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Theorem cycpmconjslem2 32854
Description: Lemma for cycpmconjs 32855. (Contributed by Thierry Arnoux, 14-Oct-2023.)
Hypotheses
Ref Expression
cycpmconjs.c 𝐢 = (𝑀 β€œ (β—‘β™― β€œ {𝑃}))
cycpmconjs.s 𝑆 = (SymGrpβ€˜π·)
cycpmconjs.n 𝑁 = (β™―β€˜π·)
cycpmconjs.m 𝑀 = (toCycβ€˜π·)
cycpmconjs.b 𝐡 = (Baseβ€˜π‘†)
cycpmconjs.a + = (+gβ€˜π‘†)
cycpmconjs.l βˆ’ = (-gβ€˜π‘†)
cycpmconjs.p (πœ‘ β†’ 𝑃 ∈ (0...𝑁))
cycpmconjs.d (πœ‘ β†’ 𝐷 ∈ Fin)
cycpmconjs.q (πœ‘ β†’ 𝑄 ∈ 𝐢)
Assertion
Ref Expression
cycpmconjslem2 (πœ‘ β†’ βˆƒπ‘ž(π‘ž:(0..^𝑁)–1-1-onto→𝐷 ∧ ((β—‘π‘ž ∘ 𝑄) ∘ π‘ž) = ((( I β†Ύ (0..^𝑃)) cyclShift 1) βˆͺ ( I β†Ύ (𝑃..^𝑁)))))
Distinct variable groups:   + ,π‘ž   𝐷,π‘ž   𝑀,π‘ž   𝑁,π‘ž   𝑃,π‘ž   𝑄,π‘ž
Allowed substitution hints:   πœ‘(π‘ž)   𝐡(π‘ž)   𝐢(π‘ž)   𝑆(π‘ž)   βˆ’ (π‘ž)

Proof of Theorem cycpmconjslem2
Dummy variables 𝑓 𝑒 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzofi 13963 . . . . 5 (0..^𝑁) ∈ Fin
2 diffi 9195 . . . . 5 ((0..^𝑁) ∈ Fin β†’ ((0..^𝑁) βˆ– dom 𝑒) ∈ Fin)
31, 2mp1i 13 . . . 4 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ ((0..^𝑁) βˆ– dom 𝑒) ∈ Fin)
4 cycpmconjs.d . . . . . 6 (πœ‘ β†’ 𝐷 ∈ Fin)
5 diffi 9195 . . . . . 6 (𝐷 ∈ Fin β†’ (𝐷 βˆ– ran 𝑒) ∈ Fin)
64, 5syl 17 . . . . 5 (πœ‘ β†’ (𝐷 βˆ– ran 𝑒) ∈ Fin)
76ad2antrr 725 . . . 4 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (𝐷 βˆ– ran 𝑒) ∈ Fin)
8 cycpmconjs.n . . . . . . . . . 10 𝑁 = (β™―β€˜π·)
9 hashcl 14339 . . . . . . . . . . 11 (𝐷 ∈ Fin β†’ (β™―β€˜π·) ∈ β„•0)
104, 9syl 17 . . . . . . . . . 10 (πœ‘ β†’ (β™―β€˜π·) ∈ β„•0)
118, 10eqeltrid 2832 . . . . . . . . 9 (πœ‘ β†’ 𝑁 ∈ β„•0)
12 hashfzo0 14413 . . . . . . . . 9 (𝑁 ∈ β„•0 β†’ (β™―β€˜(0..^𝑁)) = 𝑁)
1311, 12syl 17 . . . . . . . 8 (πœ‘ β†’ (β™―β€˜(0..^𝑁)) = 𝑁)
1413, 8eqtrdi 2783 . . . . . . 7 (πœ‘ β†’ (β™―β€˜(0..^𝑁)) = (β™―β€˜π·))
1514ad2antrr 725 . . . . . 6 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (β™―β€˜(0..^𝑁)) = (β™―β€˜π·))
16 simplr 768 . . . . . . . . . 10 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃})))
1716elin1d 4194 . . . . . . . . 9 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ 𝑒 ∈ {𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷})
18 elrabi 3674 . . . . . . . . 9 (𝑒 ∈ {𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} β†’ 𝑒 ∈ Word 𝐷)
19 wrdfin 14506 . . . . . . . . 9 (𝑒 ∈ Word 𝐷 β†’ 𝑒 ∈ Fin)
2017, 18, 193syl 18 . . . . . . . 8 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ 𝑒 ∈ Fin)
21 id 22 . . . . . . . . . . . . 13 (𝑀 = 𝑒 β†’ 𝑀 = 𝑒)
22 dmeq 5900 . . . . . . . . . . . . 13 (𝑀 = 𝑒 β†’ dom 𝑀 = dom 𝑒)
23 eqidd 2728 . . . . . . . . . . . . 13 (𝑀 = 𝑒 β†’ 𝐷 = 𝐷)
2421, 22, 23f1eq123d 6825 . . . . . . . . . . . 12 (𝑀 = 𝑒 β†’ (𝑀:dom 𝑀–1-1→𝐷 ↔ 𝑒:dom 𝑒–1-1→𝐷))
2524elrab 3680 . . . . . . . . . . 11 (𝑒 ∈ {𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ↔ (𝑒 ∈ Word 𝐷 ∧ 𝑒:dom 𝑒–1-1→𝐷))
2625simprbi 496 . . . . . . . . . 10 (𝑒 ∈ {𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} β†’ 𝑒:dom 𝑒–1-1→𝐷)
2717, 26syl 17 . . . . . . . . 9 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ 𝑒:dom 𝑒–1-1→𝐷)
28 f1fun 6789 . . . . . . . . 9 (𝑒:dom 𝑒–1-1→𝐷 β†’ Fun 𝑒)
2927, 28syl 17 . . . . . . . 8 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ Fun 𝑒)
30 hashfun 14420 . . . . . . . . 9 (𝑒 ∈ Fin β†’ (Fun 𝑒 ↔ (β™―β€˜π‘’) = (β™―β€˜dom 𝑒)))
3130biimpa 476 . . . . . . . 8 ((𝑒 ∈ Fin ∧ Fun 𝑒) β†’ (β™―β€˜π‘’) = (β™―β€˜dom 𝑒))
3220, 29, 31syl2anc 583 . . . . . . 7 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (β™―β€˜π‘’) = (β™―β€˜dom 𝑒))
3316dmexd 7905 . . . . . . . 8 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ dom 𝑒 ∈ V)
34 hashf1rn 14335 . . . . . . . 8 ((dom 𝑒 ∈ V ∧ 𝑒:dom 𝑒–1-1→𝐷) β†’ (β™―β€˜π‘’) = (β™―β€˜ran 𝑒))
3533, 27, 34syl2anc 583 . . . . . . 7 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (β™―β€˜π‘’) = (β™―β€˜ran 𝑒))
3632, 35eqtr3d 2769 . . . . . 6 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (β™―β€˜dom 𝑒) = (β™―β€˜ran 𝑒))
3715, 36oveq12d 7432 . . . . 5 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ ((β™―β€˜(0..^𝑁)) βˆ’ (β™―β€˜dom 𝑒)) = ((β™―β€˜π·) βˆ’ (β™―β€˜ran 𝑒)))
381a1i 11 . . . . . 6 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (0..^𝑁) ∈ Fin)
39 wrddm 14495 . . . . . . . 8 (𝑒 ∈ Word 𝐷 β†’ dom 𝑒 = (0..^(β™―β€˜π‘’)))
