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Theorem ablfaclem3 18695
Description: Lemma for ablfac 18696. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
Hypotheses
Ref Expression
ablfac.b 𝐵 = (Base‘𝐺)
ablfac.c 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
ablfac.1 (𝜑𝐺 ∈ Abel)
ablfac.2 (𝜑𝐵 ∈ Fin)
ablfac.o 𝑂 = (od‘𝐺)
ablfac.a 𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}
ablfac.s 𝑆 = (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})
ablfac.w 𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})
Assertion
Ref Expression
ablfaclem3 (𝜑 → (𝑊𝐵) ≠ ∅)
Distinct variable groups:   𝑠,𝑝,𝑥,𝐴   𝑔,𝑟,𝑠,𝑆   𝑔,𝑝,𝑤,𝑥,𝐵,𝑟,𝑠   𝑂,𝑝,𝑥   𝐶,𝑔,𝑝,𝑠,𝑤,𝑥   𝑊,𝑝,𝑤,𝑥   𝜑,𝑝,𝑠,𝑤,𝑥   𝑔,𝐺,𝑝,𝑟,𝑠,𝑤,𝑥
Allowed substitution hints:   𝜑(𝑔,𝑟)   𝐴(𝑤,𝑔,𝑟)   𝐶(𝑟)   𝑆(𝑥,𝑤,𝑝)   𝑂(𝑤,𝑔,𝑠,𝑟)   𝑊(𝑔,𝑠,𝑟)

Proof of Theorem ablfaclem3
Dummy variables 𝑎 𝑏 𝑐 𝑓 𝑞 𝑡 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzfid 13003 . . . 4 (𝜑 → (1...(♯‘𝐵)) ∈ Fin)
2 ablfac.a . . . . 5 𝐴 = {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)}
3 prmnn 15613 . . . . . . . 8 (𝑤 ∈ ℙ → 𝑤 ∈ ℕ)
433ad2ant2 1157 . . . . . . 7 ((𝜑𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ∈ ℕ)
5 prmz 15614 . . . . . . . . 9 (𝑤 ∈ ℙ → 𝑤 ∈ ℤ)
6 ablfac.1 . . . . . . . . . . 11 (𝜑𝐺 ∈ Abel)
7 ablgrp 18406 . . . . . . . . . . 11 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
8 ablfac.b . . . . . . . . . . . 12 𝐵 = (Base‘𝐺)
98grpbn0 17663 . . . . . . . . . . 11 (𝐺 ∈ Grp → 𝐵 ≠ ∅)
106, 7, 93syl 18 . . . . . . . . . 10 (𝜑𝐵 ≠ ∅)
11 ablfac.2 . . . . . . . . . . 11 (𝜑𝐵 ∈ Fin)
12 hashnncl 13382 . . . . . . . . . . 11 (𝐵 ∈ Fin → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
1311, 12syl 17 . . . . . . . . . 10 (𝜑 → ((♯‘𝐵) ∈ ℕ ↔ 𝐵 ≠ ∅))
1410, 13mpbird 248 . . . . . . . . 9 (𝜑 → (♯‘𝐵) ∈ ℕ)
15 dvdsle 15262 . . . . . . . . 9 ((𝑤 ∈ ℤ ∧ (♯‘𝐵) ∈ ℕ) → (𝑤 ∥ (♯‘𝐵) → 𝑤 ≤ (♯‘𝐵)))
165, 14, 15syl2anr 586 . . . . . . . 8 ((𝜑𝑤 ∈ ℙ) → (𝑤 ∥ (♯‘𝐵) → 𝑤 ≤ (♯‘𝐵)))
17163impia 1138 . . . . . . 7 ((𝜑𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ≤ (♯‘𝐵))
1814nnzd 11754 . . . . . . . . 9 (𝜑 → (♯‘𝐵) ∈ ℤ)
19183ad2ant1 1156 . . . . . . . 8 ((𝜑𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → (♯‘𝐵) ∈ ℤ)
20 fznn 12638 . . . . . . . 8 ((♯‘𝐵) ∈ ℤ → (𝑤 ∈ (1...(♯‘𝐵)) ↔ (𝑤 ∈ ℕ ∧ 𝑤 ≤ (♯‘𝐵))))
2119, 20syl 17 . . . . . . 7 ((𝜑𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → (𝑤 ∈ (1...(♯‘𝐵)) ↔ (𝑤 ∈ ℕ ∧ 𝑤 ≤ (♯‘𝐵))))
224, 17, 21mpbir2and 695 . . . . . 6 ((𝜑𝑤 ∈ ℙ ∧ 𝑤 ∥ (♯‘𝐵)) → 𝑤 ∈ (1...(♯‘𝐵)))
2322rabssdv 3890 . . . . 5 (𝜑 → {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} ⊆ (1...(♯‘𝐵)))
242, 23syl5eqss 3857 . . . 4 (𝜑𝐴 ⊆ (1...(♯‘𝐵)))
25 ssfi 8426 . . . 4 (((1...(♯‘𝐵)) ∈ Fin ∧ 𝐴 ⊆ (1...(♯‘𝐵))) → 𝐴 ∈ Fin)
261, 24, 25syl2anc 575 . . 3 (𝜑𝐴 ∈ Fin)
27 dfin5 3788 . . . . . . . 8 (Word 𝐶 ∩ (𝑊‘(𝑆𝑞))) = {𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆𝑞))}
28 ablfac.o . . . . . . . . . . . . . 14 𝑂 = (od‘𝐺)
29 ablfac.s . . . . . . . . . . . . . 14 𝑆 = (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))})
30 ssrab2 3895 . . . . . . . . . . . . . . . 16 {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} ⊆ ℙ
312, 30eqsstri 3843 . . . . . . . . . . . . . . 15 𝐴 ⊆ ℙ
3231a1i 11 . . . . . . . . . . . . . 14 (𝜑𝐴 ⊆ ℙ)
338, 28, 29, 6, 11, 32ablfac1b 18678 . . . . . . . . . . . . 13 (𝜑𝐺dom DProd 𝑆)
348fvexi 6429 . . . . . . . . . . . . . . . 16 𝐵 ∈ V
3534rabex 5018 . . . . . . . . . . . . . . 15 {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} ∈ V
3635, 29dmmpti 6241 . . . . . . . . . . . . . 14 dom 𝑆 = 𝐴
3736a1i 11 . . . . . . . . . . . . 13 (𝜑 → dom 𝑆 = 𝐴)
3833, 37dprdf2 18615 . . . . . . . . . . . 12 (𝜑𝑆:𝐴⟶(SubGrp‘𝐺))
