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Theorem cyc3co2 30836
 Description: Represent a 3-cycle as a composition of two 2-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.)
Hypotheses
Ref Expression
cycpm3.c 𝐶 = (toCyc‘𝐷)
cycpm3.s 𝑆 = (SymGrp‘𝐷)
cycpm3.d (𝜑𝐷𝑉)
cycpm3.i (𝜑𝐼𝐷)
cycpm3.j (𝜑𝐽𝐷)
cycpm3.k (𝜑𝐾𝐷)
cycpm3.1 (𝜑𝐼𝐽)
cycpm3.2 (𝜑𝐽𝐾)
cycpm3.3 (𝜑𝐾𝐼)
cyc3co2.t · = (+g𝑆)
Assertion
Ref Expression
cyc3co2 (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) = ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)))

Proof of Theorem cyc3co2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 cycpm3.c . . . . 5 𝐶 = (toCyc‘𝐷)
2 cycpm3.s . . . . 5 𝑆 = (SymGrp‘𝐷)
3 cycpm3.d . . . . 5 (𝜑𝐷𝑉)
4 cycpm3.i . . . . 5 (𝜑𝐼𝐷)
5 cycpm3.j . . . . 5 (𝜑𝐽𝐷)
6 cycpm3.k . . . . 5 (𝜑𝐾𝐷)
7 cycpm3.1 . . . . 5 (𝜑𝐼𝐽)
8 cycpm3.2 . . . . 5 (𝜑𝐽𝐾)
9 cycpm3.3 . . . . 5 (𝜑𝐾𝐼)
101, 2, 3, 4, 5, 6, 7, 8, 9cycpm3cl 30831 . . . 4 (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (Base‘𝑆))
11 eqid 2801 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
122, 11symgbasf 18500 . . . 4 ((𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (Base‘𝑆) → (𝐶‘⟨“𝐼𝐽𝐾”⟩):𝐷𝐷)
1310, 12syl 17 . . 3 (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩):𝐷𝐷)
1413ffnd 6492 . 2 (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) Fn 𝐷)
152symggrp 18524 . . . . . 6 (𝐷𝑉𝑆 ∈ Grp)
163, 15syl 17 . . . . 5 (𝜑𝑆 ∈ Grp)
179necomd 3045 . . . . . 6 (𝜑𝐼𝐾)
181, 3, 4, 6, 17, 2cycpm2cl 30816 . . . . 5 (𝜑 → (𝐶‘⟨“𝐼𝐾”⟩) ∈ (Base‘𝑆))
191, 3, 4, 5, 7, 2cycpm2cl 30816 . . . . 5 (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆))
20 cyc3co2.t . . . . . 6 · = (+g𝑆)
2111, 20grpcl 18107 . . . . 5 ((𝑆 ∈ Grp ∧ (𝐶‘⟨“𝐼𝐾”⟩) ∈ (Base‘𝑆) ∧ (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆)) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) ∈ (Base‘𝑆))
2216, 18, 19, 21syl3anc 1368 . . . 4 (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) ∈ (Base‘𝑆))
232, 11symgbasf 18500 . . . 4 (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) ∈ (Base‘𝑆) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)):𝐷𝐷)
2422, 23syl 17 . . 3 (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)):𝐷𝐷)
2524ffnd 6492 . 2 (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) Fn 𝐷)
261, 2, 3, 4, 5, 6, 7, 8, 9cyc3fv1 30833 . . . . . . . 8 (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼) = 𝐽)
2726adantr 484 . . . . . . 7 ((𝜑𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼) = 𝐽)
28 simpr 488 . . . . . . . 8 ((𝜑𝑥 = 𝐼) → 𝑥 = 𝐼)
2928fveq2d 6653 . . . . . . 7 ((𝜑𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼))
302, 11, 20symgov 18508 . . . . . . . . . . 11 (((𝐶‘⟨“𝐼𝐾”⟩) ∈ (Base‘𝑆) ∧ (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆)) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
3118, 19, 30syl2anc 587 . . . . . . . . . 10 (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
3231adantr 484 . . . . . . . . 9 ((𝜑𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
3332fveq1d 6651 . . . . . . . 8 ((𝜑𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
342, 11symgbasf 18500 . . . . . . . . . . . 12 ((𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆) → (𝐶‘⟨“𝐼𝐽”⟩):𝐷𝐷)
