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Theorem psgndiflemA 19995
Description: Lemma 2 for psgndif 19996. (Contributed by AV, 31-Jan-2019.)
Hypotheses
Ref Expression
psgnfix.p 𝑃 = (Base‘(SymGrp‘𝑁))
psgnfix.t 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
psgnfix.s 𝑆 = (SymGrp‘(𝑁 ∖ {𝐾}))
psgnfix.z 𝑍 = (SymGrp‘𝑁)
psgnfix.r 𝑅 = ran (pmTrsp‘𝑁)
Assertion
Ref Expression
psgndiflemA (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) → ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅) → (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → (-1↑(#‘𝑊)) = (-1↑(#‘𝑈)))))
Distinct variable groups:   𝐾,𝑞   𝑃,𝑞   𝑄,𝑞
Allowed substitution hints:   𝑅(𝑞)   𝑆(𝑞)   𝑇(𝑞)   𝑈(𝑞)   𝑁(𝑞)   𝑊(𝑞)   𝑍(𝑞)

Proof of Theorem psgndiflemA
Dummy variables 𝑤 𝑖 𝑛 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6229 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (#‘𝑤) = (#‘𝑊))
21eqeq1d 2653 . . . . . . . . . . 11 (𝑤 = 𝑊 → ((#‘𝑤) = (#‘𝑟) ↔ (#‘𝑊) = (#‘𝑟)))
31oveq2d 6706 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (0..^(#‘𝑤)) = (0..^(#‘𝑊)))
4 fveq1 6228 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑊 → (𝑤𝑖) = (𝑊𝑖))
54fveq1d 6231 . . . . . . . . . . . . . . 15 (𝑤 = 𝑊 → ((𝑤𝑖)‘𝑛) = ((𝑊𝑖)‘𝑛))
65eqeq1d 2653 . . . . . . . . . . . . . 14 (𝑤 = 𝑊 → (((𝑤𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛) ↔ ((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))
76ralbidv 3015 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → (∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛) ↔ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))
87anbi2d 740 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ((((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)) ↔ (((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛))))
93, 8raleqbidv 3182 . . . . . . . . . . 11 (𝑤 = 𝑊 → (∀𝑖 ∈ (0..^(#‘𝑤))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)) ↔ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛))))
102, 9anbi12d 747 . . . . . . . . . 10 (𝑤 = 𝑊 → (((#‘𝑤) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛))) ↔ ((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))))
1110rexbidv 3081 . . . . . . . . 9 (𝑤 = 𝑊 → (∃𝑟 ∈ Word 𝑅((#‘𝑤) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛))) ↔ ∃𝑟 ∈ Word 𝑅((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))))
1211rspccv 3337 . . . . . . . 8 (∀𝑤 ∈ Word 𝑇𝑟 ∈ Word 𝑅((#‘𝑤) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛))) → (𝑊 ∈ Word 𝑇 → ∃𝑟 ∈ Word 𝑅((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))))
13 psgnfix.t . . . . . . . . 9 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
14 psgnfix.r . . . . . . . . 9 𝑅 = ran (pmTrsp‘𝑁)
1513, 14pmtrdifwrdel2 17952 . . . . . . . 8 (𝐾𝑁 → ∀𝑤 ∈ Word 𝑇𝑟 ∈ Word 𝑅((#‘𝑤) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛))))
1612, 15syl11 33 . . . . . . 7 (𝑊 ∈ Word 𝑇 → (𝐾𝑁 → ∃𝑟 ∈ Word 𝑅((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))))
17163ad2ant1 1102 . . . . . 6 ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅) → (𝐾𝑁 → ∃𝑟 ∈ Word 𝑅((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))))
1817com12 32 . . . . 5 (𝐾𝑁 → ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅) → ∃𝑟 ∈ Word 𝑅((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))))
1918ad2antlr 763 . . . 4 (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) → ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅) → ∃𝑟 ∈ Word 𝑅((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))))
2019imp 444 . . 3 ((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅)) → ∃𝑟 ∈ Word 𝑅((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛))))
21 oveq2 6698 . . . . . . . . 9 ((#‘𝑊) = (#‘𝑟) → (-1↑(#‘𝑊)) = (-1↑(#‘𝑟)))
2221adantr 480 . . . . . . . 8 (((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛))) → (-1↑(#‘𝑊)) = (-1↑(#‘𝑟)))
2322ad3antlr 767 . . . . . . 7 ((((𝑟 ∈ Word 𝑅 ∧ ((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → (-1↑(#‘𝑊)) = (-1↑(#‘𝑟)))
24 psgnfix.z . . . . . . . 8 𝑍 = (SymGrp‘𝑁)
25 simplll 813 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅)) → 𝑁 ∈ Fin)
2625ad2antlr 763 . . . . . . . 8 ((((𝑟 ∈ Word 𝑅 ∧ ((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → 𝑁 ∈ Fin)
27 simplll 813 . . . . . . . 8 ((((𝑟 ∈ Word 𝑅 ∧ ((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → 𝑟 ∈ Word 𝑅)
28 simprr3 1131 . . . . . . . . 9 (((𝑟 ∈ Word 𝑅 ∧ ((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) → 𝑈 ∈ Word 𝑅)
2928adantr 480 . . . . . . . 8 ((((𝑟 ∈ Word 𝑅 ∧ ((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → 𝑈 ∈ Word 𝑅)
30 simplrl 817 . . . . . . . . . 10 ((((𝑟 ∈ Word 𝑅 ∧ ((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → ((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}))