3938ffvelrnda 6588 . . . . . . . . . . 11 ((𝜑𝑞𝐴) → (𝑆𝑞) ∈ (SubGrp‘𝐺))
40 ablfac.c . . . . . . . . . . . 12 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
41 ablfac.w . . . . . . . . . . . 12 𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})
428, 40, 6, 11, 28, 2, 29, 41ablfaclem1 18693 . . . . . . . . . . 11 ((𝑆𝑞) ∈ (SubGrp‘𝐺) → (𝑊‘(𝑆𝑞)) = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆𝑞))})
4339, 42syl 17 . . . . . . . . . 10 ((𝜑𝑞𝐴) → (𝑊‘(𝑆𝑞)) = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆𝑞))})
44 ssrab2 3895 . . . . . . . . . 10 {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆𝑞))} ⊆ Word 𝐶
4543, 44syl6eqss 3863 . . . . . . . . 9 ((𝜑𝑞𝐴) → (𝑊‘(𝑆𝑞)) ⊆ Word 𝐶)
46 sseqin2 4027 . . . . . . . . 9 ((𝑊‘(𝑆𝑞)) ⊆ Word 𝐶 ↔ (Word 𝐶 ∩ (𝑊‘(𝑆𝑞))) = (𝑊‘(𝑆𝑞)))
4745, 46sylib 209 . . . . . . . 8 ((𝜑𝑞𝐴) → (Word 𝐶 ∩ (𝑊‘(𝑆𝑞))) = (𝑊‘(𝑆𝑞)))
4827, 47syl5eqr 2865 . . . . . . 7 ((𝜑𝑞𝐴) → {𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆𝑞))} = (𝑊‘(𝑆𝑞)))
4948, 43eqtrd 2851 . . . . . 6 ((𝜑𝑞𝐴) → {𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆𝑞))} = {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆𝑞))})
50 eqid 2817 . . . . . . . . 9 (Base‘(𝐺s (𝑆𝑞))) = (Base‘(𝐺s (𝑆𝑞)))
51 eqid 2817 . . . . . . . . 9 {𝑟 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} = {𝑟 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
526adantr 468 . . . . . . . . . 10 ((𝜑𝑞𝐴) → 𝐺 ∈ Abel)
53 eqid 2817 . . . . . . . . . . 11 (𝐺s (𝑆𝑞)) = (𝐺s (𝑆𝑞))
5453subgabl 18449 . . . . . . . . . 10 ((𝐺 ∈ Abel ∧ (𝑆𝑞) ∈ (SubGrp‘𝐺)) → (𝐺s (𝑆𝑞)) ∈ Abel)
5552, 39, 54syl2anc 575 . . . . . . . . 9 ((𝜑𝑞𝐴) → (𝐺s (𝑆𝑞)) ∈ Abel)
5632sselda 3809 . . . . . . . . . 10 ((𝜑𝑞𝐴) → 𝑞 ∈ ℙ)
5753subgbas 17807 . . . . . . . . . . . . . 14 ((𝑆𝑞) ∈ (SubGrp‘𝐺) → (𝑆𝑞) = (Base‘(𝐺s (𝑆𝑞))))
5839, 57syl 17 . . . . . . . . . . . . 13 ((𝜑𝑞𝐴) → (𝑆𝑞) = (Base‘(𝐺s (𝑆𝑞))))
5958fveq2d 6419 . . . . . . . . . . . 12 ((𝜑𝑞𝐴) → (♯‘(𝑆𝑞)) = (♯‘(Base‘(𝐺s (𝑆𝑞)))))
608, 28, 29, 6, 11, 32ablfac1a 18677 . . . . . . . . . . . 12 ((𝜑𝑞𝐴) → (♯‘(𝑆𝑞)) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
6159, 60eqtr3d 2853 . . . . . . . . . . 11 ((𝜑𝑞𝐴) → (♯‘(Base‘(𝐺s (𝑆𝑞)))) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
6261oveq2d 6897 . . . . . . . . . . . . 13 ((𝜑𝑞𝐴) → (𝑞 pCnt (♯‘(Base‘(𝐺s (𝑆𝑞))))) = (𝑞 pCnt (𝑞↑(𝑞 pCnt (♯‘𝐵)))))
6314adantr 468 . . . . . . . . . . . . . . . 16 ((𝜑𝑞𝐴) → (♯‘𝐵) ∈ ℕ)
6456, 63pccld 15779 . . . . . . . . . . . . . . 15 ((𝜑𝑞𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℕ0)
6564nn0zd 11753 . . . . . . . . . . . . . 14 ((𝜑𝑞𝐴) → (𝑞 pCnt (♯‘𝐵)) ∈ ℤ)
66 pcid 15801 . . . . . . . . . . . . . 14 ((𝑞 ∈ ℙ ∧ (𝑞 pCnt (♯‘𝐵)) ∈ ℤ) → (𝑞 pCnt (𝑞↑(𝑞 pCnt (♯‘𝐵)))) = (𝑞 pCnt (♯‘𝐵)))
6756, 65, 66syl2anc 575 . . . . . . . . . . . . 13 ((𝜑𝑞𝐴) → (𝑞 pCnt (𝑞↑(𝑞 pCnt (♯‘𝐵)))) = (𝑞 pCnt (♯‘𝐵)))
6862, 67eqtrd 2851 . . . . . . . . . . . 12 ((𝜑𝑞𝐴) → (𝑞 pCnt (♯‘(Base‘(𝐺s (𝑆𝑞))))) = (𝑞 pCnt (♯‘𝐵)))
6968oveq2d 6897 . . . . . . . . . . 11 ((𝜑𝑞𝐴) → (𝑞↑(𝑞 pCnt (♯‘(Base‘(𝐺s (𝑆𝑞)))))) = (𝑞↑(𝑞 pCnt (♯‘𝐵))))
7061, 69eqtr4d 2854 . . . . . . . . . 10 ((𝜑𝑞𝐴) → (♯‘(Base‘(𝐺s (𝑆𝑞)))) = (𝑞↑(𝑞 pCnt (♯‘(Base‘(𝐺s (𝑆𝑞)))))))
7153subggrp 17806 . . . . . . . . . . . 12 ((𝑆𝑞) ∈ (SubGrp‘𝐺) → (𝐺s (𝑆𝑞)) ∈ Grp)
7239, 71syl 17 . . . . . . . . . . 11 ((𝜑𝑞𝐴) → (𝐺s (𝑆𝑞)) ∈ Grp)
7311adantr 468 . . . . . . . . . . . . 13 ((𝜑𝑞𝐴) → 𝐵 ∈ Fin)
748subgss 17804 . . . . . . . . . . . . . 14 ((𝑆𝑞) ∈ (SubGrp‘𝐺) → (𝑆𝑞) ⊆ 𝐵)
7539, 74syl 17 . . . . . . . . . . . . 13 ((𝜑𝑞𝐴) → (𝑆𝑞) ⊆ 𝐵)
76 ssfi 8426 . . . . . . . . . . . . 13 ((𝐵 ∈ Fin ∧ (𝑆𝑞) ⊆ 𝐵) → (𝑆𝑞) ∈ Fin)
7773, 75, 76syl2anc 575 . . . . . . . . . . . 12 ((𝜑𝑞𝐴) → (𝑆𝑞) ∈ Fin)
7858, 77eqeltrrd 2897 . . . . . . . . . . 11 ((𝜑𝑞𝐴) → (Base‘(𝐺s (𝑆𝑞))) ∈ Fin)