3519, 34syl 17 . . . . . . . . . . 11 (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩):𝐷𝐷)
3635ffund 6495 . . . . . . . . . 10 (𝜑 → Fun (𝐶‘⟨“𝐼𝐽”⟩))
374adantr 484 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐼) → 𝐼𝐷)
3834fdmd 6501 . . . . . . . . . . . . 13 ((𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
3919, 38syl 17 . . . . . . . . . . . 12 (𝜑 → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
4039adantr 484 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐼) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
4137, 28, 403eltr4d 2908 . . . . . . . . . 10 ((𝜑𝑥 = 𝐼) → 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩))
42 fvco 6740 . . . . . . . . . 10 ((Fun (𝐶‘⟨“𝐼𝐽”⟩) ∧ 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩)) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
4336, 41, 42syl2an2r 684 . . . . . . . . 9 ((𝜑𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
4428fveq2d 6653 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼))
451, 3, 4, 5, 7, 2cyc2fv1 30817 . . . . . . . . . . . 12 (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼) = 𝐽)
4645adantr 484 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼) = 𝐽)
4744, 46eqtrd 2836 . . . . . . . . . 10 ((𝜑𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝐽)
4847fveq2d 6653 . . . . . . . . 9 ((𝜑𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐽))
498necomd 3045 . . . . . . . . . . 11 (𝜑𝐾𝐽)
507necomd 3045 . . . . . . . . . . 11 (𝜑𝐽𝐼)
511, 2, 3, 4, 6, 5, 17, 49, 50cyc2fvx 30830 . . . . . . . . . 10 (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐽) = 𝐽)
5251adantr 484 . . . . . . . . 9 ((𝜑𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐽) = 𝐽)
5343, 48, 523eqtrd 2840 . . . . . . . 8 ((𝜑𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐽)
5433, 53eqtrd 2836 . . . . . . 7 ((𝜑𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐽)
5527, 29, 543eqtr4d 2846 . . . . . 6 ((𝜑𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
5655adantlr 714 . . . . 5 (((𝜑𝑥 ∈ {𝐼, 𝐽, 𝐾}) ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
571, 2, 3, 4, 5, 6, 7, 8, 9cyc3fv2 30834 . . . . . . . 8 (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽) = 𝐾)
5857adantr 484 . . . . . . 7 ((𝜑𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽) = 𝐾)
59 simpr 488 . . . . . . . 8 ((𝜑𝑥 = 𝐽) → 𝑥 = 𝐽)
6059fveq2d 6653 . . . . . . 7 ((𝜑𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽))
6131adantr 484 . . . . . . . . 9 ((𝜑𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
6261fveq1d 6651 . . . . . . . 8 ((𝜑𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
635adantr 484 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐽) → 𝐽𝐷)
6439adantr 484 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐽) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
6563, 59, 643eltr4d 2908 . . . . . . . . . 10 ((𝜑𝑥 = 𝐽) → 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩))
6636, 65, 42syl2an2r 684 . . . . . . . . 9 ((𝜑𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
6759fveq2d 6653 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽))
681, 3, 4, 5, 7, 2cyc2fv2 30818 . . . . . . . . . . . 12 (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼)
6968adantr 484 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼)
