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Theorem ablfaclem3 19998
Description: Lemma for ablfac 19999. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
Hypotheses
Ref Expression
ablfac.b 𝐡 = (Baseβ€˜πΊ)
ablfac.c 𝐢 = {π‘Ÿ ∈ (SubGrpβ€˜πΊ) ∣ (𝐺 β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )}
ablfac.1 (πœ‘ β†’ 𝐺 ∈ Abel)
ablfac.2 (πœ‘ β†’ 𝐡 ∈ Fin)
ablfac.o 𝑂 = (odβ€˜πΊ)
ablfac.a 𝐴 = {𝑀 ∈ β„™ ∣ 𝑀 βˆ₯ (β™―β€˜π΅)}
ablfac.s 𝑆 = (𝑝 ∈ 𝐴 ↦ {π‘₯ ∈ 𝐡 ∣ (π‘‚β€˜π‘₯) βˆ₯ (𝑝↑(𝑝 pCnt (β™―β€˜π΅)))})
ablfac.w π‘Š = (𝑔 ∈ (SubGrpβ€˜πΊ) ↦ {𝑠 ∈ Word 𝐢 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})
Assertion
Ref Expression
ablfaclem3 (πœ‘ β†’ (π‘Šβ€˜π΅) β‰  βˆ…)
Distinct variable groups:   𝑠,𝑝,π‘₯,𝐴   𝑔,π‘Ÿ,𝑠,𝑆   𝑔,𝑝,𝑀,π‘₯,𝐡,π‘Ÿ,𝑠   𝑂,𝑝,π‘₯   𝐢,𝑔,𝑝,𝑠,𝑀,π‘₯   π‘Š,𝑝,𝑀,π‘₯   πœ‘,𝑝,𝑠,𝑀,π‘₯   𝑔,𝐺,𝑝,π‘Ÿ,𝑠,𝑀,π‘₯
Allowed substitution hints:   πœ‘(𝑔,π‘Ÿ)   𝐴(𝑀,𝑔,π‘Ÿ)   𝐢(π‘Ÿ)   𝑆(π‘₯,𝑀,𝑝)   𝑂(𝑀,𝑔,𝑠,π‘Ÿ)   π‘Š(𝑔,𝑠,π‘Ÿ)
Proof of Theorem ablfaclem3
Dummy variables π‘Ž 𝑏 𝑐 𝑓 β„Ž π‘ž 𝑑 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzfid 13942 . . . 4 (πœ‘ β†’ (1...(β™―β€˜π΅)) ∈ Fin)
2 ablfac.a . . . . 5 𝐴 = {𝑀 ∈ β„™ ∣ 𝑀 βˆ₯ (β™―β€˜π΅)}
3 prmnn 16615 . . . . . . . 8 (𝑀 ∈ β„™ β†’ 𝑀 ∈ β„•)
433ad2ant2 1132 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ β„™ ∧ 𝑀 βˆ₯ (β™―β€˜π΅)) β†’ 𝑀 ∈ β„•)
5 prmz 16616 . . . . . . . . 9 (𝑀 ∈ β„™ β†’ 𝑀 ∈ β„€)
6 ablfac.1 . . . . . . . . . . 11 (πœ‘ β†’ 𝐺 ∈ Abel)
7 ablgrp 19694 . . . . . . . . . . 11 (𝐺 ∈ Abel β†’ 𝐺 ∈ Grp)
8 ablfac.b . . . . . . . . . . . 12 𝐡 = (Baseβ€˜πΊ)
98grpbn0 18887 . . . . . . . . . . 11 (𝐺 ∈ Grp β†’ 𝐡 β‰  βˆ…)
106, 7, 93syl 18 . . . . . . . . . 10 (πœ‘ β†’ 𝐡 β‰  βˆ…)
11 ablfac.2 . . . . . . . . . . 11 (πœ‘ β†’ 𝐡 ∈ Fin)
12 hashnncl 14330 . . . . . . . . . . 11 (𝐡 ∈ Fin β†’ ((β™―β€˜π΅) ∈ β„• ↔ 𝐡 β‰  βˆ…))
1311, 12syl 17 . . . . . . . . . 10 (πœ‘ β†’ ((β™―β€˜π΅) ∈ β„• ↔ 𝐡 β‰  βˆ…))
1410, 13mpbird 256 . . . . . . . . 9 (πœ‘ β†’ (β™―β€˜π΅) ∈ β„•)
15 dvdsle 16257 . . . . . . . . 9 ((𝑀 ∈ β„€ ∧ (β™―β€˜π΅) ∈ β„•) β†’ (𝑀 βˆ₯ (β™―β€˜π΅) β†’ 𝑀 ≀ (β™―β€˜π΅)))
165, 14, 15syl2anr 595 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ β„™) β†’ (𝑀 βˆ₯ (β™―β€˜π΅) β†’ 𝑀 ≀ (β™―β€˜π΅)))
17163impia 1115 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ β„™ ∧ 𝑀 βˆ₯ (β™―β€˜π΅)) β†’ 𝑀 ≀ (β™―β€˜π΅))
1814nnzd 12589 . . . . . . . . 9 (πœ‘ β†’ (β™―β€˜π΅) ∈ β„€)
19183ad2ant1 1131 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ β„™ ∧ 𝑀 βˆ₯ (β™―β€˜π΅)) β†’ (β™―β€˜π΅) ∈ β„€)
20 fznn 13573 . . . . . . . 8 ((β™―β€˜π΅) ∈ β„€ β†’ (𝑀 ∈ (1...(β™―β€˜π΅)) ↔ (𝑀 ∈ β„• ∧ 𝑀 ≀ (β™―β€˜π΅))))
2119, 20syl 17 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ β„™ ∧ 𝑀 βˆ₯ (β™―β€˜π΅)) β†’ (𝑀 ∈ (1...(β™―β€˜π΅)) ↔ (𝑀 ∈ β„• ∧ 𝑀 ≀ (β™―β€˜π΅))))
224, 17, 21mpbir2and 709 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ β„™ ∧ 𝑀 βˆ₯ (β™―β€˜π΅)) β†’ 𝑀 ∈ (1...(β™―β€˜π΅)))
2322rabssdv 4071 . . . . 5 (πœ‘ β†’ {𝑀 ∈ β„™ ∣ 𝑀 βˆ₯ (β™―β€˜π΅)} βŠ† (1...(β™―β€˜π΅)))
242, 23eqsstrid 4029 . . . 4 (πœ‘ β†’ 𝐴 βŠ† (1...(β™―β€˜π΅)))
251, 24ssfid 9269 . . 3 (πœ‘ β†’ 𝐴 ∈ Fin)
26 dfin5 3955 . . . . . . . 8 (Word 𝐢 ∩ (π‘Šβ€˜(π‘†β€˜π‘ž))) = {𝑦 ∈ Word 𝐢 ∣ 𝑦 ∈ (π‘Šβ€˜(π‘†β€˜π‘ž))}
27 ablfac.o . . . . . . . . . . . . . 14 𝑂 = (odβ€˜πΊ)
28 ablfac.s . . . . . . . . . . . . . 14 𝑆 = (𝑝 ∈ 𝐴 ↦ {π‘₯ ∈ 𝐡 ∣ (π‘‚β€˜π‘₯) βˆ₯ (𝑝↑(𝑝 pCnt (β™―β€˜π΅)))})
292ssrab3 4079 . . . . . . . . . . . . . . 15 𝐴 βŠ† β„™
3029a1i 11 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐴 βŠ† β„™)
318, 27, 28, 6, 11, 30ablfac1b 19981 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐺dom DProd 𝑆)
328fvexi 6904 . . . . . . . . . . . . . . . 16 𝐡 ∈ V
3332rabex 5331 . . . . . . . . . . . . . . 15 {π‘₯ ∈ 𝐡 ∣ (π‘‚β€˜π‘₯) βˆ₯ (𝑝↑(𝑝 pCnt (β™―β€˜π΅)))} ∈ V
3433, 28dmmpti 6693 . . . . . . . . . . . . . 14 dom 𝑆 = 𝐴
3534a1i 11 . . . . . . . . . . . . 13 (πœ‘ β†’ dom 𝑆 = 𝐴)
3631, 35dprdf2 19918 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑆:𝐴⟢(SubGrpβ€˜πΊ))
3736ffvelcdmda 7085 . . . . . . . . . . 11 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (π‘†β€˜π‘ž) ∈ (SubGrpβ€˜πΊ))
38 ablfac.c . . . . . . . . . . . 12 𝐢 = {π‘Ÿ ∈ (SubGrpβ€˜πΊ) ∣ (𝐺 β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )}