31 3simpa 1078 . . . . . . . . . . . 12 ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅) → (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)))
3231adantl 481 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅)) → (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)))
3332ad2antlr 763 . . . . . . . . . 10 ((((𝑟 ∈ Word 𝑅 ∧ ((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)))
34 simplrl 817 . . . . . . . . . . 11 (((𝑟 ∈ Word 𝑅 ∧ ((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) → (#‘𝑊) = (#‘𝑟))
3534adantr 480 . . . . . . . . . 10 ((((𝑟 ∈ Word 𝑅 ∧ ((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → (#‘𝑊) = (#‘𝑟))
36 simplrr 818 . . . . . . . . . . 11 (((𝑟 ∈ Word 𝑅 ∧ ((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) → ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))
3736adantr 480 . . . . . . . . . 10 ((((𝑟 ∈ Word 𝑅 ∧ ((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))
38 psgnfix.p . . . . . . . . . . . . 13 𝑃 = (Base‘(SymGrp‘𝑁))
39 psgnfix.s . . . . . . . . . . . . 13 𝑆 = (SymGrp‘(𝑁 ∖ {𝐾}))
4038, 13, 39, 24, 14psgndiflemB 19994 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) → ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)) → ((𝑟 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛))) → 𝑄 = (𝑍 Σg 𝑟))))
4140imp31 447 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑟 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) → 𝑄 = (𝑍 Σg 𝑟))
4241eqcomd 2657 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊))) ∧ (𝑟 ∈ Word 𝑅 ∧ (#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) → (𝑍 Σg 𝑟) = 𝑄)
4330, 33, 27, 35, 37, 42syl23anc 1373 . . . . . . . . 9 ((((𝑟 ∈ Word 𝑅 ∧ ((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → (𝑍 Σg 𝑟) = 𝑄)
44 id 22 . . . . . . . . . . 11 (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → 𝑄 = ((SymGrp‘𝑁) Σg 𝑈))
4524eqcomi 2660 . . . . . . . . . . . 12 (SymGrp‘𝑁) = 𝑍
4645oveq1i 6700 . . . . . . . . . . 11 ((SymGrp‘𝑁) Σg 𝑈) = (𝑍 Σg 𝑈)
4744, 46syl6eq 2701 . . . . . . . . . 10 (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → 𝑄 = (𝑍 Σg 𝑈))
4847adantl 481 . . . . . . . . 9 ((((𝑟 ∈ Word 𝑅 ∧ ((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → 𝑄 = (𝑍 Σg 𝑈))
4943, 48eqtrd 2685 . . . . . . . 8 ((((𝑟 ∈ Word 𝑅 ∧ ((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → (𝑍 Σg 𝑟) = (𝑍 Σg 𝑈))
5024, 14, 26, 27, 29, 49psgnuni 17965 . . . . . . 7 ((((𝑟 ∈ Word 𝑅 ∧ ((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → (-1↑(#‘𝑟)) = (-1↑(#‘𝑈)))
5123, 50eqtrd 2685 . . . . . 6 ((((𝑟 ∈ Word 𝑅 ∧ ((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) ∧ 𝑄 = ((SymGrp‘𝑁) Σg 𝑈)) → (-1↑(#‘𝑊)) = (-1↑(#‘𝑈)))
5251ex 449 . . . . 5 (((𝑟 ∈ Word 𝑅 ∧ ((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) ∧ (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅))) → (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → (-1↑(#‘𝑊)) = (-1↑(#‘𝑈))))
5352ex 449 . . . 4 ((𝑟 ∈ Word 𝑅 ∧ ((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))) → ((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅)) → (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → (-1↑(#‘𝑊)) = (-1↑(#‘𝑈)))))
5453rexlimiva 3057 . . 3 (∃𝑟 ∈ Word 𝑅((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛))) → ((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅)) → (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → (-1↑(#‘𝑊)) = (-1↑(#‘𝑈)))))
5520, 54mpcom 38 . 2 ((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅)) → (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → (-1↑(#‘𝑊)) = (-1↑(#‘𝑈))))
5655ex 449 1 (((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) → ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅) → (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → (-1↑(#‘𝑊)) = (-1↑(#‘𝑈)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  cdif 3604  {csn 4210  ran crn 5144  cres 5145  cfv 5926  (class class class)co 6690  Fincfn 7997  0cc0 9974  1c1 9975  -cneg 10305  ..^cfzo 12504  cexp 12900  #chash 13157  Word cword 13323  Basecbs 15904   Σg cgsu 16148  SymGrpcsymg 17843  pmTrspcpmtr 17907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-xor 1505  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-ot 4219  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-word 13331  df-lsw 13332  df-concat 13333  df-s1 13334  df-substr 13335  df-splice 13336  df-reverse 13337  df-s2 13639  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-tset 16007  df-0g 16149  df-gsum 16150  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-subg 17638  df-ghm 17705  df-gim 17748  df-oppg 17822  df-symg 17844  df-pmtr 17908  df-psgn 17957
This theorem is referenced by:  psgndif  19996
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