7067, 69eqtrd 2836 . . . . . . . . . 10 ((𝜑𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝐼)
7170fveq2d 6653 . . . . . . . . 9 ((𝜑𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐼))
721, 3, 4, 6, 17, 2cyc2fv1 30817 . . . . . . . . . 10 (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐼) = 𝐾)
7372adantr 484 . . . . . . . . 9 ((𝜑𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐼) = 𝐾)
7466, 71, 733eqtrd 2840 . . . . . . . 8 ((𝜑𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐾)
7562, 74eqtrd 2836 . . . . . . 7 ((𝜑𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐾)
7658, 60, 753eqtr4d 2846 . . . . . 6 ((𝜑𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
7776adantlr 714 . . . . 5 (((𝜑𝑥 ∈ {𝐼, 𝐽, 𝐾}) ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
781, 2, 3, 4, 5, 6, 7, 8, 9cyc3fv3 30835 . . . . . . . 8 (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾) = 𝐼)
7978adantr 484 . . . . . . 7 ((𝜑𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾) = 𝐼)
80 simpr 488 . . . . . . . 8 ((𝜑𝑥 = 𝐾) → 𝑥 = 𝐾)
8180fveq2d 6653 . . . . . . 7 ((𝜑𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾))
8231adantr 484 . . . . . . . . 9 ((𝜑𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
8382fveq1d 6651 . . . . . . . 8 ((𝜑𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
846adantr 484 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐾) → 𝐾𝐷)
8539adantr 484 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐾) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
8684, 80, 853eltr4d 2908 . . . . . . . . . 10 ((𝜑𝑥 = 𝐾) → 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩))
8736, 86, 42syl2an2r 684 . . . . . . . . 9 ((𝜑𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
8880fveq2d 6653 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾))
891, 2, 3, 4, 5, 6, 7, 8, 9cyc2fvx 30830 . . . . . . . . . . . 12 (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾) = 𝐾)
9089adantr 484 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾) = 𝐾)
9188, 90eqtrd 2836 . . . . . . . . . 10 ((𝜑𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝐾)
9291fveq2d 6653 . . . . . . . . 9 ((𝜑𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐾))
931, 3, 4, 6, 17, 2cyc2fv2 30818 . . . . . . . . . 10 (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐾) = 𝐼)
9493adantr 484 . . . . . . . . 9 ((𝜑𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐾) = 𝐼)
9587, 92, 943eqtrd 2840 . . . . . . . 8 ((𝜑𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐼)
9683, 95eqtrd 2836 . . . . . . 7 ((𝜑𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐼)
9779, 81, 963eqtr4d 2846 . . . . . 6 ((𝜑𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
9897adantlr 714 . . . . 5 (((𝜑𝑥 ∈ {𝐼, 𝐽, 𝐾}) ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
99 eltpi 4588 . . . . . 6 (𝑥 ∈ {𝐼, 𝐽, 𝐾} → (𝑥 = 𝐼𝑥 = 𝐽𝑥 = 𝐾))
10099adantl 485 . . . . 5 ((𝜑𝑥 ∈ {𝐼, 𝐽, 𝐾}) → (𝑥 = 𝐼𝑥 = 𝐽𝑥 = 𝐾))
10156, 77, 98, 100mpjao3dan 1428 . . . 4 ((𝜑𝑥 ∈ {𝐼, 𝐽, 𝐾}) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
102101adantlr 714 . . 3 (((𝜑𝑥𝐷) ∧ 𝑥 ∈ {𝐼, 𝐽, 𝐾}) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
