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Theorem cyc3co2 33142
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 33137 . . . 4 (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (Base‘𝑆))
11 eqid 2734 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
122, 11symgbasf 19407 . . . 4 ((𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (Base‘𝑆) → (𝐶‘⟨“𝐼𝐽𝐾”⟩):𝐷𝐷)
1310, 12syl 17 . . 3 (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩):𝐷𝐷)
1413ffnd 6737 . 2 (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) Fn 𝐷)
152symggrp 19432 . . . . . 6 (𝐷𝑉𝑆 ∈ Grp)
163, 15syl 17 . . . . 5 (𝜑𝑆 ∈ Grp)
179necomd 2993 . . . . . 6 (𝜑𝐼𝐾)
181, 3, 4, 6, 17, 2cycpm2cl 33122 . . . . 5 (𝜑 → (𝐶‘⟨“𝐼𝐾”⟩) ∈ (Base‘𝑆))
191, 3, 4, 5, 7, 2cycpm2cl 33122 . . . . 5 (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆))
20 cyc3co2.t . . . . . 6 · = (+g𝑆)
2111, 20grpcl 18971 . . . . 5 ((𝑆 ∈ Grp ∧ (𝐶‘⟨“𝐼𝐾”⟩) ∈ (Base‘𝑆) ∧ (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆)) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) ∈ (Base‘𝑆))
2216, 18, 19, 21syl3anc 1370 . . . 4 (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) ∈ (Base‘𝑆))
232, 11symgbasf 19407 . . . 4 (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) ∈ (Base‘𝑆) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)):𝐷𝐷)
2422, 23syl 17 . . 3 (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)):𝐷𝐷)
2524ffnd 6737 . 2 (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) Fn 𝐷)
261, 2, 3, 4, 5, 6, 7, 8, 9cyc3fv1 33139 . . . . . . . 8 (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼) = 𝐽)
2726adantr 480 . . . . . . 7 ((𝜑𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼) = 𝐽)
28 simpr 484 . . . . . . . 8 ((𝜑𝑥 = 𝐼) → 𝑥 = 𝐼)
2928fveq2d 6910 . . . . . . 7 ((𝜑𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼))
302, 11, 20symgov 19415 . . . . . . . . . . 11 (((𝐶‘⟨“𝐼𝐾”⟩) ∈ (Base‘𝑆) ∧ (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆)) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
3118, 19, 30syl2anc 584 . . . . . . . . . 10 (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
3231adantr 480 . . . . . . . . 9 ((𝜑𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
3332fveq1d 6908 . . . . . . . 8 ((𝜑𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
342, 11symgbasf 19407 . . . . . . . . . . . 12 ((𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆) → (𝐶‘⟨“𝐼𝐽”⟩):𝐷𝐷)
3519, 34syl 17 . . . . . . . . . . 11 (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩):𝐷𝐷)
3635ffund 6740 . . . . . . . . . 10 (𝜑 → Fun (𝐶‘⟨“𝐼𝐽”⟩))
374adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐼) → 𝐼𝐷)
3834fdmd 6746 . . . . . . . . . . . . 13 ((𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
3919, 38syl 17 . . . . . . . . . . . 12 (𝜑 → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
4039adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐼) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
4137, 28, 403eltr4d 2853 . . . . . . . . . 10 ((𝜑𝑥 = 𝐼) → 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩))
42 fvco 7006 . . . . . . . . . 10 ((Fun (𝐶‘⟨“𝐼𝐽”⟩) ∧ 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩)) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
4336, 41, 42syl2an2r 685 . . . . . . . . 9 ((𝜑𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
4428fveq2d 6910 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼))
451, 3, 4, 5, 7, 2cyc2fv1 33123 . . . . . . . . . . . 12 (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼) = 𝐽)
4645adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼) = 𝐽)
4744, 46eqtrd 2774 . . . . . . . . . 10 ((𝜑𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝐽)
4847fveq2d 6910 . . . . . . . . 9 ((𝜑𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐽))
498necomd 2993 . . . . . . . . . . 11 (𝜑𝐾𝐽)