7950pgpfi2 18229 . . . . . . . . . . 11 (((𝐺s (𝑆𝑞)) ∈ Grp ∧ (Base‘(𝐺s (𝑆𝑞))) ∈ Fin) → (𝑞 pGrp (𝐺s (𝑆𝑞)) ↔ (𝑞 ∈ ℙ ∧ (♯‘(Base‘(𝐺s (𝑆𝑞)))) = (𝑞↑(𝑞 pCnt (♯‘(Base‘(𝐺s (𝑆𝑞)))))))))
8072, 78, 79syl2anc 575 . . . . . . . . . 10 ((𝜑𝑞𝐴) → (𝑞 pGrp (𝐺s (𝑆𝑞)) ↔ (𝑞 ∈ ℙ ∧ (♯‘(Base‘(𝐺s (𝑆𝑞)))) = (𝑞↑(𝑞 pCnt (♯‘(Base‘(𝐺s (𝑆𝑞)))))))))
8156, 70, 80mpbir2and 695 . . . . . . . . 9 ((𝜑𝑞𝐴) → 𝑞 pGrp (𝐺s (𝑆𝑞)))
8250, 51, 55, 81, 78pgpfac 18692 . . . . . . . 8 ((𝜑𝑞𝐴) → ∃𝑠 ∈ Word {𝑟 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ((𝐺s (𝑆𝑞))dom DProd 𝑠 ∧ ((𝐺s (𝑆𝑞)) DProd 𝑠) = (Base‘(𝐺s (𝑆𝑞)))))
83 ssrab2 3895 . . . . . . . . . . . . . 14 {𝑟 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ (SubGrp‘(𝐺s (𝑆𝑞)))
84 sswrd 13531 . . . . . . . . . . . . . 14 ({𝑟 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ (SubGrp‘(𝐺s (𝑆𝑞))) → Word {𝑟 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ Word (SubGrp‘(𝐺s (𝑆𝑞))))
8583, 84ax-mp 5 . . . . . . . . . . . . 13 Word {𝑟 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ Word (SubGrp‘(𝐺s (𝑆𝑞)))
8685sseli 3805 . . . . . . . . . . . 12 (𝑠 ∈ Word {𝑟 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} → 𝑠 ∈ Word (SubGrp‘(𝐺s (𝑆𝑞))))
8739adantr 468 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑞𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺s (𝑆𝑞)))) → (𝑆𝑞) ∈ (SubGrp‘𝐺))
8887adantr 468 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑞𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺s (𝑆𝑞)))) ∧ (𝐺s (𝑆𝑞))dom DProd 𝑠) → (𝑆𝑞) ∈ (SubGrp‘𝐺))
8953subgdmdprd 18642 . . . . . . . . . . . . . . . . . . 19 ((𝑆𝑞) ∈ (SubGrp‘𝐺) → ((𝐺s (𝑆𝑞))dom DProd 𝑠 ↔ (𝐺dom DProd 𝑠 ∧ ran 𝑠 ⊆ 𝒫 (𝑆𝑞))))
9087, 89syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑞𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺s (𝑆𝑞)))) → ((𝐺s (𝑆𝑞))dom DProd 𝑠 ↔ (𝐺dom DProd 𝑠 ∧ ran 𝑠 ⊆ 𝒫 (𝑆𝑞))))
9190simprbda 488 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑞𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺s (𝑆𝑞)))) ∧ (𝐺s (𝑆𝑞))dom DProd 𝑠) → 𝐺dom DProd 𝑠)
9290simplbda 489 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑞𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺s (𝑆𝑞)))) ∧ (𝐺s (𝑆𝑞))dom DProd 𝑠) → ran 𝑠 ⊆ 𝒫 (𝑆𝑞))
9353, 88, 91, 92subgdprd 18643 . . . . . . . . . . . . . . . 16 ((((𝜑𝑞𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺s (𝑆𝑞)))) ∧ (𝐺s (𝑆𝑞))dom DProd 𝑠) → ((𝐺s (𝑆𝑞)) DProd 𝑠) = (𝐺 DProd 𝑠))
9458ad2antrr 708 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑞𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺s (𝑆𝑞)))) ∧ (𝐺s (𝑆𝑞))dom DProd 𝑠) → (𝑆𝑞) = (Base‘(𝐺s (𝑆𝑞))))
9594eqcomd 2823 . . . . . . . . . . . . . . . 16 ((((𝜑𝑞𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺s (𝑆𝑞)))) ∧ (𝐺s (𝑆𝑞))dom DProd 𝑠) → (Base‘(𝐺s (𝑆𝑞))) = (𝑆𝑞))
9693, 95eqeq12d 2832 . . . . . . . . . . . . . . 15 ((((𝜑𝑞𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺s (𝑆𝑞)))) ∧ (𝐺s (𝑆𝑞))dom DProd 𝑠) → (((𝐺s (𝑆𝑞)) DProd 𝑠) = (Base‘(𝐺s (𝑆𝑞))) ↔ (𝐺 DProd 𝑠) = (𝑆𝑞)))
9796biimpd 220 . . . . . . . . . . . . . 14 ((((𝜑𝑞𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺s (𝑆𝑞)))) ∧ (𝐺s (𝑆𝑞))dom DProd 𝑠) → (((𝐺s (𝑆𝑞)) DProd 𝑠) = (Base‘(𝐺s (𝑆𝑞))) → (𝐺 DProd 𝑠) = (𝑆𝑞)))
9897, 91jctild 517 . . . . . . . . . . . . 13 ((((𝜑𝑞𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺s (𝑆𝑞)))) ∧ (𝐺s (𝑆𝑞))dom DProd 𝑠) → (((𝐺s (𝑆𝑞)) DProd 𝑠) = (Base‘(𝐺s (𝑆𝑞))) → (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆𝑞))))
9998expimpd 443 . . . . . . . . . . . 12 (((𝜑𝑞𝐴) ∧ 𝑠 ∈ Word (SubGrp‘(𝐺s (𝑆𝑞)))) → (((𝐺s (𝑆𝑞))dom DProd 𝑠 ∧ ((𝐺s (𝑆𝑞)) DProd 𝑠) = (Base‘(𝐺s (𝑆𝑞)))) → (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆𝑞))))
