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Theorem ablfaclem3 20009
Description: Lemma for ablfac 20010. (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 13944 . . . 4 (πœ‘ β†’ (1...(β™―β€˜π΅)) ∈ Fin)
2 ablfac.a . . . . 5 𝐴 = {𝑀 ∈ β„™ ∣ 𝑀 βˆ₯ (β™―β€˜π΅)}
3 prmnn 16618 . . . . . . . 8 (𝑀 ∈ β„™ β†’ 𝑀 ∈ β„•)
433ad2ant2 1131 . . . . . . 7 ((πœ‘ ∧ 𝑀 ∈ β„™ ∧ 𝑀 βˆ₯ (β™―β€˜π΅)) β†’ 𝑀 ∈ β„•)
5 prmz 16619 . . . . . . . . 9 (𝑀 ∈ β„™ β†’ 𝑀 ∈ β„€)
6 ablfac.1 . . . . . . . . . . 11 (πœ‘ β†’ 𝐺 ∈ Abel)
7 ablgrp 19705 . . . . . . . . . . 11 (𝐺 ∈ Abel β†’ 𝐺 ∈ Grp)
8 ablfac.b . . . . . . . . . . . 12 𝐡 = (Baseβ€˜πΊ)
98grpbn0 18896 . . . . . . . . . . 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 14331 . . . . . . . . . . 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 257 . . . . . . . . 9 (πœ‘ β†’ (β™―β€˜π΅) ∈ β„•)
15 dvdsle 16260 . . . . . . . . 9 ((𝑀 ∈ β„€ ∧ (β™―β€˜π΅) ∈ β„•) β†’ (𝑀 βˆ₯ (β™―β€˜π΅) β†’ 𝑀 ≀ (β™―β€˜π΅)))
165, 14, 15syl2anr 596 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ β„™) β†’ (𝑀 βˆ₯ (β™―β€˜π΅) β†’ 𝑀 ≀ (β™―β€˜π΅)))
17163impia 1114 . . . . . . 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 1130 . . . . . . . 8 ((πœ‘ ∧ 𝑀 ∈ β„™ ∧ 𝑀 βˆ₯ (β™―β€˜π΅)) β†’ (β™―β€˜π΅) ∈ β„€)
20 fznn 13575 . . . . . . . 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 710 . . . . . 6 ((πœ‘ ∧ 𝑀 ∈ β„™ ∧ 𝑀 βˆ₯ (β™―β€˜π΅)) β†’ 𝑀 ∈ (1...(β™―β€˜π΅)))
2322rabssdv 4067 . . . . 5 (πœ‘ β†’ {𝑀 ∈ β„™ ∣ 𝑀 βˆ₯ (β™―β€˜π΅)} βŠ† (1...(β™―β€˜π΅)))
242, 23eqsstrid 4025 . . . 4 (πœ‘ β†’ 𝐴 βŠ† (1...(β™―β€˜π΅)))
251, 24ssfid 9269 . . 3 (πœ‘ β†’ 𝐴 ∈ Fin)
26 dfin5 3951 . . . . . . . 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 4075 . . . . . . . . . . . . . . 15 𝐴 βŠ† β„™
3029a1i 11 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐴 βŠ† β„™)
318, 27, 28, 6, 11, 30ablfac1b 19992 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐺dom DProd 𝑆)
328fvexi 6899 . . . . . . . . . . . . . . . 16 𝐡 ∈ V
3332rabex 5325 . . . . . . . . . . . . . . 15 {π‘₯ ∈ 𝐡 ∣ (π‘‚β€˜π‘₯) βˆ₯ (𝑝↑(𝑝 pCnt (β™―β€˜π΅)))} ∈ V
3433, 28dmmpti 6688 . . . . . . . . . . . . . 14 dom 𝑆 = 𝐴
3534a1i 11 . . . . . . . . . . . . 13 (πœ‘ β†’ dom 𝑆 = 𝐴)
3631, 35dprdf2 19929 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑆:𝐴⟢(SubGrpβ€˜πΊ))
3736ffvelcdmda 7080 . . . . . . . . . . 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 20007 . . . . . . . . . . 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 4072 . . . . . . . . . 10 {𝑠 ∈ Word 𝐢 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = (π‘†β€˜π‘ž))} βŠ† Word 𝐢
4341, 42eqsstrdi 4031 . . . . . . . . 9 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (π‘Šβ€˜(π‘†β€˜π‘ž)) βŠ† Word 𝐢)
44 sseqin2 4210 . . . . . . . . 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 2780 . . . . . . 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 2766 . . . . . 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 2726 . . . . . . . . 9 (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) = (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))
49 eqid 2726 . . . . . . . . 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 2726 . . . . . . . . . . 11 (𝐺 β†Ύs (π‘†β€˜π‘ž)) = (𝐺 β†Ύs (π‘†β€˜π‘ž))
5150subgabl 19756 . . . . . . . . . 10 ((𝐺 ∈ Abel ∧ (π‘†β€˜π‘ž) ∈ (SubGrpβ€˜πΊ)) β†’ (𝐺 β†Ύs (π‘†β€˜π‘ž)) ∈ Abel)
526, 37, 51syl2an2r 682 . . . . . . . . 9 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (𝐺 β†Ύs (π‘†β€˜π‘ž)) ∈ Abel)
5330sselda 3977 . . . . . . . . . 10 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ π‘ž ∈ β„™)
5450subgbas 19057 . . . . . . . . . . . . . 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 6889 . . . . . . . . . . . 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 19991 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (β™―β€˜(π‘†β€˜π‘ž)) = (π‘žβ†‘(π‘ž pCnt (β™―β€˜π΅))))