39 ablfac.w . . . . . . . . . . . 12 π‘Š = (𝑔 ∈ (SubGrpβ€˜πΊ) ↦ {𝑠 ∈ Word 𝐢 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)})
408, 38, 6, 11, 27, 2, 28, 39ablfaclem1 19996 . . . . . . . . . . 11 ((π‘†β€˜π‘ž) ∈ (SubGrpβ€˜πΊ) β†’ (π‘Šβ€˜(π‘†β€˜π‘ž)) = {𝑠 ∈ Word 𝐢 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (π‘†β€˜π‘ž))})
4137, 40syl 17 . . . . . . . . . 10 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (π‘Šβ€˜(π‘†β€˜π‘ž)) = {𝑠 ∈ Word 𝐢 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (π‘†β€˜π‘ž))})
42 ssrab2 4076 . . . . . . . . . 10 {𝑠 ∈ Word 𝐢 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (π‘†β€˜π‘ž))} βŠ† Word 𝐢
4341, 42eqsstrdi 4035 . . . . . . . . 9 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (π‘Šβ€˜(π‘†β€˜π‘ž)) βŠ† Word 𝐢)
44 sseqin2 4214 . . . . . . . . 9 ((π‘Šβ€˜(π‘†β€˜π‘ž)) βŠ† Word 𝐢 ↔ (Word 𝐢 ∩ (π‘Šβ€˜(π‘†β€˜π‘ž))) = (π‘Šβ€˜(π‘†β€˜π‘ž)))
4543, 44sylib 217 . . . . . . . 8 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (Word 𝐢 ∩ (π‘Šβ€˜(π‘†β€˜π‘ž))) = (π‘Šβ€˜(π‘†β€˜π‘ž)))
4626, 45eqtr3id 2784 . . . . . . 7 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ {𝑦 ∈ Word 𝐢 ∣ 𝑦 ∈ (π‘Šβ€˜(π‘†β€˜π‘ž))} = (π‘Šβ€˜(π‘†β€˜π‘ž)))
4746, 41eqtrd 2770 . . . . . 6 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ {𝑦 ∈ Word 𝐢 ∣ 𝑦 ∈ (π‘Šβ€˜(π‘†β€˜π‘ž))} = {𝑠 ∈ Word 𝐢 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (π‘†β€˜π‘ž))})
48 eqid 2730 . . . . . . . . 9 (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) = (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))
49 eqid 2730 . . . . . . . . 9 {π‘Ÿ ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )} = {π‘Ÿ ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )}
50 eqid 2730 . . . . . . . . . . 11 (𝐺 β†Ύs (π‘†β€˜π‘ž)) = (𝐺 β†Ύs (π‘†β€˜π‘ž))
5150subgabl 19745 . . . . . . . . . 10 ((𝐺 ∈ Abel ∧ (π‘†β€˜π‘ž) ∈ (SubGrpβ€˜πΊ)) β†’ (𝐺 β†Ύs (π‘†β€˜π‘ž)) ∈ Abel)
526, 37, 51syl2an2r 681 . . . . . . . . 9 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (𝐺 β†Ύs (π‘†β€˜π‘ž)) ∈ Abel)
5330sselda 3981 . . . . . . . . . 10 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ π‘ž ∈ β„™)
5450subgbas 19046 . . . . . . . . . . . . . 14 ((π‘†β€˜π‘ž) ∈ (SubGrpβ€˜πΊ) β†’ (π‘†β€˜π‘ž) = (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))))
5537, 54syl 17 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (π‘†β€˜π‘ž) = (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))))
5655fveq2d 6894 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (β™―β€˜(π‘†β€˜π‘ž)) = (β™―β€˜(Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))))
578, 27, 28, 6, 11, 30ablfac1a 19980 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (β™―β€˜(π‘†β€˜π‘ž)) = (π‘žβ†‘(π‘ž pCnt (β™―β€˜π΅))))
5856, 57eqtr3d 2772 . . . . . . . . . . 11 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (β™―β€˜(Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) = (π‘žβ†‘(π‘ž pCnt (β™―β€˜π΅))))
5958oveq2d 7427 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (π‘ž pCnt (β™―β€˜(Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))))) = (π‘ž pCnt (π‘žβ†‘(π‘ž pCnt (β™―β€˜π΅)))))
6014adantr 479 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (β™―β€˜π΅) ∈ β„•)
6153, 60pccld 16787 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (π‘ž pCnt (β™―β€˜π΅)) ∈ β„•0)
6261nn0zd 12588 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (π‘ž pCnt (β™―β€˜π΅)) ∈ β„€)
63 pcid 16810 . . . . . . . . . . . . . 14 ((π‘ž ∈ β„™ ∧ (π‘ž pCnt (β™―β€˜π΅)) ∈ β„€) β†’ (π‘ž pCnt (π‘žβ†‘(π‘ž pCnt (β™―β€˜π΅)))) = (π‘ž pCnt (β™―β€˜π΅)))
6453, 62, 63syl2anc 582 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (π‘ž pCnt (π‘žβ†‘(π‘ž pCnt (β™―β€˜π΅)))) = (π‘ž pCnt (β™―β€˜π΅)))
6559, 64eqtrd 2770 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (π‘ž pCnt (β™―β€˜(Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))))) = (π‘ž pCnt (β™―β€˜π΅)))
6665oveq2d 7427 . . . . . . . . . . 11 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (π‘žβ†‘(π‘ž pCnt (β™―β€˜(Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))))) = (π‘žβ†‘(π‘ž pCnt (β™―β€˜π΅))))
6758, 66eqtr4d 2773 . . . . . . . . . 10 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (β™―β€˜(Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) = (π‘žβ†‘(π‘ž pCnt (β™―β€˜(Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))))))