4017, 18, 393syl 18 . . . . . . 7 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ dom 𝑒 = (0..^(β™―β€˜π‘’)))
41 hashcl 14339 . . . . . . . . . . 11 (𝑒 ∈ Fin β†’ (β™―β€˜π‘’) ∈ β„•0)
4217, 18, 19, 414syl 19 . . . . . . . . . 10 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (β™―β€˜π‘’) ∈ β„•0)
4342nn0zd 12606 . . . . . . . . 9 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (β™―β€˜π‘’) ∈ β„€)
4410nn0zd 12606 . . . . . . . . . . 11 (πœ‘ β†’ (β™―β€˜π·) ∈ β„€)
458, 44eqeltrid 2832 . . . . . . . . . 10 (πœ‘ β†’ 𝑁 ∈ β„€)
4645ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ 𝑁 ∈ β„€)
474ad2antrr 725 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ 𝐷 ∈ Fin)
48 wrdf 14493 . . . . . . . . . . . . 13 (𝑒 ∈ Word 𝐷 β†’ 𝑒:(0..^(β™―β€˜π‘’))⟢𝐷)
4948frnd 6724 . . . . . . . . . . . 12 (𝑒 ∈ Word 𝐷 β†’ ran 𝑒 βŠ† 𝐷)
5017, 18, 493syl 18 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ ran 𝑒 βŠ† 𝐷)
51 hashss 14392 . . . . . . . . . . 11 ((𝐷 ∈ Fin ∧ ran 𝑒 βŠ† 𝐷) β†’ (β™―β€˜ran 𝑒) ≀ (β™―β€˜π·))
5247, 50, 51syl2anc 583 . . . . . . . . . 10 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (β™―β€˜ran 𝑒) ≀ (β™―β€˜π·))
538a1i 11 . . . . . . . . . 10 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ 𝑁 = (β™―β€˜π·))
5452, 35, 533brtr4d 5174 . . . . . . . . 9 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (β™―β€˜π‘’) ≀ 𝑁)
55 eluz1 12848 . . . . . . . . . 10 ((β™―β€˜π‘’) ∈ β„€ β†’ (𝑁 ∈ (β„€β‰₯β€˜(β™―β€˜π‘’)) ↔ (𝑁 ∈ β„€ ∧ (β™―β€˜π‘’) ≀ 𝑁)))
5655biimpar 477 . . . . . . . . 9 (((β™―β€˜π‘’) ∈ β„€ ∧ (𝑁 ∈ β„€ ∧ (β™―β€˜π‘’) ≀ 𝑁)) β†’ 𝑁 ∈ (β„€β‰₯β€˜(β™―β€˜π‘’)))
5743, 46, 54, 56syl12anc 836 . . . . . . . 8 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ 𝑁 ∈ (β„€β‰₯β€˜(β™―β€˜π‘’)))
58 fzoss2 13684 . . . . . . . 8 (𝑁 ∈ (β„€β‰₯β€˜(β™―β€˜π‘’)) β†’ (0..^(β™―β€˜π‘’)) βŠ† (0..^𝑁))
5957, 58syl 17 . . . . . . 7 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (0..^(β™―β€˜π‘’)) βŠ† (0..^𝑁))
6040, 59eqsstrd 4016 . . . . . 6 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ dom 𝑒 βŠ† (0..^𝑁))
61 hashssdif 14395 . . . . . 6 (((0..^𝑁) ∈ Fin ∧ dom 𝑒 βŠ† (0..^𝑁)) β†’ (β™―β€˜((0..^𝑁) βˆ– dom 𝑒)) = ((β™―β€˜(0..^𝑁)) βˆ’ (β™―β€˜dom 𝑒)))
6238, 60, 61syl2anc 583 . . . . 5 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (β™―β€˜((0..^𝑁) βˆ– dom 𝑒)) = ((β™―β€˜(0..^𝑁)) βˆ’ (β™―β€˜dom 𝑒)))
63 hashssdif 14395 . . . . . 6 ((𝐷 ∈ Fin ∧ ran 𝑒 βŠ† 𝐷) β†’ (β™―β€˜(𝐷 βˆ– ran 𝑒)) = ((β™―β€˜π·) βˆ’ (β™―β€˜ran 𝑒)))
6447, 50, 63syl2anc 583 . . . . 5 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (β™―β€˜(𝐷 βˆ– ran 𝑒)) = ((β™―β€˜π·) βˆ’ (β™―β€˜ran 𝑒)))
6537, 62, 643eqtr4d 2777 . . . 4 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (β™―β€˜((0..^𝑁) βˆ– dom 𝑒)) = (β™―β€˜(𝐷 βˆ– ran 𝑒)))
66 hasheqf1o 14332 . . . . 5 ((((0..^𝑁) βˆ– dom 𝑒) ∈ Fin ∧ (𝐷 βˆ– ran 𝑒) ∈ Fin) β†’ ((β™―β€˜((0..^𝑁) βˆ– dom 𝑒)) = (β™―β€˜(𝐷 βˆ– ran 𝑒)) ↔ βˆƒπ‘“ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)))
6766biimpa 476 . . . 4 (((((0..^𝑁) βˆ– dom 𝑒) ∈ Fin ∧ (𝐷 βˆ– ran 𝑒) ∈ Fin) ∧ (β™―β€˜((0..^𝑁) βˆ– dom 𝑒)) = (β™―β€˜(𝐷 βˆ– ran 𝑒))) β†’ βˆƒπ‘“ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒))
683, 7, 65, 67syl21anc 837 . . 3 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ βˆƒπ‘“ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒))
6927adantr 480 . . . . . . 7 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ 𝑒:dom 𝑒–1-1→𝐷)
70 f1f1orn 6844 . . . . . . 7 (𝑒:dom 𝑒–1-1→𝐷 β†’ 𝑒:dom 𝑒–1-1-ontoβ†’ran 𝑒)
7169, 70syl 17 . . . . . 6 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ 𝑒:dom 𝑒–1-1-ontoβ†’ran 𝑒)
72 simpr 484 . . . . . 6 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒))
73 disjdif 4467 . . . . . . 7 (dom 𝑒 ∩ ((0..^𝑁) βˆ– dom 𝑒)) = βˆ…
7473a1i 11 . . . . . 6 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (dom 𝑒 ∩ ((0..^𝑁) βˆ– dom 𝑒)) = βˆ…)
75 disjdif 4467 . . . . . . 7 (ran 𝑒 ∩ (𝐷 βˆ– ran 𝑒)) = βˆ…
7675a1i 11 . . . . . 6 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (ran 𝑒 ∩ (𝐷 βˆ– ran 𝑒)) = βˆ…)
77 f1oun 6852 . . . . . 6 (((𝑒:dom 𝑒–1-1-ontoβ†’ran 𝑒 ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) ∧ ((dom 𝑒 ∩ ((0..^𝑁) βˆ– dom 𝑒)) = βˆ… ∧ (ran 𝑒 ∩ (𝐷 βˆ– ran 𝑒)) = βˆ…)) β†’ (𝑒 βˆͺ 𝑓):(dom 𝑒 βˆͺ ((0..^𝑁) βˆ– dom 𝑒))–1-1-ontoβ†’(ran 𝑒 βˆͺ (𝐷 βˆ– ran 𝑒)))