10086, 99sylan2 582 . . . . . . . . . . 11 (((𝜑𝑞𝐴) ∧ 𝑠 ∈ Word {𝑟 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}) → (((𝐺s (𝑆𝑞))dom DProd 𝑠 ∧ ((𝐺s (𝑆𝑞)) DProd 𝑠) = (Base‘(𝐺s (𝑆𝑞)))) → (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆𝑞))))
101 oveq2 6889 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑦 → ((𝐺s (𝑆𝑞)) ↾s 𝑟) = ((𝐺s (𝑆𝑞)) ↾s 𝑦))
102101eleq1d 2881 . . . . . . . . . . . . . . 15 (𝑟 = 𝑦 → (((𝐺s (𝑆𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp ) ↔ ((𝐺s (𝑆𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )))
103102cbvrabv 3400 . . . . . . . . . . . . . 14 {𝑟 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} = {𝑦 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )}
10453subsubg 17826 . . . . . . . . . . . . . . . . . . 19 ((𝑆𝑞) ∈ (SubGrp‘𝐺) → (𝑦 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ↔ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑦 ⊆ (𝑆𝑞))))
10539, 104syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑞𝐴) → (𝑦 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ↔ (𝑦 ∈ (SubGrp‘𝐺) ∧ 𝑦 ⊆ (𝑆𝑞))))
106105simprbda 488 . . . . . . . . . . . . . . . . 17 (((𝜑𝑞𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺s (𝑆𝑞)))) → 𝑦 ∈ (SubGrp‘𝐺))
1071063adant3 1155 . . . . . . . . . . . . . . . 16 (((𝜑𝑞𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∧ ((𝐺s (𝑆𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )) → 𝑦 ∈ (SubGrp‘𝐺))
108393ad2ant1 1156 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑞𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∧ ((𝐺s (𝑆𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )) → (𝑆𝑞) ∈ (SubGrp‘𝐺))
109105simplbda 489 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑞𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺s (𝑆𝑞)))) → 𝑦 ⊆ (𝑆𝑞))
1101093adant3 1155 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑞𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∧ ((𝐺s (𝑆𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )) → 𝑦 ⊆ (𝑆𝑞))
111 ressabs 16158 . . . . . . . . . . . . . . . . . 18 (((𝑆𝑞) ∈ (SubGrp‘𝐺) ∧ 𝑦 ⊆ (𝑆𝑞)) → ((𝐺s (𝑆𝑞)) ↾s 𝑦) = (𝐺s 𝑦))
112108, 110, 111syl2anc 575 . . . . . . . . . . . . . . . . 17 (((𝜑𝑞𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∧ ((𝐺s (𝑆𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )) → ((𝐺s (𝑆𝑞)) ↾s 𝑦) = (𝐺s 𝑦))
113 simp3 1161 . . . . . . . . . . . . . . . . 17 (((𝜑𝑞𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∧ ((𝐺s (𝑆𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )) → ((𝐺s (𝑆𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp ))
114112, 113eqeltrrd 2897 . . . . . . . . . . . . . . . 16 (((𝜑𝑞𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∧ ((𝐺s (𝑆𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )) → (𝐺s 𝑦) ∈ (CycGrp ∩ ran pGrp ))
115 oveq2 6889 . . . . . . . . . . . . . . . . . 18 (𝑟 = 𝑦 → (𝐺s 𝑟) = (𝐺s 𝑦))
116115eleq1d 2881 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑦 → ((𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp ) ↔ (𝐺s 𝑦) ∈ (CycGrp ∩ ran pGrp )))
117116, 40elrab2 3573 . . . . . . . . . . . . . . . 16 (𝑦𝐶 ↔ (𝑦 ∈ (SubGrp‘𝐺) ∧ (𝐺s 𝑦) ∈ (CycGrp ∩ ran pGrp )))
118107, 114, 117sylanbrc 574 . . . . . . . . . . . . . . 15 (((𝜑𝑞𝐴) ∧ 𝑦 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∧ ((𝐺s (𝑆𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )) → 𝑦𝐶)
119118rabssdv 3890 . . . . . . . . . . . . . 14 ((𝜑𝑞𝐴) → {𝑦 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑦) ∈ (CycGrp ∩ ran pGrp )} ⊆ 𝐶)
120103, 119syl5eqss 3857 . . . . . . . . . . . . 13 ((𝜑𝑞𝐴) → {𝑟 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ 𝐶)
121 sswrd 13531 . . . . . . . . . . . . 13 ({𝑟 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ 𝐶 → Word {𝑟 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ Word 𝐶)