6850subggrp 19045 . . . . . . . . . . . 12 ((π‘†β€˜π‘ž) ∈ (SubGrpβ€˜πΊ) β†’ (𝐺 β†Ύs (π‘†β€˜π‘ž)) ∈ Grp)
6937, 68syl 17 . . . . . . . . . . 11 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (𝐺 β†Ύs (π‘†β€˜π‘ž)) ∈ Grp)
7011adantr 479 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ 𝐡 ∈ Fin)
718subgss 19043 . . . . . . . . . . . . . 14 ((π‘†β€˜π‘ž) ∈ (SubGrpβ€˜πΊ) β†’ (π‘†β€˜π‘ž) βŠ† 𝐡)
7237, 71syl 17 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (π‘†β€˜π‘ž) βŠ† 𝐡)
7370, 72ssfid 9269 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (π‘†β€˜π‘ž) ∈ Fin)
7455, 73eqeltrrd 2832 . . . . . . . . . . 11 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∈ Fin)
7548pgpfi2 19515 . . . . . . . . . . 11 (((𝐺 β†Ύs (π‘†β€˜π‘ž)) ∈ Grp ∧ (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∈ Fin) β†’ (π‘ž pGrp (𝐺 β†Ύs (π‘†β€˜π‘ž)) ↔ (π‘ž ∈ β„™ ∧ (β™―β€˜(Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) = (π‘žβ†‘(π‘ž pCnt (β™―β€˜(Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))))))))
7669, 74, 75syl2anc 582 . . . . . . . . . 10 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (π‘ž pGrp (𝐺 β†Ύs (π‘†β€˜π‘ž)) ↔ (π‘ž ∈ β„™ ∧ (β™―β€˜(Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) = (π‘žβ†‘(π‘ž pCnt (β™―β€˜(Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))))))))
7753, 67, 76mpbir2and 709 . . . . . . . . 9 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ π‘ž pGrp (𝐺 β†Ύs (π‘†β€˜π‘ž)))
7848, 49, 52, 77, 74pgpfac 19995 . . . . . . . 8 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ βˆƒπ‘  ∈ Word {π‘Ÿ ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )} ((𝐺 β†Ύs (π‘†β€˜π‘ž))dom DProd 𝑠 ∧ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) DProd 𝑠) = (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))))
79 ssrab2 4076 . . . . . . . . . . . . . 14 {π‘Ÿ ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )} βŠ† (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))
80 sswrd 14476 . . . . . . . . . . . . . 14 ({π‘Ÿ ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )} βŠ† (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) β†’ Word {π‘Ÿ ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )} βŠ† Word (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))))
8179, 80ax-mp 5 . . . . . . . . . . . . 13 Word {π‘Ÿ ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )} βŠ† Word (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))
8281sseli 3977 . . . . . . . . . . . 12 (𝑠 ∈ Word {π‘Ÿ ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )} β†’ 𝑠 ∈ Word (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))))
8337adantr 479 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) β†’ (π‘†β€˜π‘ž) ∈ (SubGrpβ€˜πΊ))
8483adantr 479 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) ∧ (𝐺 β†Ύs (π‘†β€˜π‘ž))dom DProd 𝑠) β†’ (π‘†β€˜π‘ž) ∈ (SubGrpβ€˜πΊ))
8550subgdmdprd 19945 . . . . . . . . . . . . . . . . . . 19 ((π‘†β€˜π‘ž) ∈ (SubGrpβ€˜πΊ) β†’ ((𝐺 β†Ύs (π‘†β€˜π‘ž))dom DProd 𝑠 ↔ (𝐺dom DProd 𝑠 ∧ ran 𝑠 βŠ† 𝒫 (π‘†β€˜π‘ž))))
8683, 85syl 17 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) β†’ ((𝐺 β†Ύs (π‘†β€˜π‘ž))dom DProd 𝑠 ↔ (𝐺dom DProd 𝑠 ∧ ran 𝑠 βŠ† 𝒫 (π‘†β€˜π‘ž))))
8786simprbda 497 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) ∧ (𝐺 β†Ύs (π‘†β€˜π‘ž))dom DProd 𝑠) β†’ 𝐺dom DProd 𝑠)
8886simplbda 498 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) ∧ (𝐺 β†Ύs (π‘†β€˜π‘ž))dom DProd 𝑠) β†’ ran 𝑠 βŠ† 𝒫 (π‘†β€˜π‘ž))
8950, 84, 87, 88subgdprd 19946 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) ∧ (𝐺 β†Ύs (π‘†β€˜π‘ž))dom DProd 𝑠) β†’ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) DProd 𝑠) = (𝐺 DProd 𝑠))
9055ad2antrr 722 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) ∧ (𝐺 β†Ύs (π‘†β€˜π‘ž))dom DProd 𝑠) β†’ (π‘†β€˜π‘ž) = (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))))
9190eqcomd 2736 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) ∧ (𝐺 β†Ύs (π‘†β€˜π‘ž))dom DProd 𝑠) β†’ (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) = (π‘†β€˜π‘ž))