7871, 72, 74, 76, 77syl22anc 838 . . . . 5 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (𝑒 βˆͺ 𝑓):(dom 𝑒 βˆͺ ((0..^𝑁) βˆ– dom 𝑒))–1-1-ontoβ†’(ran 𝑒 βˆͺ (𝐷 βˆ– ran 𝑒)))
79 eqidd 2728 . . . . . 6 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (𝑒 βˆͺ 𝑓) = (𝑒 βˆͺ 𝑓))
8060adantr 480 . . . . . . 7 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ dom 𝑒 βŠ† (0..^𝑁))
81 undif 4477 . . . . . . 7 (dom 𝑒 βŠ† (0..^𝑁) ↔ (dom 𝑒 βˆͺ ((0..^𝑁) βˆ– dom 𝑒)) = (0..^𝑁))
8280, 81sylib 217 . . . . . 6 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (dom 𝑒 βˆͺ ((0..^𝑁) βˆ– dom 𝑒)) = (0..^𝑁))
83 undif 4477 . . . . . . . 8 (ran 𝑒 βŠ† 𝐷 ↔ (ran 𝑒 βˆͺ (𝐷 βˆ– ran 𝑒)) = 𝐷)
8450, 83sylib 217 . . . . . . 7 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (ran 𝑒 βˆͺ (𝐷 βˆ– ran 𝑒)) = 𝐷)
8584adantr 480 . . . . . 6 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (ran 𝑒 βˆͺ (𝐷 βˆ– ran 𝑒)) = 𝐷)
8679, 82, 85f1oeq123d 6827 . . . . 5 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((𝑒 βˆͺ 𝑓):(dom 𝑒 βˆͺ ((0..^𝑁) βˆ– dom 𝑒))–1-1-ontoβ†’(ran 𝑒 βˆͺ (𝐷 βˆ– ran 𝑒)) ↔ (𝑒 βˆͺ 𝑓):(0..^𝑁)–1-1-onto→𝐷))
8778, 86mpbid 231 . . . 4 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (𝑒 βˆͺ 𝑓):(0..^𝑁)–1-1-onto→𝐷)
88 f1ocnv 6845 . . . . . . . . . 10 ((𝑒 βˆͺ 𝑓):(0..^𝑁)–1-1-onto→𝐷 β†’ β—‘(𝑒 βˆͺ 𝑓):𝐷–1-1-ontoβ†’(0..^𝑁))
8987, 88syl 17 . . . . . . . . 9 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ β—‘(𝑒 βˆͺ 𝑓):𝐷–1-1-ontoβ†’(0..^𝑁))
90 cycpmconjs.p . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑃 ∈ (0...𝑁))
91 cycpmconjs.c . . . . . . . . . . . . . 14 𝐢 = (𝑀 β€œ (β—‘β™― β€œ {𝑃}))
92 cycpmconjs.s . . . . . . . . . . . . . 14 𝑆 = (SymGrpβ€˜π·)
93 cycpmconjs.m . . . . . . . . . . . . . 14 𝑀 = (toCycβ€˜π·)
94 cycpmconjs.b . . . . . . . . . . . . . 14 𝐡 = (Baseβ€˜π‘†)
9591, 92, 8, 93, 94cycpmgcl 32852 . . . . . . . . . . . . 13 ((𝐷 ∈ Fin ∧ 𝑃 ∈ (0...𝑁)) β†’ 𝐢 βŠ† 𝐡)
964, 90, 95syl2anc 583 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐢 βŠ† 𝐡)
97 cycpmconjs.q . . . . . . . . . . . 12 (πœ‘ β†’ 𝑄 ∈ 𝐢)
9896, 97sseldd 3979 . . . . . . . . . . 11 (πœ‘ β†’ 𝑄 ∈ 𝐡)
9992, 94symgbasf1o 19320 . . . . . . . . . . 11 (𝑄 ∈ 𝐡 β†’ 𝑄:𝐷–1-1-onto→𝐷)
10098, 99syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝑄:𝐷–1-1-onto→𝐷)
101100ad3antrrr 729 . . . . . . . . 9 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ 𝑄:𝐷–1-1-onto→𝐷)
102 f1oco 6856 . . . . . . . . 9 ((β—‘(𝑒 βˆͺ 𝑓):𝐷–1-1-ontoβ†’(0..^𝑁) ∧ 𝑄:𝐷–1-1-onto→𝐷) β†’ (β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄):𝐷–1-1-ontoβ†’(0..^𝑁))
10389, 101, 102syl2anc 583 . . . . . . . 8 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄):𝐷–1-1-ontoβ†’(0..^𝑁))
104 f1oco 6856 . . . . . . . 8 (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄):𝐷–1-1-ontoβ†’(0..^𝑁) ∧ (𝑒 βˆͺ 𝑓):(0..^𝑁)–1-1-onto→𝐷) β†’ ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)):(0..^𝑁)–1-1-ontoβ†’(0..^𝑁))
105103, 87, 104syl2anc 583 . . . . . . 7 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)):(0..^𝑁)–1-1-ontoβ†’(0..^𝑁))
106 f1ofun 6835 . . . . . . 7 (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)):(0..^𝑁)–1-1-ontoβ†’(0..^𝑁) β†’ Fun ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)))
107 funrel 6564 . . . . . . 7 (Fun ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) β†’ Rel ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)))
108105, 106, 1073syl 18 . . . . . 6 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ Rel ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)))
109 f1odm 6837 . . . . . . . 8 (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)):(0..^𝑁)–1-1-ontoβ†’(0..^𝑁) β†’ dom ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) = (0..^𝑁))
110105, 109syl 17 . . . . . . 7 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ dom ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) = (0..^𝑁))
111 fzosplit 13689 . . . . . . . . 9 (𝑃 ∈ (0...𝑁) β†’ (0..^𝑁) = ((0..^𝑃) βˆͺ (𝑃..^𝑁)))
11290, 111syl 17 . . . . . . . 8 (πœ‘ β†’ (0..^𝑁) = ((0..^𝑃) βˆͺ (𝑃..^𝑁)))
113112ad3antrrr 729 . . . . . . 7 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (0..^𝑁) = ((0..^𝑃) βˆͺ (𝑃..^𝑁)))