5856, 57eqtr3d 2768 . . . . . . . . . . 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 7421 . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (β™―β€˜π΅) ∈ β„•)
6153, 60pccld 16792 . . . . . . . . . . . . . . 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 16815 . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . 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 2766 . . . . . . . . . . . 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 7421 . . . . . . . . . . 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 2769 . . . . . . . . . 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 19056 . . . . . . . . . . . 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 480 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ 𝐡 ∈ Fin)
718subgss 19054 . . . . . . . . . . . . . 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 2828 . . . . . . . . . . 11 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ (Baseβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž))) ∈ Fin)
7548pgpfi2 19526 . . . . . . . . . . 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 583 . . . . . . . . . 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 710 . . . . . . . . 9 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ π‘ž pGrp (𝐺 β†Ύs (π‘†β€˜π‘ž)))
7848, 49, 52, 77, 74pgpfac 20006 . . . . . . . 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 4072 . . . . . . . . . . . . . 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 14478 . . . . . . . . . . . . . 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 3973 . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . 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 19956 . . . . . . . . . . . . . . . . . . 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 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†’ 𝐺dom DProd 𝑠)
8886simplbda 499 . . . . . . . . . . . . . . . . 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 19957 . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . 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 2742 . . . . . . . . . . . . . . 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 525 . . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 592 . . . . . . . . . . 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 7413 . . . . . . . . . . . . . . . 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 2812 . . . . . . . . . . . . . . 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 3436 . . . . . . . . . . . . . 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 19076 . . . . . . . . . . . . . . . . . . 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 498 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑦 ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) β†’ 𝑦 ∈ (SubGrpβ€˜πΊ))
1031023adant3 1129 . . . . . . . . . . . . . . . 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 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†’ (π‘†β€˜π‘ž) ∈ (SubGrpβ€˜πΊ))
105101simplbda 499 . . . . . . . . . . . . . . . . . . 19 (((πœ‘ ∧ π‘ž ∈ 𝐴) ∧ 𝑦 ∈ (SubGrpβ€˜(𝐺 β†Ύs (π‘†β€˜π‘ž)))) β†’ 𝑦 βŠ† (π‘†β€˜π‘ž))
1061053adant3 1129 . . . . . . . . . . . . . . . . . 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 17203 . . . . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . . . . 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 2828 . . . . . . . . . . . . . . . 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 7413 . . . . . . . . . . . . . . . . . 18 (π‘Ÿ = 𝑦 β†’ (𝐺 β†Ύs π‘Ÿ) = (𝐺 β†Ύs 𝑦))
112111eleq1d 2812 . . . . . . . . . . . . . . . . 17 (π‘Ÿ = 𝑦 β†’ ((𝐺 β†Ύs π‘Ÿ) ∈ (CycGrp ∩ ran pGrp ) ↔ (𝐺 β†Ύs 𝑦) ∈ (CycGrp ∩ ran pGrp )))