9289, 91eqeq12d 2746 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) ∧ (𝐺 β†Ύs (π‘†β€˜π‘ž))dom DProd 𝑠) β†’ (((𝐺 β†Ύs (π‘†β€˜π‘ž)) DProd 𝑠) = (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ↔ (𝐺 DProd 𝑠) = (π‘†β€˜π‘ž)))
9392biimpd 228 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) ∧ (𝐺 β†Ύs (π‘†β€˜π‘ž))dom DProd 𝑠) β†’ (((𝐺 β†Ύs (π‘†β€˜π‘ž)) DProd 𝑠) = (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) β†’ (𝐺 DProd 𝑠) = (π‘†β€˜π‘ž)))
9493, 87jctild 524 . . . . . . . . . . . . 13 ((((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) ∧ (𝐺 β†Ύs (π‘†β€˜π‘ž))dom DProd 𝑠) β†’ (((𝐺 β†Ύs (π‘†β€˜π‘ž)) DProd 𝑠) = (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) β†’ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (π‘†β€˜π‘ž))))
9594expimpd 452 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑠 ∈ Word (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) β†’ (((𝐺 β†Ύs (π‘†β€˜π‘ž))dom DProd 𝑠 ∧ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) DProd 𝑠) = (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) β†’ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (π‘†β€˜π‘ž))))
9682, 95sylan2 591 . . . . . . . . . . 11 (((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑠 ∈ Word {π‘Ÿ ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )}) β†’ (((𝐺 β†Ύs (π‘†β€˜π‘ž))dom DProd 𝑠 ∧ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) DProd 𝑠) = (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) β†’ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (π‘†β€˜π‘ž))))
97 oveq2 7419 . . . . . . . . . . . . . . . 16 (π‘Ÿ = 𝑦 β†’ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) = ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs 𝑦))
9897eleq1d 2816 . . . . . . . . . . . . . . 15 (π‘Ÿ = 𝑦 β†’ (((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp ) ↔ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs 𝑦) ∈ (CycGrp ∩ ran pGrp )))
9998cbvrabv 3440 . . . . . . . . . . . . . 14 {π‘Ÿ ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )} = {𝑦 ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs 𝑦) ∈ (CycGrp ∩ ran pGrp )}
10050subsubg 19065 . . . . . . . . . . . . . . . . . . 19 ((π‘†β€˜π‘ž) ∈ (SubGrpβ€˜πΊ) β†’ (𝑦 ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ↔ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ 𝑦 βŠ† (π‘†β€˜π‘ž))))
10137, 100syl 17 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (𝑦 ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ↔ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ 𝑦 βŠ† (π‘†β€˜π‘ž))))
102101simprbda 497 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑦 ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) β†’ 𝑦 ∈ (SubGrpβ€˜πΊ))
1031023adant3 1130 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑦 ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∧ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs 𝑦) ∈ (CycGrp ∩ ran pGrp )) β†’ 𝑦 ∈ (SubGrpβ€˜πΊ))
104373ad2ant1 1131 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑦 ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∧ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs 𝑦) ∈ (CycGrp ∩ ran pGrp )) β†’ (π‘†β€˜π‘ž) ∈ (SubGrpβ€˜πΊ))
105101simplbda 498 . . . . . . . . . . . . . . . . . . 19 (((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑦 ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) β†’ 𝑦 βŠ† (π‘†β€˜π‘ž))
1061053adant3 1130 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑦 ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∧ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs 𝑦) ∈ (CycGrp ∩ ran pGrp )) β†’ 𝑦 βŠ† (π‘†β€˜π‘ž))
107 ressabs 17198 . . . . . . . . . . . . . . . . . 18 (((π‘†β€˜π‘ž) ∈ (SubGrpβ€˜πΊ) ∧ 𝑦 βŠ† (π‘†β€˜π‘ž)) β†’ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs 𝑦) = (𝐺 β†Ύs 𝑦))
108104, 106, 107syl2anc 582 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑦 ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∧ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs 𝑦) ∈ (CycGrp ∩ ran pGrp )) β†’ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs 𝑦) = (𝐺 β†Ύs 𝑦))
109 simp3 1136 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑦 ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∧ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs 𝑦) ∈ (CycGrp ∩ ran pGrp )) β†’ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs 𝑦) ∈ (CycGrp ∩ ran pGrp ))