114110, 113eqtrd 2767 . . . . . 6 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ dom ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) = ((0..^𝑃) βˆͺ (𝑃..^𝑁)))
115 fzodisj 13690 . . . . . . 7 ((0..^𝑃) ∩ (𝑃..^𝑁)) = βˆ…
116 reldisjun 6030 . . . . . . 7 ((Rel ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) ∧ dom ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) = ((0..^𝑃) βˆͺ (𝑃..^𝑁)) ∧ ((0..^𝑃) ∩ (𝑃..^𝑁)) = βˆ…) β†’ ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) = ((((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) β†Ύ (0..^𝑃)) βˆͺ (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) β†Ύ (𝑃..^𝑁))))
117115, 116mp3an3 1447 . . . . . 6 ((Rel ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) ∧ dom ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) = ((0..^𝑃) βˆͺ (𝑃..^𝑁))) β†’ ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) = ((((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) β†Ύ (0..^𝑃)) βˆͺ (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) β†Ύ (𝑃..^𝑁))))
118108, 114, 117syl2anc 583 . . . . 5 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) = ((((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) β†Ύ (0..^𝑃)) βˆͺ (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) β†Ύ (𝑃..^𝑁))))
119 resco 6248 . . . . . . . 8 (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) β†Ύ (0..^𝑃)) = ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ ((𝑒 βˆͺ 𝑓) β†Ύ (0..^𝑃)))
120119a1i 11 . . . . . . 7 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) β†Ύ (0..^𝑃)) = ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ ((𝑒 βˆͺ 𝑓) β†Ύ (0..^𝑃))))
12117, 18syl 17 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ 𝑒 ∈ Word 𝐷)
122 wrdfn 14502 . . . . . . . . . . . 12 (𝑒 ∈ Word 𝐷 β†’ 𝑒 Fn (0..^(β™―β€˜π‘’)))
123121, 122syl 17 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ 𝑒 Fn (0..^(β™―β€˜π‘’)))
12416elin2d 4195 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ 𝑒 ∈ (β—‘β™― β€œ {𝑃}))
125 hashf 14321 . . . . . . . . . . . . . . . 16 β™―:V⟢(β„•0 βˆͺ {+∞})
126 ffn 6716 . . . . . . . . . . . . . . . 16 (β™―:V⟢(β„•0 βˆͺ {+∞}) β†’ β™― Fn V)
127 fniniseg 7063 . . . . . . . . . . . . . . . 16 (β™― Fn V β†’ (𝑒 ∈ (β—‘β™― β€œ {𝑃}) ↔ (𝑒 ∈ V ∧ (β™―β€˜π‘’) = 𝑃)))
128125, 126, 127mp2b 10 . . . . . . . . . . . . . . 15 (𝑒 ∈ (β—‘β™― β€œ {𝑃}) ↔ (𝑒 ∈ V ∧ (β™―β€˜π‘’) = 𝑃))
129128simprbi 496 . . . . . . . . . . . . . 14 (𝑒 ∈ (β—‘β™― β€œ {𝑃}) β†’ (β™―β€˜π‘’) = 𝑃)
130124, 129syl 17 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (β™―β€˜π‘’) = 𝑃)
131130oveq2d 7430 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (0..^(β™―β€˜π‘’)) = (0..^𝑃))
132131fneq2d 6642 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (𝑒 Fn (0..^(β™―β€˜π‘’)) ↔ 𝑒 Fn (0..^𝑃)))
133123, 132mpbid 231 . . . . . . . . . 10 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ 𝑒 Fn (0..^𝑃))
134133adantr 480 . . . . . . . . 9 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ 𝑒 Fn (0..^𝑃))
135 f1ofn 6834 . . . . . . . . . 10 (𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒) β†’ 𝑓 Fn ((0..^𝑁) βˆ– dom 𝑒))
13672, 135syl 17 . . . . . . . . 9 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ 𝑓 Fn ((0..^𝑁) βˆ– dom 𝑒))
13740, 131eqtrd 2767 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ dom 𝑒 = (0..^𝑃))
138137ineq1d 4207 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (dom 𝑒 ∩ ((0..^𝑁) βˆ– dom 𝑒)) = ((0..^𝑃) ∩ ((0..^𝑁) βˆ– dom 𝑒)))
13973a1i 11 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ (dom 𝑒 ∩ ((0..^𝑁) βˆ– dom 𝑒)) = βˆ…)
140138, 139eqtr3d 2769 . . . . . . . . . 10 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ ((0..^𝑃) ∩ ((0..^𝑁) βˆ– dom 𝑒)) = βˆ…)
141140adantr 480 . . . . . . . . 9 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((0..^𝑃) ∩ ((0..^𝑁) βˆ– dom 𝑒)) = βˆ…)
142 fnunres1 6660 . . . . . . . . 9 ((𝑒 Fn (0..^𝑃) ∧ 𝑓 Fn ((0..^𝑁) βˆ– dom 𝑒) ∧ ((0..^𝑃) ∩ ((0..^𝑁) βˆ– dom 𝑒)) = βˆ…) β†’ ((𝑒 βˆͺ 𝑓) β†Ύ (0..^𝑃)) = 𝑒)
143134, 136, 141, 142syl3anc 1369 . . . . . . . 8 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((𝑒 βˆͺ 𝑓) β†Ύ (0..^𝑃)) = 𝑒)
144143coeq2d 5859 . . . . . . 7 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ ((𝑒 βˆͺ 𝑓) β†Ύ (0..^𝑃))) = ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ 𝑒))