113112, 38elrab2 3681 . . . . . . . . . . . . . . . 16 (𝑦 ∈ 𝐢 ↔ (𝑦 ∈ (SubGrpβ€˜πΊ) ∧ (𝐺 β†Ύs 𝑦) ∈ (CycGrp ∩ ran pGrp )))
114103, 110, 113sylanbrc 582 . . . . . . . . . . . . . . 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 4067 . . . . . . . . . . . . . 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 4025 . . . . . . . . . . . . 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 14478 . . . . . . . . . . . . 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 3977 . . . . . . . . . . 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 525 . . . . . . . . . 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 453 . . . . . . . . 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 3158 . . . . . . . 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 4380 . . . . . . 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 3002 . . . . 5 ((πœ‘ ∧ π‘ž ∈ 𝐴) β†’ {𝑦 ∈ Word 𝐢 ∣ 𝑦 ∈ (π‘Šβ€˜(π‘†β€˜π‘ž))} β‰  βˆ…)
127 rabn0 4380 . . . . 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 3140 . . 3 (πœ‘ β†’ βˆ€π‘ž ∈ 𝐴 βˆƒπ‘¦ ∈ Word 𝐢𝑦 ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)))
130 eleq1 2815 . . . 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 583 . 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 4633 . . . . . . . . 9 (π‘ž = 𝑦 β†’ {π‘ž} = {𝑦})
134 fveq2 6885 . . . . . . . . . 10 (π‘ž = 𝑦 β†’ (π‘“β€˜π‘ž) = (π‘“β€˜π‘¦))
135134dmeqd 5899 . . . . . . . . 9 (π‘ž = 𝑦 β†’ dom (π‘“β€˜π‘ž) = dom (π‘“β€˜π‘¦))
136133, 135xpeq12d 5700 . . . . . . . 8 (π‘ž = 𝑦 β†’ ({π‘ž} Γ— dom (π‘“β€˜π‘ž)) = ({𝑦} Γ— dom (π‘“β€˜π‘¦)))
137136cbviunv 5036 . . . . . . 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 768 . . . . . . . . . . . . 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 7080 . . . . . . . . . . . 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 14475 . . . . . . . . . . . 12 ((π‘“β€˜π‘¦) ∈ Word 𝐢 β†’ (π‘“β€˜π‘¦):(0..^(β™―β€˜(π‘“β€˜π‘¦)))⟢𝐢)
142 fdm 6720 . . . . . . . . . . . 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 13945 . . . . . . . . . . 11 (0..^(β™―β€˜(π‘“β€˜π‘¦))) ∈ Fin
145143, 144eqeltrdi 2835 . . . . . . . . . 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 586 . . . . . . . . 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 3140 . . . . . . . 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 682 . . . . . . 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 2831 . . . . . 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 14321 . . . . . 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 14395 . . . . . 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 13945 . . . . . 6 (0..^(β™―β€˜βˆͺ π‘ž ∈ 𝐴 ({π‘ž} Γ— dom (π‘“β€˜π‘ž)))) ∈ Fin
156 hashen 14312 . . . . . 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 586 . . . . 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 480 . . . . . 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 480 . . . . . 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 5144 . . . . . . . 8 (𝑀 = π‘Ž β†’ (𝑀 βˆ₯ (β™―β€˜π΅) ↔ π‘Ž βˆ₯ (β™―β€˜π΅)))
164163cbvrabv 3436 . . . . . . 7 {𝑀 ∈ β„™ ∣ 𝑀 βˆ₯ (β™―β€˜π΅)} = {π‘Ž ∈ β„™ ∣ π‘Ž βˆ₯ (β™―β€˜π΅)}
1652, 164eqtri 2754 . . . . . 6 𝐴 = {π‘Ž ∈ β„™ ∣ π‘Ž βˆ₯ (β™―β€˜π΅)}
166 fveq2 6885 . . . . . . . . . . 11 (π‘₯ = 𝑐 β†’ (π‘‚β€˜π‘₯) = (π‘‚β€˜π‘))
167166breq1d 5151 . . . . . . . . . 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 3436 . . . . . . . . 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 7412 . . . . . . . . . . . 12 (𝑝 = 𝑏 β†’ (𝑝 pCnt (β™―β€˜π΅)) = (𝑏 pCnt (β™―β€˜π΅)))
171169, 170oveq12d 7423 . . . . . . . . . . 11 (𝑝 = 𝑏 β†’ (𝑝↑(𝑝 pCnt (β™―β€˜π΅))) = (𝑏↑(𝑏 pCnt (β™―β€˜π΅))))