122120, 121syl 17 . . . . . . . . . . . 12 ((𝜑𝑞𝐴) → Word {𝑟 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ⊆ Word 𝐶)
123122sselda 3809 . . . . . . . . . . 11 (((𝜑𝑞𝐴) ∧ 𝑠 ∈ Word {𝑟 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}) → 𝑠 ∈ Word 𝐶)
124100, 123jctild 517 . . . . . . . . . 10 (((𝜑𝑞𝐴) ∧ 𝑠 ∈ Word {𝑟 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )}) → (((𝐺s (𝑆𝑞))dom DProd 𝑠 ∧ ((𝐺s (𝑆𝑞)) DProd 𝑠) = (Base‘(𝐺s (𝑆𝑞)))) → (𝑠 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆𝑞)))))
125124expimpd 443 . . . . . . . . 9 ((𝜑𝑞𝐴) → ((𝑠 ∈ Word {𝑟 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ∧ ((𝐺s (𝑆𝑞))dom DProd 𝑠 ∧ ((𝐺s (𝑆𝑞)) DProd 𝑠) = (Base‘(𝐺s (𝑆𝑞))))) → (𝑠 ∈ Word 𝐶 ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆𝑞)))))
126125reximdv2 3212 . . . . . . . 8 ((𝜑𝑞𝐴) → (∃𝑠 ∈ Word {𝑟 ∈ (SubGrp‘(𝐺s (𝑆𝑞))) ∣ ((𝐺s (𝑆𝑞)) ↾s 𝑟) ∈ (CycGrp ∩ ran pGrp )} ((𝐺s (𝑆𝑞))dom DProd 𝑠 ∧ ((𝐺s (𝑆𝑞)) DProd 𝑠) = (Base‘(𝐺s (𝑆𝑞)))) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆𝑞))))
12782, 126mpd 15 . . . . . . 7 ((𝜑𝑞𝐴) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆𝑞)))
128 rabn0 4169 . . . . . . 7 ({𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆𝑞))} ≠ ∅ ↔ ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆𝑞)))
129127, 128sylibr 225 . . . . . 6 ((𝜑𝑞𝐴) → {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (𝑆𝑞))} ≠ ∅)
13049, 129eqnetrd 3056 . . . . 5 ((𝜑𝑞𝐴) → {𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆𝑞))} ≠ ∅)
131 rabn0 4169 . . . . 5 ({𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆𝑞))} ≠ ∅ ↔ ∃𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆𝑞)))
132130, 131sylib 209 . . . 4 ((𝜑𝑞𝐴) → ∃𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆𝑞)))
133132ralrimiva 3165 . . 3 (𝜑 → ∀𝑞𝐴𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆𝑞)))
134 eleq1 2884 . . . 4 (𝑦 = (𝑓𝑞) → (𝑦 ∈ (𝑊‘(𝑆𝑞)) ↔ (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞))))
135134ac6sfi 8450 . . 3 ((𝐴 ∈ Fin ∧ ∀𝑞𝐴𝑦 ∈ Word 𝐶𝑦 ∈ (𝑊‘(𝑆𝑞))) → ∃𝑓(𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞))))
13626, 133, 135syl2anc 575 . 2 (𝜑 → ∃𝑓(𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞))))
137 sneq 4391 . . . . . . . . 9 (𝑞 = 𝑦 → {𝑞} = {𝑦})
138 fveq2 6415 . . . . . . . . . 10 (𝑞 = 𝑦 → (𝑓𝑞) = (𝑓𝑦))
139138dmeqd 5538 . . . . . . . . 9 (𝑞 = 𝑦 → dom (𝑓𝑞) = dom (𝑓𝑦))
140137, 139xpeq12d 5352 . . . . . . . 8 (𝑞 = 𝑦 → ({𝑞} × dom (𝑓𝑞)) = ({𝑦} × dom (𝑓𝑦)))
141140cbviunv 4762 . . . . . . 7 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)) = 𝑦𝐴 ({𝑦} × dom (𝑓𝑦))
14226adantr 468 . . . . . . . 8 ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞)))) → 𝐴 ∈ Fin)
143 snfi 8284 . . . . . . . . . 10 {𝑦} ∈ Fin
144 simprl 778 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞)))) → 𝑓:𝐴⟶Word 𝐶)
145144ffvelrnda 6588 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞)))) ∧ 𝑦𝐴) → (𝑓𝑦) ∈ Word 𝐶)
146 wrdf 13528 . . . . . . . . . . . 12 ((𝑓𝑦) ∈ Word 𝐶 → (𝑓𝑦):(0..^(♯‘(𝑓𝑦)))⟶𝐶)
147 fdm 6271 . . . . . . . . . . . 12 ((𝑓𝑦):(0..^(♯‘(𝑓𝑦)))⟶𝐶 → dom (𝑓𝑦) = (0..^(♯‘(𝑓𝑦))))
148145, 146, 1473syl 18 . . . . . . . . . . 11 (((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞)))) ∧ 𝑦𝐴) → dom (𝑓𝑦) = (0..^(♯‘(𝑓𝑦))))
149 fzofi 13004 . . . . . . . . . . 11 (0..^(♯‘(𝑓𝑦))) ∈ Fin
150148, 149syl6eqel 2904 . . . . . . . . . 10 (((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞)))) ∧ 𝑦𝐴) → dom (𝑓𝑦) ∈ Fin)
151 xpfi 8477 . . . . . . . . . 10 (({𝑦} ∈ Fin ∧ dom (𝑓𝑦) ∈ Fin) → ({𝑦} × dom (𝑓𝑦)) ∈ Fin)
152143, 150, 151sylancr 577 . . . . . . . . 9 (((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞)))) ∧ 𝑦𝐴) → ({𝑦} × dom (𝑓𝑦)) ∈ Fin)