10335adantr 484 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → (𝐶‘⟨“𝐼𝐽”⟩):𝐷𝐷)
104103ffund 6495 . . . . . . 7 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → Fun (𝐶‘⟨“𝐼𝐽”⟩))
105 simpr 488 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾}))
106105eldifad 3896 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → 𝑥𝐷)
10739adantr 484 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
108106, 107eleqtrrd 2896 . . . . . . 7 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩))
109104, 108, 42syl2anc 587 . . . . . 6 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
1103adantr 484 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → 𝐷𝑉)
1114, 5s2cld 14228 . . . . . . . . 9 (𝜑 → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
112111adantr 484 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
1134, 5, 7s2f1 30651 . . . . . . . . 9 (𝜑 → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1𝐷)
114113adantr 484 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1𝐷)
115 tpid1g 4668 . . . . . . . . . . . . 13 (𝐼𝐷𝐼 ∈ {𝐼, 𝐽, 𝐾})
1164, 115syl 17 . . . . . . . . . . . 12 (𝜑𝐼 ∈ {𝐼, 𝐽, 𝐾})
117 tpid2g 4670 . . . . . . . . . . . . 13 (𝐽𝐷𝐽 ∈ {𝐼, 𝐽, 𝐾})
1185, 117syl 17 . . . . . . . . . . . 12 (𝜑𝐽 ∈ {𝐼, 𝐽, 𝐾})
119116, 118prssd 4718 . . . . . . . . . . 11 (𝜑 → {𝐼, 𝐽} ⊆ {𝐼, 𝐽, 𝐾})
1204, 5s2rn 30650 . . . . . . . . . . . 12 (𝜑 → ran ⟨“𝐼𝐽”⟩ = {𝐼, 𝐽})
121120eqcomd 2807 . . . . . . . . . . 11 (𝜑 → {𝐼, 𝐽} = ran ⟨“𝐼𝐽”⟩)
1224, 5, 6s3rn 30652 . . . . . . . . . . . 12 (𝜑 → ran ⟨“𝐼𝐽𝐾”⟩ = {𝐼, 𝐽, 𝐾})
123122eqcomd 2807 . . . . . . . . . . 11 (𝜑 → {𝐼, 𝐽, 𝐾} = ran ⟨“𝐼𝐽𝐾”⟩)
124119, 121, 1233sstr3d 3964 . . . . . . . . . 10 (𝜑 → ran ⟨“𝐼𝐽”⟩ ⊆ ran ⟨“𝐼𝐽𝐾”⟩)
125124adantr 484 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ran ⟨“𝐼𝐽”⟩ ⊆ ran ⟨“𝐼𝐽𝐾”⟩)
126105eldifbd 3897 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ {𝐼, 𝐽, 𝐾})
127122adantr 484 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ran ⟨“𝐼𝐽𝐾”⟩ = {𝐼, 𝐽, 𝐾})
128126, 127neleqtrrd 2915 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ ran ⟨“𝐼𝐽𝐾”⟩)
129125, 128ssneldd 3921 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ ran ⟨“𝐼𝐽”⟩)
1301, 110, 112, 114, 106, 129cycpmfv3 30811 . . . . . . 7 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝑥)
131130fveq2d 6653 . . . . . 6 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝑥))
1324, 6s2cld 14228 . . . . . . . 8 (𝜑 → ⟨“𝐼𝐾”⟩ ∈ Word 𝐷)
133132adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐾”⟩ ∈ Word 𝐷)
1344, 6, 17s2f1 30651 . . . . . . . 8 (𝜑 → ⟨“𝐼𝐾”⟩:dom ⟨“𝐼𝐾”⟩–1-1𝐷)
135134adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐾”⟩:dom ⟨“𝐼𝐾”⟩–1-1𝐷)
136 tpid3g 4671 . . . . . . . . . . . 12 (𝐾𝐷𝐾 ∈ {𝐼, 𝐽, 𝐾})
1376, 136syl 17 . . . . . . . . . . 11 (𝜑𝐾 ∈ {𝐼, 𝐽, 𝐾})
138116, 137prssd 4718 . . . . . . . . . 10 (𝜑 → {𝐼, 𝐾} ⊆ {𝐼, 𝐽, 𝐾})
139138adantr 484 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → {𝐼, 𝐾} ⊆ {𝐼, 𝐽, 𝐾})
1404, 6s2rn 30650 . . . . . . . . . . 11 (𝜑 → ran ⟨“𝐼𝐾”⟩ = {𝐼, 𝐾})
141140eqcomd 2807 . . . . . . . . . 10 (𝜑 → {𝐼, 𝐾} = ran ⟨“𝐼𝐾”⟩)
142141adantr 484 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → {𝐼, 𝐾} = ran ⟨“𝐼𝐾”⟩)
143123adantr 484 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → {𝐼, 𝐽, 𝐾} = ran ⟨“𝐼𝐽𝐾”⟩)
144139, 142, 1433sstr3d 3964 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ran ⟨“𝐼𝐾”⟩ ⊆ ran ⟨“𝐼𝐽𝐾”⟩)
