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Theorem psgndiflemA 19766
 Description: Lemma 2 for psgndif 19767. (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 psgnfix.t . . . . . . . . . 10 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
2 psgnfix.r . . . . . . . . . 10 𝑅 = ran (pmTrsp‘𝑁)
31, 2pmtrdifwrdel2 17729 . . . . . . . . 9 (𝐾𝑁 → ∀𝑤 ∈ Word 𝑇𝑟 ∈ Word 𝑅((#‘𝑤) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛))))
4 fveq2 6103 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → (#‘𝑤) = (#‘𝑊))
54eqeq1d 2612 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ((#‘𝑤) = (#‘𝑟) ↔ (#‘𝑊) = (#‘𝑟)))
64oveq2d 6565 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → (0..^(#‘𝑤)) = (0..^(#‘𝑊)))
7 fveq1 6102 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑊 → (𝑤𝑖) = (𝑊𝑖))
87fveq1d 6105 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑊 → ((𝑤𝑖)‘𝑛) = ((𝑊𝑖)‘𝑛))
98eqeq1d 2612 . . . . . . . . . . . . . . 15 (𝑤 = 𝑊 → (((𝑤𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛) ↔ ((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))
109ralbidv 2969 . . . . . . . . . . . . . 14 (𝑤 = 𝑊 → (∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛) ↔ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))
1110anbi2d 736 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → ((((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)) ↔ (((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛))))
126, 11raleqbidv 3129 . . . . . . . . . . . 12 (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))))
135, 12anbi12d 743 . . . . . . . . . . 11 (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)))))
1413rexbidv 3034 . . . . . . . . . 10 (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)))))
1514rspccv 3279 . . . . . . . . 9 (∀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)))))
163, 15syl 17 . . . . . . . 8 (𝐾𝑁 → (𝑊 ∈ Word 𝑇 → ∃𝑟 ∈ Word 𝑅((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))))
1716com12 32 . . . . . . 7 (𝑊 ∈ Word 𝑇 → (𝐾𝑁 → ∃𝑟 ∈ Word 𝑅((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛)))))
18173ad2ant1 1075 . . . . . 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)))))
1918com12 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)))))
2019ad2antlr 759 . . . 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)))))
2120imp 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))))
22 oveq2 6557 . . . . . . . . 9 ((#‘𝑊) = (#‘𝑟) → (-1↑(#‘𝑊)) = (-1↑(#‘𝑟)))
2322adantr 480 . . . . . . . 8 (((#‘𝑊) = (#‘𝑟) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(((𝑟𝑖)‘𝐾) = 𝐾 ∧ ∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑟𝑖)‘𝑛))) → (-1↑(#‘𝑊)) = (-1↑(#‘𝑟)))
2423ad3antlr 763 . . . . . . 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)))
25 psgnfix.z . . . . . . . 8 𝑍 = (SymGrp‘𝑁)
26 simplll 794 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}) ∧ (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅)) → 𝑁 ∈ Fin)
2726ad2antlr 759 . . . . . . . 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)
28 simplll 794 . . . . . . . 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)
29 simprr3 1104 . . . . . . . . 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)
3029adantr 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)
31 simplrl 796 . . . . . . . . . 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}))
32 3simpa 1051 . . . . . . . . . . . 12 ((𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊) ∧ 𝑈 ∈ Word 𝑅) → (𝑊 ∈ Word 𝑇 ∧ (𝑄 ↾ (𝑁 ∖ {𝐾})) = (𝑆 Σg 𝑊)))
3332adantl 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)))
3433ad2antlr 759 . . . . . . . . . 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)))
35 simplrl 796 . . . . . . . . . . 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))
3635adantr 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))
37 simplrr 797 . . . . . . . . . . 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)))
3837adantr 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)))
39 psgnfix.p . . . . . . . . . . . . 13 𝑃 = (Base‘(SymGrp‘𝑁))
40 psgnfix.s . . . . . . . . . . . . 13 𝑆 = (SymGrp‘(𝑁 ∖ {𝐾}))
4139, 1, 40, 25, 2psgndiflemB 19765 . . . . . . . . . . . 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))))
4241imp31 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))
4342eqcomd 2616 . . . . . . . . . 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)
4431, 34, 28, 36, 38, 43syl23anc 1325 . . . . . . . . 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)
45 id 22 . . . . . . . . . . 11 (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → 𝑄 = ((SymGrp‘𝑁) Σg 𝑈))
4625eqcomi 2619 . . . . . . . . . . . 12 (SymGrp‘𝑁) = 𝑍
4746oveq1i 6559 . . . . . . . . . . 11 ((SymGrp‘𝑁) Σg 𝑈) = (𝑍 Σg 𝑈)
4845, 47syl6eq 2660 . . . . . . . . . 10 (𝑄 = ((SymGrp‘𝑁) Σg 𝑈) → 𝑄 = (𝑍 Σg 𝑈))
4948adantl 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))
5044, 49eqtrd 2644 . . . . . . . 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))
5125, 2, 27, 28, 30, 50psgnuni 17742 . . . . . . 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)))
5224, 51eqtrd 2644 . . . . . 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)))
5352ex 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))))
5453ex 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)))))
5554rexlimiva 3010 . . 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)))))
5621, 55mpcom 37 . 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))))
5756ex 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 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900   ∖ cdif 3537  {csn 4125  ran crn 5039   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  0cc0 9815  1c1 9816  -cneg 10146  ..^cfzo 12334  ↑cexp 12722  #chash 12979  Word cword 13146  Basecbs 15695   Σg cgsu 15924  SymGrpcsymg 17620  pmTrspcpmtr 17684 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-xor 1457  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-splice 13159  df-reverse 13160  df-s2 13444  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-tset 15787  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-subg 17414  df-ghm 17481  df-gim 17524  df-oppg 17599  df-symg 17621  df-pmtr 17685  df-psgn 17734 This theorem is referenced by:  psgndif  19767
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