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Theorem pgpfaclem3 19134
Description: Lemma for pgpfac 19135. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
Hypotheses
Ref Expression
pgpfac.b 𝐵 = (Base‘𝐺)
pgpfac.c 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
pgpfac.g (𝜑𝐺 ∈ Abel)
pgpfac.p (𝜑𝑃 pGrp 𝐺)
pgpfac.f (𝜑𝐵 ∈ Fin)
pgpfac.u (𝜑𝑈 ∈ (SubGrp‘𝐺))
pgpfac.a (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))
Assertion
Ref Expression
pgpfaclem3 (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
Distinct variable groups:   𝑡,𝑠,𝐶   𝑠,𝑟,𝑡,𝐺   𝜑,𝑡   𝐵,𝑠,𝑡   𝑈,𝑟,𝑠,𝑡
Allowed substitution hints:   𝜑(𝑠,𝑟)   𝐵(𝑟)   𝐶(𝑟)   𝑃(𝑡,𝑠,𝑟)

Proof of Theorem pgpfaclem3
Dummy variables 𝑤 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wrd0 13877 . . 3 ∅ ∈ Word 𝐶
2 pgpfac.g . . . . . 6 (𝜑𝐺 ∈ Abel)
3 ablgrp 18840 . . . . . 6 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
4 eqid 2818 . . . . . . 7 (0g𝐺) = (0g𝐺)
54dprd0 19082 . . . . . 6 (𝐺 ∈ Grp → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g𝐺)}))
62, 3, 53syl 18 . . . . 5 (𝜑 → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g𝐺)}))
76adantr 481 . . . 4 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g𝐺)}))
8 pgpfac.u . . . . . . . . 9 (𝜑𝑈 ∈ (SubGrp‘𝐺))
94subg0cl 18225 . . . . . . . . 9 (𝑈 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑈)
108, 9syl 17 . . . . . . . 8 (𝜑 → (0g𝐺) ∈ 𝑈)
1110adantr 481 . . . . . . 7 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → (0g𝐺) ∈ 𝑈)
12 eqid 2818 . . . . . . . . . . 11 (𝐺s 𝑈) = (𝐺s 𝑈)
1312subgbas 18221 . . . . . . . . . 10 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 = (Base‘(𝐺s 𝑈)))
148, 13syl 17 . . . . . . . . 9 (𝜑𝑈 = (Base‘(𝐺s 𝑈)))
1514adantr 481 . . . . . . . 8 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → 𝑈 = (Base‘(𝐺s 𝑈)))
1612subggrp 18220 . . . . . . . . . . 11 (𝑈 ∈ (SubGrp‘𝐺) → (𝐺s 𝑈) ∈ Grp)
178, 16syl 17 . . . . . . . . . 10 (𝜑 → (𝐺s 𝑈) ∈ Grp)
18 grpmnd 18048 . . . . . . . . . 10 ((𝐺s 𝑈) ∈ Grp → (𝐺s 𝑈) ∈ Mnd)
19 eqid 2818 . . . . . . . . . . 11 (Base‘(𝐺s 𝑈)) = (Base‘(𝐺s 𝑈))
20 eqid 2818 . . . . . . . . . . 11 (gEx‘(𝐺s 𝑈)) = (gEx‘(𝐺s 𝑈))
2119, 20gex1 18645 . . . . . . . . . 10 ((𝐺s 𝑈) ∈ Mnd → ((gEx‘(𝐺s 𝑈)) = 1 ↔ (Base‘(𝐺s 𝑈)) ≈ 1o))
2217, 18, 213syl 18 . . . . . . . . 9 (𝜑 → ((gEx‘(𝐺s 𝑈)) = 1 ↔ (Base‘(𝐺s 𝑈)) ≈ 1o))
2322biimpa 477 . . . . . . . 8 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → (Base‘(𝐺s 𝑈)) ≈ 1o)
2415, 23eqbrtrd 5079 . . . . . . 7 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → 𝑈 ≈ 1o)
25 en1eqsn 8736 . . . . . . 7 (((0g𝐺) ∈ 𝑈𝑈 ≈ 1o) → 𝑈 = {(0g𝐺)})
2611, 24, 25syl2anc 584 . . . . . 6 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → 𝑈 = {(0g𝐺)})
2726eqeq2d 2829 . . . . 5 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → ((𝐺 DProd ∅) = 𝑈 ↔ (𝐺 DProd ∅) = {(0g𝐺)}))
2827anbi2d 628 . . . 4 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → ((𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈) ↔ (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g𝐺)})))
297, 28mpbird 258 . . 3 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈))
30 breq2 5061 . . . . 5 (𝑠 = ∅ → (𝐺dom DProd 𝑠𝐺dom DProd ∅))
31 oveq2 7153 . . . . . 6 (𝑠 = ∅ → (𝐺 DProd 𝑠) = (𝐺 DProd ∅))
3231eqeq1d 2820 . . . . 5 (𝑠 = ∅ → ((𝐺 DProd 𝑠) = 𝑈 ↔ (𝐺 DProd ∅) = 𝑈))
3330, 32anbi12d 630 . . . 4 (𝑠 = ∅ → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈) ↔ (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈)))
3433rspcev 3620 . . 3 ((∅ ∈ Word 𝐶 ∧ (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈)) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