507necomd 2993 . . . . . . . . . . 11 (𝜑𝐽𝐼)
511, 2, 3, 4, 6, 5, 17, 49, 50cyc2fvx 33136 . . . . . . . . . 10 (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐽) = 𝐽)
5251adantr 480 . . . . . . . . 9 ((𝜑𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐽) = 𝐽)
5343, 48, 523eqtrd 2778 . . . . . . . 8 ((𝜑𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐽)
5433, 53eqtrd 2774 . . . . . . 7 ((𝜑𝑥 = 𝐼) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐽)
5527, 29, 543eqtr4d 2784 . . . . . 6 ((𝜑𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
5655adantlr 715 . . . . 5 (((𝜑𝑥 ∈ {𝐼, 𝐽, 𝐾}) ∧ 𝑥 = 𝐼) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
571, 2, 3, 4, 5, 6, 7, 8, 9cyc3fv2 33140 . . . . . . . 8 (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽) = 𝐾)
5857adantr 480 . . . . . . 7 ((𝜑𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽) = 𝐾)
59 simpr 484 . . . . . . . 8 ((𝜑𝑥 = 𝐽) → 𝑥 = 𝐽)
6059fveq2d 6910 . . . . . . 7 ((𝜑𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽))
6131adantr 480 . . . . . . . . 9 ((𝜑𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
6261fveq1d 6908 . . . . . . . 8 ((𝜑𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
635adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐽) → 𝐽𝐷)
6439adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐽) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
6563, 59, 643eltr4d 2853 . . . . . . . . . 10 ((𝜑𝑥 = 𝐽) → 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩))
6636, 65, 42syl2an2r 685 . . . . . . . . 9 ((𝜑𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
6759fveq2d 6910 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽))
681, 3, 4, 5, 7, 2cyc2fv2 33124 . . . . . . . . . . . 12 (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼)
6968adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼)
7067, 69eqtrd 2774 . . . . . . . . . 10 ((𝜑𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝐼)
7170fveq2d 6910 . . . . . . . . 9 ((𝜑𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐼))
721, 3, 4, 6, 17, 2cyc2fv1 33123 . . . . . . . . . 10 (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐼) = 𝐾)
7372adantr 480 . . . . . . . . 9 ((𝜑𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐼) = 𝐾)
7466, 71, 733eqtrd 2778 . . . . . . . 8 ((𝜑𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐾)
7562, 74eqtrd 2774 . . . . . . 7 ((𝜑𝑥 = 𝐽) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐾)
7658, 60, 753eqtr4d 2784 . . . . . 6 ((𝜑𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
7776adantlr 715 . . . . 5 (((𝜑𝑥 ∈ {𝐼, 𝐽, 𝐾}) ∧ 𝑥 = 𝐽) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
781, 2, 3, 4, 5, 6, 7, 8, 9cyc3fv3 33141 . . . . . . . 8 (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾) = 𝐼)
7978adantr 480 . . . . . . 7 ((𝜑𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾) = 𝐼)
80 simpr 484 . . . . . . . 8 ((𝜑𝑥 = 𝐾) → 𝑥 = 𝐾)
8180fveq2d 6910 . . . . . . 7 ((𝜑𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾))
8231adantr 480 . . . . . . . . 9 ((𝜑𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
8382fveq1d 6908 . . . . . . . 8 ((𝜑𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
846adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐾) → 𝐾𝐷)
8539adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐾) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
8684, 80, 853eltr4d 2853 . . . . . . . . . 10 ((𝜑𝑥 = 𝐾) → 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩))
8736, 86, 42syl2an2r 685 . . . . . . . . 9 ((𝜑𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
8880fveq2d 6910 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾))
891, 2, 3, 4, 5, 6, 7, 8, 9cyc2fvx 33136 . . . . . . . . . . . 12 (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾) = 𝐾)
9089adantr 480 . . . . . . . . . . 11 ((𝜑𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾) = 𝐾)
9188, 90eqtrd 2774 . . . . . . . . . 10 ((𝜑𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝐾)
9291fveq2d 6910 . . . . . . . . 9 ((𝜑𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐾))
931, 3, 4, 6, 17, 2cyc2fv2 33124 . . . . . . . . . 10 (𝜑 → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐾) = 𝐼)
9493adantr 480 . . . . . . . . 9 ((𝜑𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝐾) = 𝐼)
9587, 92, 943eqtrd 2778 . . . . . . . 8 ((𝜑𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐼)
9683, 95eqtrd 2774 . . . . . . 7 ((𝜑𝑥 = 𝐾) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝐼)
9779, 81, 963eqtr4d 2784 . . . . . 6 ((𝜑𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
9897adantlr 715 . . . . 5 (((𝜑𝑥 ∈ {𝐼, 𝐽, 𝐾}) ∧ 𝑥 = 𝐾) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
99 eltpi 4692 . . . . . 6 (𝑥 ∈ {𝐼, 𝐽, 𝐾} → (𝑥 = 𝐼𝑥 = 𝐽𝑥 = 𝐾))
10099adantl 481 . . . . 5 ((𝜑𝑥 ∈ {𝐼, 𝐽, 𝐾}) → (𝑥 = 𝐼𝑥 = 𝐽𝑥 = 𝐾))
10156, 77, 98, 100mpjao3dan 1431 . . . 4 ((𝜑𝑥 ∈ {𝐼, 𝐽, 𝐾}) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
102101adantlr 715 . . 3 (((𝜑𝑥𝐷) ∧ 𝑥 ∈ {𝐼, 𝐽, 𝐾}) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
10335adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → (𝐶‘⟨“𝐼𝐽”⟩):𝐷𝐷)
104103ffund 6740 . . . . . . 7 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → Fun (𝐶‘⟨“𝐼𝐽”⟩))
105 simpr 484 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾}))
106105eldifad 3974 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → 𝑥𝐷)
10739adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → dom (𝐶‘⟨“𝐼𝐽”⟩) = 𝐷)
108106, 107eleqtrrd 2841 . . . . . . 7 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → 𝑥 ∈ dom (𝐶‘⟨“𝐼𝐽”⟩))
109104, 108, 42syl2anc 584 . . . . . 6 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)))
1103adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → 𝐷𝑉)
1114, 5s2cld 14906 . . . . . . . . 9 (𝜑 → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
112111adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽”⟩ ∈ Word 𝐷)
1134, 5, 7s2f1 32913 . . . . . . . . 9 (𝜑 → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1𝐷)
114113adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1𝐷)
115 tpid1g 4773 . . . . . . . . . . . . 13 (𝐼𝐷𝐼 ∈ {𝐼, 𝐽, 𝐾})
1164, 115syl 17 . . . . . . . . . . . 12 (𝜑𝐼 ∈ {𝐼, 𝐽, 𝐾})
117 tpid2g 4775 . . . . . . . . . . . . 13 (𝐽𝐷𝐽 ∈ {𝐼, 𝐽, 𝐾})
1185, 117syl 17 . . . . . . . . . . . 12 (𝜑𝐽 ∈ {𝐼, 𝐽, 𝐾})
119116, 118prssd 4826 . . . . . . . . . . 11 (𝜑 → {𝐼, 𝐽} ⊆ {𝐼, 𝐽, 𝐾})
1204, 5s2rn 14998 . . . . . . . . . . . 12 (𝜑 → ran ⟨“𝐼𝐽”⟩ = {𝐼, 𝐽})
121120eqcomd 2740 . . . . . . . . . . 11 (𝜑 → {𝐼, 𝐽} = ran ⟨“𝐼𝐽”⟩)
1224, 5, 6s3rn 14999 . . . . . . . . . . . 12 (𝜑 → ran ⟨“𝐼𝐽𝐾”⟩ = {𝐼, 𝐽, 𝐾})
123122eqcomd 2740 . . . . . . . . . . 11 (𝜑 → {𝐼, 𝐽, 𝐾} = ran ⟨“𝐼𝐽𝐾”⟩)
124119, 121, 1233sstr3d 4041 . . . . . . . . . 10 (𝜑 → ran ⟨“𝐼𝐽”⟩ ⊆ ran ⟨“𝐼𝐽𝐾”⟩)
125124adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ran ⟨“𝐼𝐽”⟩ ⊆ ran ⟨“𝐼𝐽𝐾”⟩)
126105eldifbd 3975 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ {𝐼, 𝐽, 𝐾})
127122adantr 480 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ran ⟨“𝐼𝐽𝐾”⟩ = {𝐼, 𝐽, 𝐾})
128126, 127neleqtrrd 2861 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ ran ⟨“𝐼𝐽𝐾”⟩)
129125, 128ssneldd 3997 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ ran ⟨“𝐼𝐽”⟩)
1301, 110, 112, 114, 106, 129cycpmfv3 33117 . . . . . . 7 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥) = 𝑥)
131130fveq2d 6910 . . . . . 6 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐾”⟩)‘((𝐶‘⟨“𝐼𝐽”⟩)‘𝑥)) = ((𝐶‘⟨“𝐼𝐾”⟩)‘𝑥))
1324, 6s2cld 14906 . . . . . . . 8 (𝜑 → ⟨“𝐼𝐾”⟩ ∈ Word 𝐷)
133132adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐾”⟩ ∈ Word 𝐷)
1344, 6, 17s2f1 32913 . . . . . . . 8 (𝜑 → ⟨“𝐼𝐾”⟩:dom ⟨“𝐼𝐾”⟩–1-1𝐷)
135134adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐾”⟩:dom ⟨“𝐼𝐾”⟩–1-1𝐷)
