tset 17160  df-0g 17331  df-gsum 17332  df-mre 17474  df-mrc 17475  df-acs 17477  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-mhm 18609  df-submnd 18610  df-efmnd 18687  df-grp 18759  df-minusg 18760  df-subg 18933  df-ghm 19014  df-gim 19057  df-oppg 19132  df-symg 19157  df-pmtr 19232  df-psgn 19281
This theorem is referenced by:  psgndif  21029
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