145 resco 6248 . . . . . . . . . . 11 ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) β†Ύ ran 𝑒) = (β—‘(𝑒 βˆͺ 𝑓) ∘ (𝑄 β†Ύ ran 𝑒))
146 resco 6248 . . . . . . . . . . . . 13 ((◑𝑒 ∘ (π‘€β€˜π‘’)) β†Ύ ran 𝑒) = (◑𝑒 ∘ ((π‘€β€˜π‘’) β†Ύ ran 𝑒))
147146a1i 11 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((◑𝑒 ∘ (π‘€β€˜π‘’)) β†Ύ ran 𝑒) = (◑𝑒 ∘ ((π‘€β€˜π‘’) β†Ύ ran 𝑒)))
148 cnvun 6141 . . . . . . . . . . . . . . 15 β—‘(𝑒 βˆͺ 𝑓) = (◑𝑒 βˆͺ ◑𝑓)
149148reseq1i 5975 . . . . . . . . . . . . . 14 (β—‘(𝑒 βˆͺ 𝑓) β†Ύ ran 𝑒) = ((◑𝑒 βˆͺ ◑𝑓) β†Ύ ran 𝑒)
150 f1ocnv 6845 . . . . . . . . . . . . . . . 16 (𝑒:dom 𝑒–1-1-ontoβ†’ran 𝑒 β†’ ◑𝑒:ran 𝑒–1-1-ontoβ†’dom 𝑒)
151 f1ofn 6834 . . . . . . . . . . . . . . . 16 (◑𝑒:ran 𝑒–1-1-ontoβ†’dom 𝑒 β†’ ◑𝑒 Fn ran 𝑒)
15271, 150, 1513syl 18 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ◑𝑒 Fn ran 𝑒)
153 f1ocnv 6845 . . . . . . . . . . . . . . . 16 (𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒) β†’ ◑𝑓:(𝐷 βˆ– ran 𝑒)–1-1-ontoβ†’((0..^𝑁) βˆ– dom 𝑒))
154 f1ofn 6834 . . . . . . . . . . . . . . . 16 (◑𝑓:(𝐷 βˆ– ran 𝑒)–1-1-ontoβ†’((0..^𝑁) βˆ– dom 𝑒) β†’ ◑𝑓 Fn (𝐷 βˆ– ran 𝑒))
15572, 153, 1543syl 18 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ◑𝑓 Fn (𝐷 βˆ– ran 𝑒))
156 fnunres1 6660 . . . . . . . . . . . . . . 15 ((◑𝑒 Fn ran 𝑒 ∧ ◑𝑓 Fn (𝐷 βˆ– ran 𝑒) ∧ (ran 𝑒 ∩ (𝐷 βˆ– ran 𝑒)) = βˆ…) β†’ ((◑𝑒 βˆͺ ◑𝑓) β†Ύ ran 𝑒) = ◑𝑒)
157152, 155, 76, 156syl3anc 1369 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((◑𝑒 βˆͺ ◑𝑓) β†Ύ ran 𝑒) = ◑𝑒)
158149, 157eqtr2id 2780 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ◑𝑒 = (β—‘(𝑒 βˆͺ 𝑓) β†Ύ ran 𝑒))
159 simplr 768 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (π‘€β€˜π‘’) = 𝑄)
160159reseq1d 5978 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((π‘€β€˜π‘’) β†Ύ ran 𝑒) = (𝑄 β†Ύ ran 𝑒))
161158, 160coeq12d 5861 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (◑𝑒 ∘ ((π‘€β€˜π‘’) β†Ύ ran 𝑒)) = ((β—‘(𝑒 βˆͺ 𝑓) β†Ύ ran 𝑒) ∘ (𝑄 β†Ύ ran 𝑒)))
16247adantr 480 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ 𝐷 ∈ Fin)
163121adantr 480 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ 𝑒 ∈ Word 𝐷)
16493, 162, 163, 69tocycfvres1 32809 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((π‘€β€˜π‘’) β†Ύ ran 𝑒) = ((𝑒 cyclShift 1) ∘ ◑𝑒))
165160, 164eqtr3d 2769 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (𝑄 β†Ύ ran 𝑒) = ((𝑒 cyclShift 1) ∘ ◑𝑒))
166165rneqd 5934 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ran (𝑄 β†Ύ ran 𝑒) = ran ((𝑒 cyclShift 1) ∘ ◑𝑒))
167 1zzd 12615 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ 1 ∈ β„€)
168 cshf1o 32665 . . . . . . . . . . . . . . . . . 18 ((𝑒 ∈ Word 𝐷 ∧ 𝑒:dom 𝑒–1-1→𝐷 ∧ 1 ∈ β„€) β†’ (𝑒 cyclShift 1):dom 𝑒–1-1-ontoβ†’ran 𝑒)
169163, 69, 167, 168syl3anc 1369 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (𝑒 cyclShift 1):dom 𝑒–1-1-ontoβ†’ran 𝑒)
17071, 150syl 17 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ◑𝑒:ran 𝑒–1-1-ontoβ†’dom 𝑒)
171 f1oco 6856 . . . . . . . . . . . . . . . . 17 (((𝑒 cyclShift 1):dom 𝑒–1-1-ontoβ†’ran 𝑒 ∧ ◑𝑒:ran 𝑒–1-1-ontoβ†’dom 𝑒) β†’ ((𝑒 cyclShift 1) ∘ ◑𝑒):ran 𝑒–1-1-ontoβ†’ran 𝑒)
172169, 170, 171syl2anc 583 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((𝑒 cyclShift 1) ∘ ◑𝑒):ran 𝑒–1-1-ontoβ†’ran 𝑒)
173 f1ofo 6840 . . . . . . . . . . . . . . . 16 (((𝑒 cyclShift 1) ∘ ◑𝑒):ran 𝑒–1-1-ontoβ†’ran 𝑒 β†’ ((𝑒 cyclShift 1) ∘ ◑𝑒):ran 𝑒–ontoβ†’ran 𝑒)
174 forn 6808 . . . . . . . . . . . . . . . 16 (((𝑒 cyclShift 1) ∘ ◑𝑒):ran 𝑒–ontoβ†’ran 𝑒 β†’ ran ((𝑒 cyclShift 1) ∘ ◑𝑒) = ran 𝑒)
175172, 173, 1743syl 18 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ran ((𝑒 cyclShift 1) ∘ ◑𝑒) = ran 𝑒)
176166, 175eqtrd 2767 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ran (𝑄 β†Ύ ran 𝑒) = ran 𝑒)
177 ssid 4000 . . . . . . . . . . . . . 14 ran 𝑒 βŠ† ran 𝑒
178176, 177eqsstrdi 4032 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ran (𝑄 β†Ύ ran 𝑒) βŠ† ran 𝑒)
179 cores 6247 . . . . . . . . . . . . 13 (ran (𝑄 β†Ύ ran 𝑒) βŠ† ran 𝑒 β†’ ((β—‘(𝑒 βˆͺ 𝑓) β†Ύ ran 𝑒) ∘ (𝑄 β†Ύ ran 𝑒)) = (β—‘(𝑒 βˆͺ 𝑓) ∘ (𝑄 β†Ύ ran 𝑒)))
180178, 179syl 17 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((β—‘(𝑒 βˆͺ 𝑓) β†Ύ ran 𝑒) ∘ (𝑄 β†Ύ ran 𝑒)) = (β—‘(𝑒 βˆͺ 𝑓) ∘ (𝑄 β†Ύ ran 𝑒)))
181147, 161, 1803eqtrrd 2772 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (β—‘(𝑒 βˆͺ 𝑓) ∘ (𝑄 β†Ύ ran 𝑒)) = ((◑𝑒 ∘ (π‘€β€˜π‘’)) β†Ύ ran 𝑒))