136 tpid3g 4776 . . . . . . . . . . . 12 (𝐾𝐷𝐾 ∈ {𝐼, 𝐽, 𝐾})
1376, 136syl 17 . . . . . . . . . . 11 (𝜑𝐾 ∈ {𝐼, 𝐽, 𝐾})
138116, 137prssd 4826 . . . . . . . . . 10 (𝜑 → {𝐼, 𝐾} ⊆ {𝐼, 𝐽, 𝐾})
139138adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → {𝐼, 𝐾} ⊆ {𝐼, 𝐽, 𝐾})
1404, 6s2rn 14998 . . . . . . . . . . 11 (𝜑 → ran ⟨“𝐼𝐾”⟩ = {𝐼, 𝐾})
141140eqcomd 2740 . . . . . . . . . 10 (𝜑 → {𝐼, 𝐾} = ran ⟨“𝐼𝐾”⟩)
142141adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → {𝐼, 𝐾} = ran ⟨“𝐼𝐾”⟩)
143123adantr 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → {𝐼, 𝐽, 𝐾} = ran ⟨“𝐼𝐽𝐾”⟩)
144139, 142, 1433sstr3d 4041 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ran ⟨“𝐼𝐾”⟩ ⊆ ran ⟨“𝐼𝐽𝐾”⟩)
145144, 128ssneldd 3997 . . . . . . 7 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ¬ 𝑥 ∈ ran ⟨“𝐼𝐾”⟩)
1461, 110, 133, 135, 106, 145cycpmfv3 33117 . . . . . 6 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐾”⟩)‘𝑥) = 𝑥)
147109, 131, 1463eqtrd 2778 . . . . 5 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = 𝑥)
14831adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)) = ((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩)))
149148fveq1d 6908 . . . . 5 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) ∘ (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
1504, 5, 6s3cld 14907 . . . . . . 7 (𝜑 → ⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷)
151150adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽𝐾”⟩ ∈ Word 𝐷)
1524, 5, 6, 7, 8, 9s3f1 32915 . . . . . . 7 (𝜑 → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1𝐷)
153152adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1𝐷)
1541, 110, 151, 153, 106, 128cycpmfv3 33117 . . . . 5 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = 𝑥)
155147, 149, 1543eqtr4rd 2785 . . . 4 ((𝜑𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
156155adantlr 715 . . 3 (((𝜑𝑥𝐷) ∧ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
157 tpssi 4842 . . . . . . . 8 ((𝐼𝐷𝐽𝐷𝐾𝐷) → {𝐼, 𝐽, 𝐾} ⊆ 𝐷)
1584, 5, 6, 157syl3anc 1370 . . . . . . 7 (𝜑 → {𝐼, 𝐽, 𝐾} ⊆ 𝐷)
159 undif 4487 . . . . . . 7 ({𝐼, 𝐽, 𝐾} ⊆ 𝐷 ↔ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) = 𝐷)
160158, 159sylib 218 . . . . . 6 (𝜑 → ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) = 𝐷)
161160eleq2d 2824 . . . . 5 (𝜑 → (𝑥 ∈ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) ↔ 𝑥𝐷))
162161biimpar 477 . . . 4 ((𝜑𝑥𝐷) → 𝑥 ∈ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})))
163 elun 4162 . . . 4 (𝑥 ∈ ({𝐼, 𝐽, 𝐾} ∪ (𝐷 ∖ {𝐼, 𝐽, 𝐾})) ↔ (𝑥 ∈ {𝐼, 𝐽, 𝐾} ∨ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})))
164162, 163sylib 218 . . 3 ((𝜑𝑥𝐷) → (𝑥 ∈ {𝐼, 𝐽, 𝐾} ∨ 𝑥 ∈ (𝐷 ∖ {𝐼, 𝐽, 𝐾})))
165102, 156, 164mpjaodan 960 . 2 ((𝜑𝑥𝐷) → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝑥) = (((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))‘𝑥))
16614, 25, 165eqfnfvd 7053 1 (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) = ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  w3o 1085   = wceq 1536  wcel 2105  wne 2937  cdif 3959  cun 3960  wss 3962  {cpr 4632  {ctp 4634  dom cdm 5688  ran crn 5689  ccom 5692  Fun wfun 6556  wf 6558  1-1wf1 6559  cfv 6562  (class class class)co 7430  Word cword 14548  ⟨“cs2 14876  ⟨“cs3 14877  Basecbs 17244  +gcplusg 17297  Grpcgrp 18963  SymGrpcsymg 19400  toCycctocyc 33108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-fz 13544  df-fzo 13691  df-fl 13828  df-mod 13906  df-hash 14366  df-word 14549  df-concat 14605  df-s1 14630  df-substr 14675  df-pfx 14705  df-csh 14823  df-s2 14883  df-s3 14884  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-tset 17316  df-0g 17487  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-efmnd 18894  df-grp 18966  df-symg 19401  df-tocyc 33109
This theorem is referenced by:  cyc3evpm  33152  cyc3genpmlem  33153
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