351, 29, 34sylancr 587 . 2 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
3612subgabl 18885 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝐺s 𝑈) ∈ Abel)
372, 8, 36syl2anc 584 . . . . 5 (𝜑 → (𝐺s 𝑈) ∈ Abel)
38 pgpfac.f . . . . . . . 8 (𝜑𝐵 ∈ Fin)
39 pgpfac.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
4039subgss 18218 . . . . . . . . 9 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈𝐵)
418, 40syl 17 . . . . . . . 8 (𝜑𝑈𝐵)
4238, 41ssfid 8729 . . . . . . 7 (𝜑𝑈 ∈ Fin)
4314, 42eqeltrrd 2911 . . . . . 6 (𝜑 → (Base‘(𝐺s 𝑈)) ∈ Fin)
4419, 20gexcl2 18643 . . . . . 6 (((𝐺s 𝑈) ∈ Grp ∧ (Base‘(𝐺s 𝑈)) ∈ Fin) → (gEx‘(𝐺s 𝑈)) ∈ ℕ)
4517, 43, 44syl2anc 584 . . . . 5 (𝜑 → (gEx‘(𝐺s 𝑈)) ∈ ℕ)
46 eqid 2818 . . . . . 6 (od‘(𝐺s 𝑈)) = (od‘(𝐺s 𝑈))
4719, 20, 46gexex 18902 . . . . 5 (((𝐺s 𝑈) ∈ Abel ∧ (gEx‘(𝐺s 𝑈)) ∈ ℕ) → ∃𝑥 ∈ (Base‘(𝐺s 𝑈))((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))
4837, 45, 47syl2anc 584 . . . 4 (𝜑 → ∃𝑥 ∈ (Base‘(𝐺s 𝑈))((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))
4948adantr 481 . . 3 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) → ∃𝑥 ∈ (Base‘(𝐺s 𝑈))((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))
50 eqid 2818 . . . . 5 (mrCls‘(SubGrp‘(𝐺s 𝑈))) = (mrCls‘(SubGrp‘(𝐺s 𝑈)))
51 eqid 2818 . . . . 5 ((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) = ((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})
52 eqid 2818 . . . . 5 (0g‘(𝐺s 𝑈)) = (0g‘(𝐺s 𝑈))
53 eqid 2818 . . . . 5 (LSSum‘(𝐺s 𝑈)) = (LSSum‘(𝐺s 𝑈))
54 pgpfac.p . . . . . . 7 (𝜑𝑃 pGrp 𝐺)
55 subgpgp 18651 . . . . . . 7 ((𝑃 pGrp 𝐺𝑈 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺s 𝑈))
5654, 8, 55syl2anc 584 . . . . . 6 (𝜑𝑃 pGrp (𝐺s 𝑈))
5756ad2antrr 722 . . . . 5 (((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) → 𝑃 pGrp (𝐺s 𝑈))
5837ad2antrr 722 . . . . 5 (((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) → (𝐺s 𝑈) ∈ Abel)
5943ad2antrr 722 . . . . 5 (((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) → (Base‘(𝐺s 𝑈)) ∈ Fin)
60 simprr 769 . . . . 5 (((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) → ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))
61 simprl 767 . . . . 5 (((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) → 𝑥 ∈ (Base‘(𝐺s 𝑈)))
6250, 51, 19, 46, 20, 52, 53, 57, 58, 59, 60, 61pgpfac1 19131 . . . 4 (((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) → ∃𝑤 ∈ (SubGrp‘(𝐺s 𝑈))((((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})(LSSum‘(𝐺s 𝑈))𝑤) = (Base‘(𝐺s 𝑈))))
63 pgpfac.c . . . . 5 𝐶 = {𝑟 ∈ (SubGrp‘𝐺) ∣ (𝐺s 𝑟) ∈ (CycGrp ∩ ran pGrp )}
642ad3antrrr 726 . . . . 5 ((((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})(LSSum‘(𝐺s 𝑈))𝑤) = (Base‘(𝐺s 𝑈))))) → 𝐺 ∈ Abel)
6554ad3antrrr 726 . . . . 5 ((((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})(LSSum‘(𝐺s 𝑈))𝑤) = (Base‘(𝐺s 𝑈))))) → 𝑃 pGrp 𝐺)
6638ad3antrrr 726 . . . . 5 ((((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})(LSSum‘(𝐺s 𝑈))𝑤) = (Base‘(𝐺s 𝑈))))) → 𝐵 ∈ Fin)
678ad3antrrr 726 . . . . 5 ((((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})(LSSum‘(𝐺s 𝑈))𝑤) = (Base‘(𝐺s 𝑈))))) → 𝑈 ∈ (SubGrp‘𝐺))
68 pgpfac.a . . . . . 6 (𝜑 → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))
6968ad3antrrr 726 . . . . 5 ((((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})(LSSum‘(𝐺s 𝑈))𝑤) = (Base‘(𝐺s 𝑈))))) → ∀𝑡 ∈ (SubGrp‘𝐺)(𝑡𝑈 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑡)))
70 simpllr 772 . . . . 5 ((((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})(LSSum‘(𝐺s 𝑈))𝑤) = (Base‘(𝐺s 𝑈))))) → (gEx‘(𝐺s 𝑈)) ≠ 1)
71 simplrl 773 . . . . . 6 ((((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})(LSSum‘(𝐺s 𝑈))𝑤) = (Base‘(𝐺s 𝑈))))) → 𝑥 ∈ (Base‘(𝐺s 𝑈)))