182145, 181eqtrid 2779 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) β†Ύ ran 𝑒) = ((◑𝑒 ∘ (π‘€β€˜π‘’)) β†Ύ ran 𝑒))
183182coeq1d 5858 . . . . . . . . 9 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) β†Ύ ran 𝑒) ∘ 𝑒) = (((◑𝑒 ∘ (π‘€β€˜π‘’)) β†Ύ ran 𝑒) ∘ 𝑒))
184 cores 6247 . . . . . . . . . 10 (ran 𝑒 βŠ† ran 𝑒 β†’ (((◑𝑒 ∘ (π‘€β€˜π‘’)) β†Ύ ran 𝑒) ∘ 𝑒) = ((◑𝑒 ∘ (π‘€β€˜π‘’)) ∘ 𝑒))
185177, 184mp1i 13 . . . . . . . . 9 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (((◑𝑒 ∘ (π‘€β€˜π‘’)) β†Ύ ran 𝑒) ∘ 𝑒) = ((◑𝑒 ∘ (π‘€β€˜π‘’)) ∘ 𝑒))
186183, 185eqtrd 2767 . . . . . . . 8 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) β†Ύ ran 𝑒) ∘ 𝑒) = ((◑𝑒 ∘ (π‘€β€˜π‘’)) ∘ 𝑒))
187 cores 6247 . . . . . . . . 9 (ran 𝑒 βŠ† ran 𝑒 β†’ (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) β†Ύ ran 𝑒) ∘ 𝑒) = ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ 𝑒))
188177, 187mp1i 13 . . . . . . . 8 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) β†Ύ ran 𝑒) ∘ 𝑒) = ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ 𝑒))
189130adantr 480 . . . . . . . . 9 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (β™―β€˜π‘’) = 𝑃)
19091, 92, 8, 93, 162, 163, 69, 189cycpmconjslem1 32853 . . . . . . . 8 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((◑𝑒 ∘ (π‘€β€˜π‘’)) ∘ 𝑒) = (( I β†Ύ (0..^𝑃)) cyclShift 1))
191186, 188, 1903eqtr3d 2775 . . . . . . 7 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ 𝑒) = (( I β†Ύ (0..^𝑃)) cyclShift 1))
192120, 144, 1913eqtrd 2771 . . . . . 6 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) β†Ύ (0..^𝑃)) = (( I β†Ύ (0..^𝑃)) cyclShift 1))
193 resco 6248 . . . . . . . 8 (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) β†Ύ (𝑃..^𝑁)) = ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ ((𝑒 βˆͺ 𝑓) β†Ύ (𝑃..^𝑁)))
194137adantr 480 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ dom 𝑒 = (0..^𝑃))
195194difeq2d 4118 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((0..^𝑁) βˆ– dom 𝑒) = ((0..^𝑁) βˆ– (0..^𝑃)))
196 fzodif1 32545 . . . . . . . . . . . . . 14 (𝑃 ∈ (0...𝑁) β†’ ((0..^𝑁) βˆ– (0..^𝑃)) = (𝑃..^𝑁))
19790, 196syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ ((0..^𝑁) βˆ– (0..^𝑃)) = (𝑃..^𝑁))
198197ad3antrrr 729 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((0..^𝑁) βˆ– (0..^𝑃)) = (𝑃..^𝑁))
199195, 198eqtrd 2767 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((0..^𝑁) βˆ– dom 𝑒) = (𝑃..^𝑁))
200199reseq2d 5979 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((𝑒 βˆͺ 𝑓) β†Ύ ((0..^𝑁) βˆ– dom 𝑒)) = ((𝑒 βˆͺ 𝑓) β†Ύ (𝑃..^𝑁)))
201 fnunres2 6661 . . . . . . . . . . 11 ((𝑒 Fn (0..^𝑃) ∧ 𝑓 Fn ((0..^𝑁) βˆ– dom 𝑒) ∧ ((0..^𝑃) ∩ ((0..^𝑁) βˆ– dom 𝑒)) = βˆ…) β†’ ((𝑒 βˆͺ 𝑓) β†Ύ ((0..^𝑁) βˆ– dom 𝑒)) = 𝑓)
202134, 136, 141, 201syl3anc 1369 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((𝑒 βˆͺ 𝑓) β†Ύ ((0..^𝑁) βˆ– dom 𝑒)) = 𝑓)
203200, 202eqtr3d 2769 . . . . . . . . 9 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((𝑒 βˆͺ 𝑓) β†Ύ (𝑃..^𝑁)) = 𝑓)
204203coeq2d 5859 . . . . . . . 8 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ ((𝑒 βˆͺ 𝑓) β†Ύ (𝑃..^𝑁))) = ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ 𝑓))
205193, 204eqtrid 2779 . . . . . . 7 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) β†Ύ (𝑃..^𝑁)) = ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ 𝑓))
206148reseq1i 5975 . . . . . . . . . . . 12 (β—‘(𝑒 βˆͺ 𝑓) β†Ύ (𝐷 βˆ– ran 𝑒)) = ((◑𝑒 βˆͺ ◑𝑓) β†Ύ (𝐷 βˆ– ran 𝑒))
207 fnunres2 6661 . . . . . . . . . . . . 13 ((◑𝑒 Fn ran 𝑒 ∧ ◑𝑓 Fn (𝐷 βˆ– ran 𝑒) ∧ (ran 𝑒 ∩ (𝐷 βˆ– ran 𝑒)) = βˆ…) β†’ ((◑𝑒 βˆͺ ◑𝑓) β†Ύ (𝐷 βˆ– ran 𝑒)) = ◑𝑓)
208152, 155, 76, 207syl3anc 1369 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((◑𝑒 βˆͺ ◑𝑓) β†Ύ (𝐷 βˆ– ran 𝑒)) = ◑𝑓)
209206, 208eqtrid 2779 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (β—‘(𝑒 βˆͺ 𝑓) β†Ύ (𝐷 βˆ– ran 𝑒)) = ◑𝑓)
210159reseq1d 5978 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((π‘€β€˜π‘’) β†Ύ (𝐷 βˆ– ran 𝑒)) = (𝑄 β†Ύ (𝐷 βˆ– ran 𝑒)))
21193, 162, 163, 69tocycfvres2 32810 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((π‘€β€˜π‘’) β†Ύ (𝐷 βˆ– ran 𝑒)) = ( I β†Ύ (𝐷 βˆ– ran 𝑒)))
212210, 211eqtr3d 2769 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (𝑄 β†Ύ (𝐷 βˆ– ran 𝑒)) = ( I β†Ύ (𝐷 βˆ– ran 𝑒)))