153152ralrimiva 3165 . . . . . . . 8 ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞)))) → ∀𝑦𝐴 ({𝑦} × dom (𝑓𝑦)) ∈ Fin)
154 iunfi 8500 . . . . . . . 8 ((𝐴 ∈ Fin ∧ ∀𝑦𝐴 ({𝑦} × dom (𝑓𝑦)) ∈ Fin) → 𝑦𝐴 ({𝑦} × dom (𝑓𝑦)) ∈ Fin)
155142, 153, 154syl2anc 575 . . . . . . 7 ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞)))) → 𝑦𝐴 ({𝑦} × dom (𝑓𝑦)) ∈ Fin)
156141, 155syl5eqel 2900 . . . . . 6 ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞)))) → 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)) ∈ Fin)
157 hashcl 13372 . . . . . 6 ( 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)) ∈ Fin → (♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))) ∈ ℕ0)
158 hashfzo0 13441 . . . . . 6 ((♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))) ∈ ℕ0 → (♯‘(0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))) = (♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))
159156, 157, 1583syl 18 . . . . 5 ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞)))) → (♯‘(0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))) = (♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))
160 fzofi 13004 . . . . . 6 (0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)))) ∈ Fin
161 hashen 13362 . . . . . 6 (((0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)))) ∈ Fin ∧ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)) ∈ Fin) → ((♯‘(0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))) = (♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))) ↔ (0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)))) ≈ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))
162160, 156, 161sylancr 577 . . . . 5 ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞)))) → ((♯‘(0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))) = (♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))) ↔ (0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)))) ≈ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))
163159, 162mpbid 223 . . . 4 ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞)))) → (0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)))) ≈ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)))
164 bren 8208 . . . 4 ((0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)))) ≈ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)) ↔ ∃ :(0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))–1-1-onto 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)))
165163, 164sylib 209 . . 3 ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞)))) → ∃ :(0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))–1-1-onto 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)))
1666adantr 468 . . . . . 6 ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞))) ∧ :(0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))–1-1-onto 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)))) → 𝐺 ∈ Abel)
16711adantr 468 . . . . . 6 ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞))) ∧ :(0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))–1-1-onto 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)))) → 𝐵 ∈ Fin)
168 breq1 4858 . . . . . . . 8 (𝑤 = 𝑎 → (𝑤 ∥ (♯‘𝐵) ↔ 𝑎 ∥ (♯‘𝐵)))
169168cbvrabv 3400 . . . . . . 7 {𝑤 ∈ ℙ ∣ 𝑤 ∥ (♯‘𝐵)} = {𝑎 ∈ ℙ ∣ 𝑎 ∥ (♯‘𝐵)}
1702, 169eqtri 2839 . . . . . 6 𝐴 = {𝑎 ∈ ℙ ∣ 𝑎 ∥ (♯‘𝐵)}
171 fveq2 6415 . . . . . . . . . . 11 (𝑥 = 𝑐 → (𝑂𝑥) = (𝑂𝑐))
172171breq1d 4865 . . . . . . . . . 10 (𝑥 = 𝑐 → ((𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂𝑐) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))))
173172cbvrabv 3400 . . . . . . . . 9 {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑐𝐵 ∣ (𝑂𝑐) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}
174 id 22 . . . . . . . . . . . 12 (𝑝 = 𝑏𝑝 = 𝑏)