110108, 109eqeltrrd 2832 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑦 ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∧ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs 𝑦) ∈ (CycGrp ∩ ran pGrp )) β†’ (𝐺 β†Ύs 𝑦) ∈ (CycGrp ∩ ran pGrp ))
111 oveq2 7419 . . . . . . . . . . . . . . . . . 18 (π‘Ÿ = 𝑦 β†’ (𝐺 β†Ύs π‘Ÿ) = (𝐺 β†Ύs 𝑦))
112111eleq1d 2816 . . . . . . . . . . . . . . . . 17 (π‘Ÿ = 𝑦 β†’ ((𝐺 β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp ) ↔ (𝐺 β†Ύs 𝑦) ∈ (CycGrp ∩ ran pGrp )))
113112, 38elrab2 3685 . . . . . . . . . . . . . . . 16 (𝑦 ∈ 𝐢 ↔ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ (𝐺 β†Ύs 𝑦) ∈ (CycGrp ∩ ran pGrp )))
114103, 110, 113sylanbrc 581 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑦 ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∧ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs 𝑦) ∈ (CycGrp ∩ ran pGrp )) β†’ 𝑦 ∈ 𝐢)
115114rabssdv 4071 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ {𝑦 ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs 𝑦) ∈ (CycGrp ∩ ran pGrp )} βŠ† 𝐢)
11699, 115eqsstrid 4029 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ {π‘Ÿ ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )} βŠ† 𝐢)
117 sswrd 14476 . . . . . . . . . . . . 13 ({π‘Ÿ ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )} βŠ† 𝐢 β†’ Word {π‘Ÿ ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )} βŠ† Word 𝐢)
118116, 117syl 17 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ Word {π‘Ÿ ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )} βŠ† Word 𝐢)
119118sselda 3981 . . . . . . . . . . 11 (((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑠 ∈ Word {π‘Ÿ ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )}) β†’ 𝑠 ∈ Word 𝐢)
12096, 119jctild 524 . . . . . . . . . 10 (((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑠 ∈ Word {π‘Ÿ ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )}) β†’ (((𝐺 β†Ύs (π‘†β€˜π‘ž))dom DProd 𝑠 ∧ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) DProd 𝑠) = (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) β†’ (𝑠 ∈ Word 𝐢 ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (π‘†β€˜π‘ž)))))
121120expimpd 452 . . . . . . . . 9 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ ((𝑠 ∈ Word {π‘Ÿ ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )} ∧ ((𝐺 β†Ύs (π‘†β€˜π‘ž))dom DProd 𝑠 ∧ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) DProd 𝑠) = (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))))) β†’ (𝑠 ∈ Word 𝐢 ∧ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (π‘†β€˜π‘ž)))))
122121reximdv2 3162 . . . . . . . 8 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (βˆƒπ‘  ∈ Word {π‘Ÿ ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∣ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp )} ((𝐺 β†Ύs (π‘†β€˜π‘ž))dom DProd 𝑠 ∧ ((𝐺 β†Ύs (π‘†β€˜π‘ž)) DProd 𝑠) = (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) β†’ βˆƒπ‘  ∈ Word 𝐢(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (π‘†β€˜π‘ž))))
12378, 122mpd 15 . . . . . . 7 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ βˆƒπ‘  ∈ Word 𝐢(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (π‘†β€˜π‘ž)))
124 rabn0 4384 . . . . . . 7 ({𝑠 ∈ Word 𝐢 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (π‘†β€˜π‘ž))} β‰  βˆ… ↔ βˆƒπ‘  ∈ Word 𝐢(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (π‘†β€˜π‘ž)))
125123, 124sylibr 233 . . . . . 6 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ {𝑠 ∈ Word 𝐢 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (π‘†β€˜π‘ž))} β‰  βˆ…)
12647, 125eqnetrd 3006 . . . . 5 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ {𝑦 ∈ Word 𝐢 ∣ 𝑦 ∈ (π‘Šβ€˜(π‘†β€˜π‘ž))} β‰  βˆ…)
127 rabn0 4384 . . . . 5 ({𝑦 ∈ Word 𝐢 ∣ 𝑦 ∈ (π‘Šβ€˜(π‘†β€˜π‘ž))} β‰  βˆ… ↔ βˆƒπ‘¦ ∈ Word 𝐢𝑦 ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))
128126, 127sylib 217 . . . 4 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ βˆƒπ‘¦ ∈ Word 𝐢𝑦 ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))
129128ralrimiva 3144 . . 3 (πœ‘ β†’ βˆ€π‘ž ∈ 𝐴 βˆƒπ‘¦ ∈ Word 𝐢𝑦 ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))