7267, 13syl 17 . . . . . 6 ((((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})(LSSum‘(𝐺s 𝑈))𝑤) = (Base‘(𝐺s 𝑈))))) → 𝑈 = (Base‘(𝐺s 𝑈)))
7371, 72eleqtrrd 2913 . . . . 5 ((((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})(LSSum‘(𝐺s 𝑈))𝑤) = (Base‘(𝐺s 𝑈))))) → 𝑥𝑈)
74 simplrr 774 . . . . 5 ((((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})(LSSum‘(𝐺s 𝑈))𝑤) = (Base‘(𝐺s 𝑈))))) → ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))
75 simprl 767 . . . . 5 ((((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})(LSSum‘(𝐺s 𝑈))𝑤) = (Base‘(𝐺s 𝑈))))) → 𝑤 ∈ (SubGrp‘(𝐺s 𝑈)))
76 simprrl 777 . . . . 5 ((((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})(LSSum‘(𝐺s 𝑈))𝑤) = (Base‘(𝐺s 𝑈))))) → (((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺s 𝑈))})
77 simprrr 778 . . . . . 6 ((((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})(LSSum‘(𝐺s 𝑈))𝑤) = (Base‘(𝐺s 𝑈))))) → (((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})(LSSum‘(𝐺s 𝑈))𝑤) = (Base‘(𝐺s 𝑈)))
7877, 72eqtr4d 2856 . . . . 5 ((((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})(LSSum‘(𝐺s 𝑈))𝑤) = (Base‘(𝐺s 𝑈))))) → (((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})(LSSum‘(𝐺s 𝑈))𝑤) = 𝑈)
7939, 63, 64, 65, 66, 67, 69, 12, 50, 46, 20, 52, 53, 70, 73, 74, 75, 76, 78pgpfaclem2 19133 . . . 4 ((((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) ∧ (𝑤 ∈ (SubGrp‘(𝐺s 𝑈)) ∧ ((((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) ∩ 𝑤) = {(0g‘(𝐺s 𝑈))} ∧ (((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})(LSSum‘(𝐺s 𝑈))𝑤) = (Base‘(𝐺s 𝑈))))) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
8062, 79rexlimddv 3288 . . 3 (((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
8149, 80rexlimddv 3288 . 2 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
8235, 81pm2.61dane 3101 1 (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  {crab 3139  cin 3932  wss 3933  wpss 3934  c0 4288  {csn 4557   class class class wbr 5057  dom cdm 5548  ran crn 5549  cfv 6348  (class class class)co 7145  1oc1o 8084  cen 8494  Fincfn 8497  1c1 10526  cn 11626  Word cword 13849  Basecbs 16471  s cress 16472  0gc0g 16701  mrClscmrc 16842  Mndcmnd 17899  Grpcgrp 18041  SubGrpcsubg 18211  odcod 18581  gExcgex 18582   pGrp cpgp 18583  LSSumclsm 18688  Abelcabl 18836  CycGrpccyg 18925   DProd cdprd 19044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-disj 5023  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-rpss 7438  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-tpos 7881  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-omul 8096  df-er 8278  df-ec 8280  df-qs 8284  df-map 8397  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-acn 9359  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-fz 12881  df-fzo 13022  df-fl 13150  df-mod 13226  df-seq 13358  df-exp 13418  df-fac 13622  df-bc 13651  df-hash 13679  df-word 13850  df-concat 13911  df-s1 13938  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-sum 15031  df-dvds 15596  df-gcd 15832  df-prm 16004  df-pc 16162  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-0g 16703  df-gsum 16704  df-mre 16845  df-mrc 16846  df-acs 16848  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-mhm 17944  df-submnd 17945  df-grp 18044  df-minusg 18045  df-sbg 18046  df-mulg 18163  df-subg 18214  df-eqg 18216  df-ghm 18294  df-gim 18337  df-ga 18358  df-cntz 18385  df-oppg 18412  df-od 18585  df-gex 18586  df-pgp 18587  df-lsm 18690  df-pj1 18691  df-cmn 18837  df-abl 18838  df-cyg 18926  df-dprd 19046
This theorem is referenced by:  pgpfac  19135
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