213209, 212coeq12d 5861 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((β—‘(𝑒 βˆͺ 𝑓) β†Ύ (𝐷 βˆ– ran 𝑒)) ∘ (𝑄 β†Ύ (𝐷 βˆ– ran 𝑒))) = (◑𝑓 ∘ ( I β†Ύ (𝐷 βˆ– ran 𝑒))))
214212rneqd 5934 . . . . . . . . . . . . 13 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ran (𝑄 β†Ύ (𝐷 βˆ– ran 𝑒)) = ran ( I β†Ύ (𝐷 βˆ– ran 𝑒)))
215 rnresi 6072 . . . . . . . . . . . . . 14 ran ( I β†Ύ (𝐷 βˆ– ran 𝑒)) = (𝐷 βˆ– ran 𝑒)
216215eqimssi 4038 . . . . . . . . . . . . 13 ran ( I β†Ύ (𝐷 βˆ– ran 𝑒)) βŠ† (𝐷 βˆ– ran 𝑒)
217214, 216eqsstrdi 4032 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ran (𝑄 β†Ύ (𝐷 βˆ– ran 𝑒)) βŠ† (𝐷 βˆ– ran 𝑒))
218 cores 6247 . . . . . . . . . . . 12 (ran (𝑄 β†Ύ (𝐷 βˆ– ran 𝑒)) βŠ† (𝐷 βˆ– ran 𝑒) β†’ ((β—‘(𝑒 βˆͺ 𝑓) β†Ύ (𝐷 βˆ– ran 𝑒)) ∘ (𝑄 β†Ύ (𝐷 βˆ– ran 𝑒))) = (β—‘(𝑒 βˆͺ 𝑓) ∘ (𝑄 β†Ύ (𝐷 βˆ– ran 𝑒))))
219217, 218syl 17 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((β—‘(𝑒 βˆͺ 𝑓) β†Ύ (𝐷 βˆ– ran 𝑒)) ∘ (𝑄 β†Ύ (𝐷 βˆ– ran 𝑒))) = (β—‘(𝑒 βˆͺ 𝑓) ∘ (𝑄 β†Ύ (𝐷 βˆ– ran 𝑒))))
220 resco 6248 . . . . . . . . . . 11 ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) β†Ύ (𝐷 βˆ– ran 𝑒)) = (β—‘(𝑒 βˆͺ 𝑓) ∘ (𝑄 β†Ύ (𝐷 βˆ– ran 𝑒)))
221219, 220eqtr4di 2785 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((β—‘(𝑒 βˆͺ 𝑓) β†Ύ (𝐷 βˆ– ran 𝑒)) ∘ (𝑄 β†Ύ (𝐷 βˆ– ran 𝑒))) = ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) β†Ύ (𝐷 βˆ– ran 𝑒)))
222213, 221eqtr3d 2769 . . . . . . . . 9 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (◑𝑓 ∘ ( I β†Ύ (𝐷 βˆ– ran 𝑒))) = ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) β†Ύ (𝐷 βˆ– ran 𝑒)))
223222coeq1d 5858 . . . . . . . 8 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((◑𝑓 ∘ ( I β†Ύ (𝐷 βˆ– ran 𝑒))) ∘ 𝑓) = (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) β†Ύ (𝐷 βˆ– ran 𝑒)) ∘ 𝑓))
224 f1of 6833 . . . . . . . . . . . 12 (◑𝑓:(𝐷 βˆ– ran 𝑒)–1-1-ontoβ†’((0..^𝑁) βˆ– dom 𝑒) β†’ ◑𝑓:(𝐷 βˆ– ran 𝑒)⟢((0..^𝑁) βˆ– dom 𝑒))
22572, 153, 2243syl 18 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ◑𝑓:(𝐷 βˆ– ran 𝑒)⟢((0..^𝑁) βˆ– dom 𝑒))
226 fcoi1 6765 . . . . . . . . . . 11 (◑𝑓:(𝐷 βˆ– ran 𝑒)⟢((0..^𝑁) βˆ– dom 𝑒) β†’ (◑𝑓 ∘ ( I β†Ύ (𝐷 βˆ– ran 𝑒))) = ◑𝑓)
227225, 226syl 17 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (◑𝑓 ∘ ( I β†Ύ (𝐷 βˆ– ran 𝑒))) = ◑𝑓)
228227coeq1d 5858 . . . . . . . . 9 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((◑𝑓 ∘ ( I β†Ύ (𝐷 βˆ– ran 𝑒))) ∘ 𝑓) = (◑𝑓 ∘ 𝑓))
229 f1ococnv1 6862 . . . . . . . . . 10 (𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒) β†’ (◑𝑓 ∘ 𝑓) = ( I β†Ύ ((0..^𝑁) βˆ– dom 𝑒)))
23072, 229syl 17 . . . . . . . . 9 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (◑𝑓 ∘ 𝑓) = ( I β†Ύ ((0..^𝑁) βˆ– dom 𝑒)))
231199reseq2d 5979 . . . . . . . . 9 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ( I β†Ύ ((0..^𝑁) βˆ– dom 𝑒)) = ( I β†Ύ (𝑃..^𝑁)))
232228, 230, 2313eqtrd 2771 . . . . . . . 8 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((◑𝑓 ∘ ( I β†Ύ (𝐷 βˆ– ran 𝑒))) ∘ 𝑓) = ( I β†Ύ (𝑃..^𝑁)))
233 f1of 6833 . . . . . . . . . 10 (𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒) β†’ 𝑓:((0..^𝑁) βˆ– dom 𝑒)⟢(𝐷 βˆ– ran 𝑒))
234 frn 6723 . . . . . . . . . 10 (𝑓:((0..^𝑁) βˆ– dom 𝑒)⟢(𝐷 βˆ– ran 𝑒) β†’ ran 𝑓 βŠ† (𝐷 βˆ– ran 𝑒))
23572, 233, 2343syl 18 . . . . . . . . 9 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ran 𝑓 βŠ† (𝐷 βˆ– ran 𝑒))
236 cores 6247 . . . . . . . . 9 (ran 𝑓 βŠ† (𝐷 βˆ– ran 𝑒) β†’ (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) β†Ύ (𝐷 βˆ– ran 𝑒)) ∘ 𝑓) = ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ 𝑓))
237235, 236syl 17 . . . . . . . 8 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) β†Ύ (𝐷 βˆ– ran 𝑒)) ∘ 𝑓) = ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ 𝑓))
238223, 232, 2373eqtr3rd 2776 . . . . . . 7 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ 𝑓) = ( I β†Ύ (𝑃..^𝑁)))
239205, 238eqtrd 2767 . . . . . 6 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) β†Ύ (𝑃..^𝑁)) = ( I β†Ύ (𝑃..^𝑁)))
240192, 239uneq12d 4160 . . . . 5 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) β†Ύ (0..^𝑃)) βˆͺ (((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) β†Ύ (𝑃..^𝑁))) = ((( I β†Ύ (0..^𝑃)) cyclShift 1) βˆͺ ( I β†Ύ (𝑃..^𝑁))))
241118, 240eqtrd 2767 . . . 4 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) = ((( I β†Ύ (0..^𝑃)) cyclShift 1) βˆͺ ( I β†Ύ (𝑃..^𝑁))))