130 eleq1 2819 . . . 4 (𝑦 = (π‘“β€˜π‘ž) β†’ (𝑦 ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)) ↔ (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž))))
131130ac6sfi 9289 . . 3 ((𝐴 ∈ Fin ∧ βˆ€π‘ž ∈ 𝐴 βˆƒπ‘¦ ∈ Word 𝐢𝑦 ∈ (π‘Šβ€˜(π‘†β€˜π‘ž))) β†’ βˆƒπ‘“(𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž))))
13225, 129, 131syl2anc 582 . 2 (πœ‘ β†’ βˆƒπ‘“(𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž))))
133 sneq 4637 . . . . . . . . 9 (π‘ž = 𝑦 β†’ {π‘ž} = {𝑦})
134 fveq2 6890 . . . . . . . . . 10 (π‘ž = 𝑦 β†’ (π‘“β€˜π‘ž) = (π‘“β€˜π‘¦))
135134dmeqd 5904 . . . . . . . . 9 (π‘ž = 𝑦 β†’ dom (π‘“β€˜π‘ž) = dom (π‘“β€˜π‘¦))
136133, 135xpeq12d 5706 . . . . . . . 8 (π‘ž = 𝑦 β†’ ({π‘ž} Γ— dom (π‘“β€˜π‘ž)) = ({𝑦} Γ— dom (π‘“β€˜π‘¦)))
137136cbviunv 5042 . . . . . . 7 βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)) = βˆͺ 𝑦 ∈ 𝐴 ({𝑦} Γ— dom (π‘“β€˜π‘¦))
138 snfi 9046 . . . . . . . . . 10 {𝑦} ∈ Fin
139 simprl 767 . . . . . . . . . . . . 13 ((πœ‘ ∧ (𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))) β†’ 𝑓:𝐴⟢Word 𝐢)
140139ffvelcdmda 7085 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))) ∧ 𝑦 ∈ 𝐴) β†’ (π‘“β€˜π‘¦) ∈ Word 𝐢)
141 wrdf 14473 . . . . . . . . . . . 12 ((π‘“β€˜π‘¦) ∈ Word 𝐢 β†’ (π‘“β€˜π‘¦):(0..^(β™―β€˜(π‘“β€˜π‘¦)))⟢𝐢)
142 fdm 6725 . . . . . . . . . . . 12 ((π‘“β€˜π‘¦):(0..^(β™―β€˜(π‘“β€˜π‘¦)))⟢𝐢 β†’ dom (π‘“β€˜π‘¦) = (0..^(β™―β€˜(π‘“β€˜π‘¦))))
143140, 141, 1423syl 18 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))) ∧ 𝑦 ∈ 𝐴) β†’ dom (π‘“β€˜π‘¦) = (0..^(β™―β€˜(π‘“β€˜π‘¦))))
144 fzofi 13943 . . . . . . . . . . 11 (0..^(β™―β€˜(π‘“β€˜π‘¦))) ∈ Fin
145143, 144eqeltrdi 2839 . . . . . . . . . 10 (((πœ‘ ∧ (𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))) ∧ 𝑦 ∈ 𝐴) β†’ dom (π‘“β€˜π‘¦) ∈ Fin)
146 xpfi 9319 . . . . . . . . . 10 (({𝑦} ∈ Fin ∧ dom (π‘“β€˜π‘¦) ∈ Fin) β†’ ({𝑦} Γ— dom (π‘“β€˜π‘¦)) ∈ Fin)
147138, 145, 146sylancr 585 . . . . . . . . 9 (((πœ‘ ∧ (𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))) ∧ 𝑦 ∈ 𝐴) β†’ ({𝑦} Γ— dom (π‘“β€˜π‘¦)) ∈ Fin)
148147ralrimiva 3144 . . . . . . . 8 ((πœ‘ ∧ (𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))) β†’ βˆ€π‘¦ ∈ 𝐴 ({𝑦} Γ— dom (π‘“β€˜π‘¦)) ∈ Fin)
149 iunfi 9342 . . . . . . . 8 ((𝐴 ∈ Fin ∧ βˆ€π‘¦ ∈ 𝐴 ({𝑦} Γ— dom (π‘“β€˜π‘¦)) ∈ Fin) β†’ βˆͺ 𝑦 ∈ 𝐴 ({𝑦} Γ— dom (π‘“β€˜π‘¦)) ∈ Fin)
15025, 148, 149syl2an2r 681 . . . . . . 7 ((πœ‘ ∧ (𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))) β†’ βˆͺ 𝑦 ∈ 𝐴 ({𝑦} Γ— dom (π‘“β€˜π‘¦)) ∈ Fin)
151137, 150eqeltrid 2835 . . . . . 6 ((πœ‘ ∧ (𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))) β†’ βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)) ∈ Fin)
152 hashcl 14320 . . . . . 6 (βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)) ∈ Fin β†’ (β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))) ∈ β„•0)
153 hashfzo0 14394 . . . . . 6 ((β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))) ∈ β„•0 β†’ (β™―β€˜(0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))) = (β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))
154151, 152, 1533syl 18 . . . . 5 ((πœ‘ ∧ (𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))) β†’ (β™―β€˜(0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))) = (β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))
155 fzofi 13943 . . . . . 6 (0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)))) ∈ Fin
156 hashen 14311 . . . . . 6 (((0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)))) ∈ Fin ∧ βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)) ∈ Fin) β†’ ((β™―β€˜(0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))) = (β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))) ↔ (0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)))) β‰ˆ βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))
157155, 151, 156sylancr 585 . . . . 5 ((πœ‘ ∧ (𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))) β†’ ((β™―β€˜(0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))) = (β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))) ↔ (0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)))) β‰ˆ βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))