175 oveq1 6888 . . . . . . . . . . . 12 (𝑝 = 𝑏 → (𝑝 pCnt (♯‘𝐵)) = (𝑏 pCnt (♯‘𝐵)))
176174, 175oveq12d 6899 . . . . . . . . . . 11 (𝑝 = 𝑏 → (𝑝↑(𝑝 pCnt (♯‘𝐵))) = (𝑏↑(𝑏 pCnt (♯‘𝐵))))
177176breq2d 4867 . . . . . . . . . 10 (𝑝 = 𝑏 → ((𝑂𝑐) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵))) ↔ (𝑂𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))))
178177rabbidv 3390 . . . . . . . . 9 (𝑝 = 𝑏 → {𝑐𝐵 ∣ (𝑂𝑐) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑐𝐵 ∣ (𝑂𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))})
179173, 178syl5eq 2863 . . . . . . . 8 (𝑝 = 𝑏 → {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))} = {𝑐𝐵 ∣ (𝑂𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))})
180179cbvmptv 4955 . . . . . . 7 (𝑝𝐴 ↦ {𝑥𝐵 ∣ (𝑂𝑥) ∥ (𝑝↑(𝑝 pCnt (♯‘𝐵)))}) = (𝑏𝐴 ↦ {𝑐𝐵 ∣ (𝑂𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))})
18129, 180eqtri 2839 . . . . . 6 𝑆 = (𝑏𝐴 ↦ {𝑐𝐵 ∣ (𝑂𝑐) ∥ (𝑏↑(𝑏 pCnt (♯‘𝐵)))})
182 breq2 4859 . . . . . . . . . 10 (𝑠 = 𝑡 → (𝐺dom DProd 𝑠𝐺dom DProd 𝑡))
183 oveq2 6889 . . . . . . . . . . 11 (𝑠 = 𝑡 → (𝐺 DProd 𝑠) = (𝐺 DProd 𝑡))
184183eqeq1d 2819 . . . . . . . . . 10 (𝑠 = 𝑡 → ((𝐺 DProd 𝑠) = 𝑔 ↔ (𝐺 DProd 𝑡) = 𝑔))
185182, 184anbi12d 618 . . . . . . . . 9 (𝑠 = 𝑡 → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔) ↔ (𝐺dom DProd 𝑡 ∧ (𝐺 DProd 𝑡) = 𝑔)))
186185cbvrabv 3400 . . . . . . . 8 {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)} = {𝑡 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑡 ∧ (𝐺 DProd 𝑡) = 𝑔)}
187186mpteq2i 4946 . . . . . . 7 (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑠 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)}) = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑡 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑡 ∧ (𝐺 DProd 𝑡) = 𝑔)})
18841, 187eqtri 2839 . . . . . 6 𝑊 = (𝑔 ∈ (SubGrp‘𝐺) ↦ {𝑡 ∈ Word 𝐶 ∣ (𝐺dom DProd 𝑡 ∧ (𝐺 DProd 𝑡) = 𝑔)})
189 simprll 788 . . . . . 6 ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞))) ∧ :(0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))–1-1-onto 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)))) → 𝑓:𝐴⟶Word 𝐶)
190 simprlr 789 . . . . . . 7 ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞))) ∧ :(0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))–1-1-onto 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)))) → ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞)))
191 2fveq3 6420 . . . . . . . . 9 (𝑞 = 𝑦 → (𝑊‘(𝑆𝑞)) = (𝑊‘(𝑆𝑦)))
192138, 191eleq12d 2890 . . . . . . . 8 (𝑞 = 𝑦 → ((𝑓𝑞) ∈ (𝑊‘(𝑆𝑞)) ↔ (𝑓𝑦) ∈ (𝑊‘(𝑆𝑦))))
193192cbvralv 3371 . . . . . . 7 (∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞)) ↔ ∀𝑦𝐴 (𝑓𝑦) ∈ (𝑊‘(𝑆𝑦)))
194190, 193sylib 209 . . . . . 6 ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞))) ∧ :(0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))–1-1-onto 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)))) → ∀𝑦𝐴 (𝑓𝑦) ∈ (𝑊‘(𝑆𝑦)))
195 simprr 780 . . . . . 6 ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞))) ∧ :(0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))–1-1-onto 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)))) → :(0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))–1-1-onto 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)))
1968, 40, 166, 167, 28, 170, 181, 188, 189, 194, 141, 195ablfaclem2 18694 . . . . 5 ((𝜑 ∧ ((𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞))) ∧ :(0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))–1-1-onto 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)))) → (𝑊𝐵) ≠ ∅)
197196expr 446 . . . 4 ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞)))) → (:(0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))–1-1-onto 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)) → (𝑊𝐵) ≠ ∅))