145144, 128ssneldd 3921 . . . . . . 7 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ ran ⟨“𝐼𝐾”⟩)
1461, 110, 133, 135, 106, 145cycpmfv3 30811 . . . . . 6 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝑥) = 𝑥)
147109, 131, 1463eqtrd 2840 . . . . 5 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝑥)
14831adantr 484 . . . . . 6 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
149148fveq1d 6651 . . . . 5 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
1504, 5, 6s3cld 14229 . . . . . . 7 (𝜑 → ⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷)
151150adantr 484 . . . . . 6 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷)
1524, 5, 6, 7, 8, 9s3f1 30653 . . . . . . 7 (𝜑 → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1𝐷)
153152adantr 484 . . . . . 6 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1𝐷)
1541, 110, 151, 153, 106, 128cycpmfv3 30811 . . . . 5 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = 𝑥)
155147, 149, 1543eqtr4rd 2847 . . . 4 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
156155adantlr 714 . . 3 (((𝜑𝑥𝐷) ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
157 tpssi 4732 . . . . . . . 8 ((𝐼𝐷𝐽𝐷𝐾𝐷) → {𝐼, 𝐽, 𝐾} ⊆ 𝐷)
1584, 5, 6, 157syl3anc 1368 . . . . . . 7 (𝜑 → {𝐼, 𝐽, 𝐾} ⊆ 𝐷)
159 undif 4391 . . . . . . 7 ({𝐼, 𝐽, 𝐾} ⊆ 𝐷 ↔ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) = 𝐷)
160158, 159sylib 221 . . . . . 6 (𝜑 → ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) = 𝐷)
161160eleq2d 2878 . . . . 5 (𝜑 → (𝑥 ∈ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) ↔ 𝑥𝐷))
162161biimpar 481 . . . 4 ((𝜑𝑥𝐷) → 𝑥 ∈ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})))
163 elun 4079 . . . 4 (𝑥 ∈ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) ↔ (𝑥 ∈ {𝐼, 𝐽, 𝐾} ∨ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})))
164162, 163sylib 221 . . 3 ((𝜑𝑥𝐷) → (𝑥 ∈ {𝐼, 𝐽, 𝐾} ∨ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})))
165102, 156, 164mpjaodan 956 . 2 ((𝜑𝑥𝐷) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
16614, 25, 165eqfnfvd 6786 1 (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) = ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   ∨ w3o 1083   = wceq 1538   ∈ wcel 2112   ≠ wne 2990   ∖ cdif 3881   ∪ cun 3882   ⊆ wss 3884  {cpr 4530  {ctp 4532  dom cdm 5523  ran crn 5524   ∘ ccom 5527  Fun wfun 6322  ⟶wf 6324  –1-1→wf1 6325  ‘cfv 6328  (class class class)co 7139  Word cword 13861  ⟨“cs2 14198  ⟨“cs3 14199  Basecbs 16479  +gcplusg 16561  Grpcgrp 18099  SymGrpcsymg 18491  toCycctocyc 30802 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-inf 8895  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-fz 12890  df-fzo 13033  df-fl 13161  df-mod 13237  df-hash 13691  df-word 13862  df-concat 13918  df-s1 13945  df-substr 13998  df-pfx 14028  df-csh 14146  df-s2 14205  df-s3 14206  df-struct 16481  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-ress 16487  df-plusg 16574  df-tset 16580  df-0g 16711  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-efmnd 18030  df-grp 18102  df-symg 18492  df-tocyc 30803 This theorem is referenced by:  cyc3evpm  30846  cyc3genpmlem  30847
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