158154, 157mpbid 231 . . . 4 ((πœ‘ ∧ (𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))) β†’ (0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)))) β‰ˆ βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)))
159 bren 8951 . . . 4 ((0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)))) β‰ˆ βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)) ↔ βˆƒβ„Ž β„Ž:(0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))–1-1-ontoβ†’βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)))
160158, 159sylib 217 . . 3 ((πœ‘ ∧ (𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))) β†’ βˆƒβ„Ž β„Ž:(0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))–1-1-ontoβ†’βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)))
1616adantr 479 . . . . . 6 ((πœ‘ ∧ ((𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž))) ∧ β„Ž:(0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))–1-1-ontoβ†’βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)))) β†’ 𝐺 ∈ Abel)
16211adantr 479 . . . . . 6 ((πœ‘ ∧ ((𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž))) ∧ β„Ž:(0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))–1-1-ontoβ†’βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)))) β†’ 𝐡 ∈ Fin)
163 breq1 5150 . . . . . . . 8 (𝑀 = π‘Ž β†’ (𝑀 βˆ₯ (β™―β€˜π΅) ↔ π‘Ž βˆ₯ (β™―β€˜π΅)))
164163cbvrabv 3440 . . . . . . 7 {𝑀 ∈ β„™ ∣ 𝑀 βˆ₯ (β™―β€˜π΅)} = {π‘Ž ∈ β„™ ∣ π‘Ž βˆ₯ (β™―β€˜π΅)}
1652, 164eqtri 2758 . . . . . 6 𝐴 = {π‘Ž ∈ β„™ ∣ π‘Ž βˆ₯ (β™―β€˜π΅)}
166 fveq2 6890 . . . . . . . . . . 11 (π‘₯ = 𝑐 β†’ (π‘‚β€˜π‘₯) = (π‘‚β€˜π‘))
167166breq1d 5157 . . . . . . . . . 10 (π‘₯ = 𝑐 β†’ ((π‘‚β€˜π‘₯) βˆ₯ (𝑝↑(𝑝 pCnt (β™―β€˜π΅))) ↔ (π‘‚β€˜π‘) βˆ₯ (𝑝↑(𝑝 pCnt (β™―β€˜π΅)))))
168167cbvrabv 3440 . . . . . . . . 9 {π‘₯ ∈ 𝐡 ∣ (π‘‚β€˜π‘₯) βˆ₯ (𝑝↑(𝑝 pCnt (β™―β€˜π΅)))} = {𝑐 ∈ 𝐡 ∣ (π‘‚β€˜π‘) βˆ₯ (𝑝↑(𝑝 pCnt (β™―β€˜π΅)))}
169 id 22 . . . . . . . . . . . 12 (𝑝 = 𝑏 β†’ 𝑝 = 𝑏)
170 oveq1 7418 . . . . . . . . . . . 12 (𝑝 = 𝑏 β†’ (𝑝 pCnt (β™―β€˜π΅)) = (𝑏 pCnt (β™―β€˜π΅)))
171169, 170oveq12d 7429 . . . . . . . . . . 11 (𝑝 = 𝑏 β†’ (𝑝↑(𝑝 pCnt (β™―β€˜π΅))) = (𝑏↑(𝑏 pCnt (β™―β€˜π΅))))
172171breq2d 5159 . . . . . . . . . 10 (𝑝 = 𝑏 β†’ ((π‘‚β€˜π‘) βˆ₯ (𝑝↑(𝑝 pCnt (β™―β€˜π΅))) ↔ (π‘‚β€˜π‘) βˆ₯ (𝑏↑(𝑏 pCnt (β™―β€˜π΅)))))
173172rabbidv 3438 . . . . . . . . 9 (𝑝 = 𝑏 β†’ {𝑐 ∈ 𝐡 ∣ (π‘‚β€˜π‘) βˆ₯ (𝑝↑(𝑝 pCnt (β™―β€˜π΅)))} = {𝑐 ∈ 𝐡 ∣ (π‘‚β€˜π‘) βˆ₯ (𝑏↑(𝑏 pCnt (β™―β€˜π΅)))})
174168, 173eqtrid 2782 . . . . . . . 8 (𝑝 = 𝑏 β†’ {π‘₯ ∈ 𝐡 ∣ (π‘‚β€˜π‘₯) βˆ₯ (𝑝↑(𝑝 pCnt (β™―β€˜π΅)))} = {𝑐 ∈ 𝐡 ∣ (π‘‚β€˜π‘) βˆ₯ (𝑏↑(𝑏 pCnt (β™―β€˜π΅)))})
175174cbvmptv 5260 . . . . . . 7 (𝑝 ∈ 𝐴 ↦ {π‘₯ ∈ 𝐡 ∣ (π‘‚β€˜π‘₯) βˆ₯ (𝑝↑(𝑝 pCnt (β™―β€˜π΅)))}) = (𝑏 ∈ 𝐴 ↦ {𝑐 ∈ 𝐡 ∣ (π‘‚β€˜π‘) βˆ₯ (𝑏↑(𝑏 pCnt (β™―β€˜π΅)))})
17628, 175eqtri 2758 . . . . . 6 𝑆 = (𝑏 ∈ 𝐴 ↦ {𝑐 ∈ 𝐡 ∣ (π‘‚β€˜π‘) βˆ₯ (𝑏↑(𝑏 pCnt (β™―β€˜π΅)))})
177 breq2 5151 . . . . . . . . . 10 (𝑠 = 𝑑 β†’ (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑑))
178 oveq2 7419 . . . . . . . . . . 11 (𝑠 = 𝑑 β†’ (𝐺 DProd 𝑠) = (𝐺 DProd 𝑑))
179178eqeq1d 2732 . . . . . . . . . 10 (𝑠 = 𝑑 β†’ ((𝐺 DProd 𝑠) = 𝑔 ↔ (𝐺 DProd 𝑑) = 𝑔))
180177, 179anbi12d 629 . . . . . . . . 9 (𝑠 = 𝑑 β†’ ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔) ↔ (𝐺dom DProd 𝑑 ∧ (𝐺 DProd 𝑑) = 𝑔)))
181180cbvrabv 3440 . . . . . . . 8 {𝑠 ∈ Word 𝐢 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)} = {𝑑 ∈ Word 𝐢 ∣ (𝐺dom DProd 𝑑 ∧ (𝐺 DProd 𝑑) = 𝑔)}
182181mpteq2i 5252 . . . . . . 7 (𝑔 ∈ (SubGrpβ€˜πΊ) ↦ {𝑠 ∈ Word 𝐢 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)}) = (𝑔 ∈ (SubGrpβ€˜πΊ) ↦ {𝑑 ∈ Word 𝐢 ∣ (𝐺dom DProd 𝑑 ∧ (𝐺 DProd 𝑑) = 𝑔)})
18339, 182eqtri 2758 . . . . . 6 π‘Š = (𝑔 ∈ (SubGrpβ€˜πΊ) ↦ {𝑑 ∈ Word 𝐢 ∣ (𝐺dom DProd 𝑑 ∧ (𝐺 DProd 𝑑) = 𝑔)})
184 simprll 775 . . . . . 6 ((πœ‘ ∧ ((𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž))) ∧ β„Ž:(0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))–1-1-ontoβ†’βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)))) β†’ 𝑓:𝐴⟢Word 𝐢)
185 simprlr 776 . . . . . . 7 ((πœ‘ ∧ ((𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž))) ∧ β„Ž:(0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))–1-1-ontoβ†’βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)))) β†’ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))
186 2fveq3 6895 . . . . . . . . 9 (π‘ž = 𝑦 β†’ (π‘Šβ€˜(π‘†β€˜π‘ž)) = (π‘Šβ€˜(π‘†β€˜π‘¦)))