198197exlimdv 2024 . . 3 ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞)))) → (∃ :(0..^(♯‘ 𝑞𝐴 ({𝑞} × dom (𝑓𝑞))))–1-1-onto 𝑞𝐴 ({𝑞} × dom (𝑓𝑞)) → (𝑊𝐵) ≠ ∅))
199165, 198mpd 15 . 2 ((𝜑 ∧ (𝑓:𝐴⟶Word 𝐶 ∧ ∀𝑞𝐴 (𝑓𝑞) ∈ (𝑊‘(𝑆𝑞)))) → (𝑊𝐵) ≠ ∅)
200136, 199exlimddv 2026 1 (𝜑 → (𝑊𝐵) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wex 1859  wcel 2157  wne 2989  wral 3107  wrex 3108  {crab 3111  cin 3779  wss 3780  c0 4127  𝒫 cpw 4362  {csn 4381   ciun 4723   class class class wbr 4855  cmpt 4934   × cxp 5320  dom cdm 5322  ran crn 5323  wf 6104  1-1-ontowf1o 6107  cfv 6108  (class class class)co 6881  cen 8196  Fincfn 8199  0cc0 10228  1c1 10229  cle 10367  cn 11312  0cn0 11566  cz 11650  ...cfz 12556  ..^cfzo 12696  cexp 13090  chash 13344  Word cword 13509  cdvds 15210  cprime 15610   pCnt cpc 15765  Basecbs 16075  s cress 16076  Grpcgrp 17634  SubGrpcsubg 17797  odcod 18152   pGrp cpgp 18154  Abelcabl 18402  CycGrpccyg 18487   DProd cdprd 18601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4975  ax-sep 4986  ax-nul 4994  ax-pow 5046  ax-pr 5107  ax-un 7186  ax-inf2 8792  ax-cnex 10284  ax-resscn 10285  ax-1cn 10286  ax-icn 10287  ax-addcl 10288  ax-addrcl 10289  ax-mulcl 10290  ax-mulrcl 10291  ax-mulcom 10292  ax-addass 10293  ax-mulass 10294  ax-distr 10295  ax-i2m1 10296  ax-1ne0 10297  ax-1rid 10298  ax-rnegex 10299  ax-rrecex 10300  ax-cnre 10301  ax-pre-lttri 10302  ax-pre-lttrn 10303  ax-pre-ltadd 10304  ax-pre-mulgt0 10305  ax-pre-sup 10306
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-iin 4726  df-disj 4824  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5230  df-eprel 5235  df-po 5243  df-so 5244  df-fr 5281  df-se 5282  df-we 5283  df-xp 5328  df-rel 5329  df-cnv 5330  df-co 5331  df-dm 5332  df-rn 5333  df-res 5334  df-ima 5335  df-pred 5904  df-ord 5950  df-on 5951  df-lim 5952  df-suc 5953  df-iota 6071  df-fun 6110  df-fn 6111  df-f 6112  df-f1 6113  df-fo 6114  df-f1o 6115  df-fv 6116  df-isom 6117  df-riota 6842  df-ov 6884  df-oprab 6885  df-mpt2 6886  df-of 7134  df-rpss 7174  df-om 7303  df-1st 7405  df-2nd 7406  df-supp 7537  df-tpos 7594  df-wrecs 7649  df-recs 7711  df-rdg 7749  df-1o 7803  df-2o 7804  df-oadd 7807  df-omul 7808  df-er 7986  df-ec 7988  df-qs 7992  df-map 8101  df-pm 8102  df-ixp 8153  df-en 8200  df-dom 8201  df-sdom 8202  df-fin 8203  df-fsupp 8522  df-sup 8594  df-inf 8595  df-oi 8661  df-card 9055  df-acn 9058  df-cda 9282  df-pnf 10368  df-mnf 10369  df-xr 10370  df-ltxr 10371  df-le 10372  df-sub 10560  df-neg 10561  df-div 10977  df-nn 11313  df-2 11371  df-3 11372  df-n0 11567  df-xnn0 11637  df-z 11651  df-uz 11912  df-q 12015  df-rp 12054  df-fz 12557  df-fzo 12697  df-fl 12824  df-mod 12900  df-seq 13032  df-exp 13091  df-fac 13288  df-bc 13317  df-hash 13345  df-word 13517  df-concat 13519  df-s1 13520  df-cj 14069  df-re 14070  df-im 14071  df-sqrt 14205  df-abs 14206  df-clim 14449  df-sum 14647  df-dvds 15211  df-gcd 15443  df-prm 15611  df-pc 15766  df-ndx 16078  df-slot 16079  df-base 16081  df-sets 16082  df-ress 16083  df-plusg 16173  df-0g 16314  df-gsum 16315  df-mre 16458  df-mrc 16459  df-acs 16461  df-mgm 17454  df-sgrp 17496  df-mnd 17507  df-mhm 17547  df-submnd 17548  df-grp 17637  df-minusg 17638  df-sbg 17639  df-mulg 17753  df-subg 17800  df-eqg 17802  df-ghm 17867  df-gim 17910  df-ga 17931  df-cntz 17958  df-oppg 17984  df-od 18156  df-gex 18157  df-pgp 18158  df-lsm 18259  df-pj1 18260  df-cmn 18403  df-abl 18404  df-cyg 18488  df-dprd 18603
This theorem is referenced by:  ablfac  18696
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