172171breq2d 5153 . . . . . . . . . 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 3434 . . . . . . . . 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 2778 . . . . . . . 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 5254 . . . . . . 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 2754 . . . . . 6 𝑆 = (𝑏 ∈ 𝐴 ↦ {𝑐 ∈ 𝐡 ∣ (π‘‚β€˜π‘) βˆ₯ (𝑏↑(𝑏 pCnt (β™―β€˜π΅)))})
177 breq2 5145 . . . . . . . . . 10 (𝑠 = 𝑑 β†’ (𝐺dom DProd 𝑠 ↔ 𝐺dom DProd 𝑑))
178 oveq2 7413 . . . . . . . . . . 11 (𝑠 = 𝑑 β†’ (𝐺 DProd 𝑠) = (𝐺 DProd 𝑑))
179178eqeq1d 2728 . . . . . . . . . 10 (𝑠 = 𝑑 β†’ ((𝐺 DProd 𝑠) = 𝑔 ↔ (𝐺 DProd 𝑑) = 𝑔))
180177, 179anbi12d 630 . . . . . . . . 9 (𝑠 = 𝑑 β†’ ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔) ↔ (𝐺dom DProd 𝑑 ∧ (𝐺 DProd 𝑑) = 𝑔)))
181180cbvrabv 3436 . . . . . . . 8 {𝑠 ∈ Word 𝐢 ∣ (𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑔)} = {𝑑 ∈ Word 𝐢 ∣ (𝐺dom DProd 𝑑 ∧ (𝐺 DProd 𝑑) = 𝑔)}
182181mpteq2i 5246 . . . . . . 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 2754 . . . . . 6 π‘Š = (𝑔 ∈ (SubGrpβ€˜πΊ) ↦ {𝑑 ∈ Word 𝐢 ∣ (𝐺dom DProd 𝑑 ∧ (𝐺 DProd 𝑑) = 𝑔)})
184 simprll 776 . . . . . 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 777 . . . . . . 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 6890 . . . . . . . . 9 (π‘ž = 𝑦 β†’ (π‘Šβ€˜(π‘†β€˜π‘ž)) = (π‘Šβ€˜(π‘†β€˜π‘¦)))
187134, 186eleq12d 2821 . . . . . . . 8 (π‘ž = 𝑦 β†’ ((π‘“β€˜π‘ž) ∈ (π‘Šβ€˜(π‘†β€˜π‘ž)) ↔ (π‘“β€˜π‘¦) ∈ (π‘Šβ€˜(π‘†β€˜π‘¦))))
188187cbvralvw 3228 . . . . . . 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 770 . . . . . 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 20008 . . . . 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 456 . . . 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 1928 . . 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 1930 1 (πœ‘ β†’ (π‘Šβ€˜π΅) β‰  βˆ…)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098   β‰  wne 2934  βˆ€wral 3055  βˆƒwrex 3064  {crab 3426   ∩ cin 3942   βŠ† wss 3943  βˆ…c0 4317  π’« cpw 4597  {csn 4623  βˆͺ ciun 4990   class class class wbr 5141   ↦ cmpt 5224   Γ— cxp 5667  dom cdm 5669  ran crn 5670  βŸΆwf 6533  β€“1-1-ontoβ†’wf1o 6536  β€˜cfv 6537  (class class class)co 7405   β‰ˆ cen 8938  Fincfn 8941  0cc0 11112  1c1 11113   ≀ cle 11253  β„•cn 12216  β„•0cn0 12476  β„€cz 12562  ...cfz 13490  ..^cfzo 13633  β†‘cexp 14032  β™―chash 14295  Word cword 14470   βˆ₯ cdvds 16204  β„™cprime 16615   pCnt cpc 16778  Basecbs 17153   β†Ύs cress 17182  Grpcgrp 18863  SubGrpcsubg 19047  odcod 19444   pGrp cpgp 19446  Abelcabl 19701  CycGrpccyg 19797   DProd cdprd 19915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  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 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-disj 5107  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-rpss 7710  df-om 7853  df-1st 7974  df-2nd 7975  df-supp 8147  df-tpos 8212  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-2o 8468  df-oadd 8471  df-omul 8472  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 12981  df-fz 13491  df-fzo 13634  df-fl 13763  df-mod 13841  df-seq 13973  df-exp 14033  df-fac 14239  df-bc 14268  df-hash 14296  df-word 14471  df-concat 14527  df-s1 14552  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15438  df-sum 15639  df-dvds 16205  df-gcd 16443  df-prm 16616  df-pc 16779  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-0g 17396  df-gsum 17397  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-mhm 18713  df-submnd 18714  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18996  df-subg 19050  df-eqg 19052  df-ghm 19139  df-gim 19184  df-ga 19206  df-cntz 19233  df-oppg 19262  df-od 19448  df-gex 19449  df-pgp 19450  df-lsm 19556  df-pj1 19557  df-cmn 19702  df-abl 19703  df-cyg 19798  df-dprd 19917
This theorem is referenced by:  ablfac  20010
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