242 vex 3473 . . . . . 6 𝑒 ∈ V
243 vex 3473 . . . . . 6 𝑓 ∈ V
244242, 243unex 7742 . . . . 5 (𝑒 βˆͺ 𝑓) ∈ V
245 f1oeq1 6821 . . . . . 6 (π‘ž = (𝑒 βˆͺ 𝑓) β†’ (π‘ž:(0..^𝑁)–1-1-onto→𝐷 ↔ (𝑒 βˆͺ 𝑓):(0..^𝑁)–1-1-onto→𝐷))
246 cnveq 5870 . . . . . . . . 9 (π‘ž = (𝑒 βˆͺ 𝑓) β†’ β—‘π‘ž = β—‘(𝑒 βˆͺ 𝑓))
247246coeq1d 5858 . . . . . . . 8 (π‘ž = (𝑒 βˆͺ 𝑓) β†’ (β—‘π‘ž ∘ 𝑄) = (β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄))
248 id 22 . . . . . . . 8 (π‘ž = (𝑒 βˆͺ 𝑓) β†’ π‘ž = (𝑒 βˆͺ 𝑓))
249247, 248coeq12d 5861 . . . . . . 7 (π‘ž = (𝑒 βˆͺ 𝑓) β†’ ((β—‘π‘ž ∘ 𝑄) ∘ π‘ž) = ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)))
250249eqeq1d 2729 . . . . . 6 (π‘ž = (𝑒 βˆͺ 𝑓) β†’ (((β—‘π‘ž ∘ 𝑄) ∘ π‘ž) = ((( I β†Ύ (0..^𝑃)) cyclShift 1) βˆͺ ( I β†Ύ (𝑃..^𝑁))) ↔ ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) = ((( I β†Ύ (0..^𝑃)) cyclShift 1) βˆͺ ( I β†Ύ (𝑃..^𝑁)))))
251245, 250anbi12d 630 . . . . 5 (π‘ž = (𝑒 βˆͺ 𝑓) β†’ ((π‘ž:(0..^𝑁)–1-1-onto→𝐷 ∧ ((β—‘π‘ž ∘ 𝑄) ∘ π‘ž) = ((( I β†Ύ (0..^𝑃)) cyclShift 1) βˆͺ ( I β†Ύ (𝑃..^𝑁)))) ↔ ((𝑒 βˆͺ 𝑓):(0..^𝑁)–1-1-onto→𝐷 ∧ ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) = ((( I β†Ύ (0..^𝑃)) cyclShift 1) βˆͺ ( I β†Ύ (𝑃..^𝑁))))))
252244, 251spcev 3591 . . . 4 (((𝑒 βˆͺ 𝑓):(0..^𝑁)–1-1-onto→𝐷 ∧ ((β—‘(𝑒 βˆͺ 𝑓) ∘ 𝑄) ∘ (𝑒 βˆͺ 𝑓)) = ((( I β†Ύ (0..^𝑃)) cyclShift 1) βˆͺ ( I β†Ύ (𝑃..^𝑁)))) β†’ βˆƒπ‘ž(π‘ž:(0..^𝑁)–1-1-onto→𝐷 ∧ ((β—‘π‘ž ∘ 𝑄) ∘ π‘ž) = ((( I β†Ύ (0..^𝑃)) cyclShift 1) βˆͺ ( I β†Ύ (𝑃..^𝑁)))))
25387, 241, 252syl2anc 583 . . 3 ((((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) ∧ 𝑓:((0..^𝑁) βˆ– dom 𝑒)–1-1-ontoβ†’(𝐷 βˆ– ran 𝑒)) β†’ βˆƒπ‘ž(π‘ž:(0..^𝑁)–1-1-onto→𝐷 ∧ ((β—‘π‘ž ∘ 𝑄) ∘ π‘ž) = ((( I β†Ύ (0..^𝑃)) cyclShift 1) βˆͺ ( I β†Ύ (𝑃..^𝑁)))))
25468, 253exlimddv 1931 . 2 (((πœ‘ ∧ 𝑒 ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))) ∧ (π‘€β€˜π‘’) = 𝑄) β†’ βˆƒπ‘ž(π‘ž:(0..^𝑁)–1-1-onto→𝐷 ∧ ((β—‘π‘ž ∘ 𝑄) ∘ π‘ž) = ((( I β†Ύ (0..^𝑃)) cyclShift 1) βˆͺ ( I β†Ύ (𝑃..^𝑁)))))
255 nfcv 2898 . . 3 Ⅎ𝑒𝑀
25693, 92, 94tocycf 32816 . . . 4 (𝐷 ∈ Fin β†’ 𝑀:{𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷}⟢𝐡)
257 ffn 6716 . . . 4 (𝑀:{𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷}⟢𝐡 β†’ 𝑀 Fn {𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷})
2584, 256, 2573syl 18 . . 3 (πœ‘ β†’ 𝑀 Fn {𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷})
25997, 91eleqtrdi 2838 . . 3 (πœ‘ β†’ 𝑄 ∈ (𝑀 β€œ (β—‘β™― β€œ {𝑃})))
260255, 258, 259fvelimad 6960 . 2 (πœ‘ β†’ βˆƒπ‘’ ∈ ({𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷} ∩ (β—‘β™― β€œ {𝑃}))(π‘€β€˜π‘’) = 𝑄)
261254, 260r19.29a 3157 1 (πœ‘ β†’ βˆƒπ‘ž(π‘ž:(0..^𝑁)–1-1-onto→𝐷 ∧ ((β—‘π‘ž ∘ 𝑄) ∘ π‘ž) = ((( I β†Ύ (0..^𝑃)) cyclShift 1) βˆͺ ( I β†Ύ (𝑃..^𝑁)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534  βˆƒwex 1774   ∈ wcel 2099  {crab 3427  Vcvv 3469   βˆ– cdif 3941   βˆͺ cun 3942   ∩ cin 3943   βŠ† wss 3944  βˆ…c0 4318  {csn 4624   class class class wbr 5142   I cid 5569  β—‘ccnv 5671  dom cdm 5672  ran crn 5673   β†Ύ cres 5674   β€œ cima 5675   ∘ ccom 5676  Rel wrel 5677  Fun wfun 6536   Fn wfn 6537  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€“ontoβ†’wfo 6540  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7414  Fincfn 8955  0cc0 11130  1c1 11131  +∞cpnf 11267   ≀ cle 11271   βˆ’ cmin 11466  β„•0cn0 12494  β„€cz 12580  β„€β‰₯cuz 12844  ...cfz 13508  ..^cfzo 13651  β™―chash 14313  Word cword 14488   cyclShift ccsh 14762  Basecbs 17171  +gcplusg 17224  -gcsg 18883  SymGrpcsymg 19312  toCycctocyc 32805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-er 8718  df-map 8838  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-inf 9458  df-dju 9916  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-4 12299  df-5 12300  df-6 12301  df-7 12302  df-8 12303  df-9 12304  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-rp 12999  df-fz 13509  df-fzo 13652  df-fl 13781  df-mod 13859  df-hash 14314  df-word 14489  df-concat 14545  df-substr 14615  df-pfx 14645  df-csh 14763  df-struct 17107  df-sets 17124  df-slot 17142  df-ndx 17154  df-base 17172  df-ress 17201  df-plusg 17237  df-tset 17243  df-efmnd 18812  df-symg 19313  df-tocyc 32806
This theorem is referenced by:  cycpmconjs  32855
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