187134, 186eleq12d 2825 . . . . . . . 8 (π‘ž = 𝑦 β†’ ((π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)) ↔ (π‘“β€˜π‘¦) ∈ (π‘Šβ€˜(π‘†β€˜π‘¦))))
188187cbvralvw 3232 . . . . . . 7 (βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)) ↔ βˆ€π‘¦ ∈ 𝐴 (π‘“β€˜π‘¦) ∈ (π‘Šβ€˜(π‘†β€˜π‘¦)))
189185, 188sylib 217 . . . . . 6 ((πœ‘ ∧ ((𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž))) ∧ β„Ž:(0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))–1-1-ontoβ†’βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)))) β†’ βˆ€π‘¦ ∈ 𝐴 (π‘“β€˜π‘¦) ∈ (π‘Šβ€˜(π‘†β€˜π‘¦)))
190 simprr 769 . . . . . 6 ((πœ‘ ∧ ((𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž))) ∧ β„Ž:(0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))–1-1-ontoβ†’βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)))) β†’ β„Ž:(0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))–1-1-ontoβ†’βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)))
1918, 38, 161, 162, 27, 165, 176, 183, 184, 189, 137, 190ablfaclem2 19997 . . . . 5 ((πœ‘ ∧ ((𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž))) ∧ β„Ž:(0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))–1-1-ontoβ†’βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)))) β†’ (π‘Šβ€˜π΅) β‰  βˆ…)
192191expr 455 . . . 4 ((πœ‘ ∧ (𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))) β†’ (β„Ž:(0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))–1-1-ontoβ†’βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)) β†’ (π‘Šβ€˜π΅) β‰  βˆ…))
193192exlimdv 1934 . . 3 ((πœ‘ ∧ (𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))) β†’ (βˆƒβ„Ž β„Ž:(0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž))))–1-1-ontoβ†’βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)) β†’ (π‘Šβ€˜π΅) β‰  βˆ…))
194160, 193mpd 15 . 2 ((πœ‘ ∧ (𝑓:𝐴⟢Word 𝐢 ∧ βˆ€π‘ž ∈ 𝐴 (π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))) β†’ (π‘Šβ€˜π΅) β‰  βˆ…)
195132, 194exlimddv 1936 1 (πœ‘ β†’ (π‘Šβ€˜π΅) β‰  βˆ…)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068  {crab 3430   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601  {csn 4627  βˆͺ ciun 4996   class class class wbr 5147   ↦ cmpt 5230   Γ— cxp 5673  dom cdm 5675  ran crn 5676  βŸΆwf 6538  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7411   β‰ˆ cen 8938  Fincfn 8941  0cc0 11112  1c1 11113   ≀ cle 11253  β„•cn 12216  β„•0cn0 12476  β„€cz 12562  ...cfz 13488  ..^cfzo 13631  β†‘cexp 14031  β™―chash 14294  Word cword 14468   βˆ₯ cdvds 16201  β„™cprime 16612   pCnt cpc 16773  Basecbs 17148   β†Ύs cress 17177  Grpcgrp 18855  SubGrpcsubg 19036  odcod 19433   pGrp cpgp 19435  Abelcabl 19690  CycGrpccyg 19786   DProd cdprd 19904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-disj 5113  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  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-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-rpss 7715  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-tpos 8213  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-omul 8473  df-er 8705  df-ec 8707  df-qs 8711  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-sup 9439  df-inf 9440  df-oi 9507  df-dju 9898  df-card 9936  df-acn 9939  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-q 12937  df-rp 12979  df-fz 13489  df-fzo 13632  df-fl 13761  df-mod 13839  df-seq 13971  df-exp 14032  df-fac 14238  df-bc 14267  df-hash 14295  df-word 14469  df-concat 14525  df-s1 14550  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436  df-sum 15637  df-dvds 16202  df-gcd 16440  df-prm 16613  df-pc 16774  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-0g 17391  df-gsum 17392  df-mre 17534  df-mrc 17535  df-acs 17537  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18705  df-submnd 18706  df-grp 18858  df-minusg 18859  df-sbg 18860  df-mulg 18987  df-subg 19039  df-eqg 19041  df-ghm 19128  df-gim 19173  df-ga 19195  df-cntz 19222  df-oppg 19251  df-od 19437  df-gex 19438  df-pgp 19439  df-lsm 19545  df-pj1 19546  df-cmn 19691  df-abl 19692  df-cyg 19787  df-dprd 19906
This theorem is referenced by:  ablfac  19999
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