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Theorem pgpfaclem3 20031
Description: Lemma for pgpfac 20032. (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 14513 . . 3 ∅ ∈ Word 𝐶
2 pgpfac.g . . . . . 6 (𝜑𝐺 ∈ Abel)
3 ablgrp 19731 . . . . . 6 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
4 eqid 2727 . . . . . . 7 (0g𝐺) = (0g𝐺)
54dprd0 19979 . . . . . 6 (𝐺 ∈ Grp → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g𝐺)}))
62, 3, 53syl 18 . . . . 5 (𝜑 → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g𝐺)}))
76adantr 480 . . . 4 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g𝐺)}))
8 pgpfac.u . . . . . . . . 9 (𝜑𝑈 ∈ (SubGrp‘𝐺))
94subg0cl 19080 . . . . . . . . 9 (𝑈 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑈)
108, 9syl 17 . . . . . . . 8 (𝜑 → (0g𝐺) ∈ 𝑈)
1110adantr 480 . . . . . . 7 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → (0g𝐺) ∈ 𝑈)
12 eqid 2727 . . . . . . . . . . 11 (𝐺s 𝑈) = (𝐺s 𝑈)
1312subgbas 19076 . . . . . . . . . 10 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 = (Base‘(𝐺s 𝑈)))
148, 13syl 17 . . . . . . . . 9 (𝜑𝑈 = (Base‘(𝐺s 𝑈)))
1514adantr 480 . . . . . . . 8 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → 𝑈 = (Base‘(𝐺s 𝑈)))
1612subggrp 19075 . . . . . . . . . . 11 (𝑈 ∈ (SubGrp‘𝐺) → (𝐺s 𝑈) ∈ Grp)
178, 16syl 17 . . . . . . . . . 10 (𝜑 → (𝐺s 𝑈) ∈ Grp)
18 grpmnd 18888 . . . . . . . . . 10 ((𝐺s 𝑈) ∈ Grp → (𝐺s 𝑈) ∈ Mnd)
19 eqid 2727 . . . . . . . . . . 11 (Base‘(𝐺s 𝑈)) = (Base‘(𝐺s 𝑈))
20 eqid 2727 . . . . . . . . . . 11 (gEx‘(𝐺s 𝑈)) = (gEx‘(𝐺s 𝑈))
2119, 20gex1 19537 . . . . . . . . . 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 476 . . . . . . . 8 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → (Base‘(𝐺s 𝑈)) ≈ 1o)
2415, 23eqbrtrd 5164 . . . . . . 7 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → 𝑈 ≈ 1o)
25 en1eqsn 9290 . . . . . . 7 (((0g𝐺) ∈ 𝑈𝑈 ≈ 1o) → 𝑈 = {(0g𝐺)})
2611, 24, 25syl2anc 583 . . . . . 6 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → 𝑈 = {(0g𝐺)})
2726eqeq2d 2738 . . . . 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 257 . . 3 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈))
30 breq2 5146 . . . . 5 (𝑠 = ∅ → (𝐺dom DProd 𝑠𝐺dom DProd ∅))
31 oveq2 7422 . . . . . 6 (𝑠 = ∅ → (𝐺 DProd 𝑠) = (𝐺 DProd ∅))
3231eqeq1d 2729 . . . . 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 3607 . . 3 ((∅ ∈ Word 𝐶 ∧ (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈)) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
351, 29, 34sylancr 586 . 2 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
3612subgabl 19782 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝐺s 𝑈) ∈ Abel)
372, 8, 36syl2anc 583 . . . . 5 (𝜑 → (𝐺s 𝑈) ∈ Abel)
38 pgpfac.f . . . . . . . 8 (𝜑𝐵 ∈ Fin)
39 pgpfac.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
4039subgss 19073 . . . . . . . . 9 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈𝐵)
418, 40syl 17 . . . . . . . 8 (𝜑𝑈𝐵)
4238, 41ssfid 9283 . . . . . . 7 (𝜑𝑈 ∈ Fin)
4314, 42eqeltrrd 2829 . . . . . 6 (𝜑 → (Base‘(𝐺s 𝑈)) ∈ Fin)
4419, 20gexcl2 19535 . . . . . 6 (((𝐺s 𝑈) ∈ Grp ∧ (Base‘(𝐺s 𝑈)) ∈ Fin) → (gEx‘(𝐺s 𝑈)) ∈ ℕ)
4517, 43, 44syl2anc 583 . . . . 5 (𝜑 → (gEx‘(𝐺s 𝑈)) ∈ ℕ)
46 eqid 2727 . . . . . 6 (od‘(𝐺s 𝑈)) = (od‘(𝐺s 𝑈))
4719, 20, 46gexex 19799 . . . . 5 (((𝐺s 𝑈) ∈ Abel ∧ (gEx‘(𝐺s 𝑈)) ∈ ℕ) → ∃𝑥 ∈ (Base‘(𝐺s 𝑈))((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))
4837, 45, 47syl2anc 583 . . . 4 (𝜑 → ∃𝑥 ∈ (Base‘(𝐺s 𝑈))((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))
4948adantr 480 . . 3 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) → ∃𝑥 ∈ (Base‘(𝐺s 𝑈))((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))
50 eqid 2727 . . . . 5 (mrCls‘(SubGrp‘(𝐺s 𝑈))) = (mrCls‘(SubGrp‘(𝐺s 𝑈)))
51 eqid 2727 . . . . 5 ((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) = ((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})
52 eqid 2727 . . . . 5 (0g‘(𝐺s 𝑈)) = (0g‘(𝐺s 𝑈))
53 eqid 2727 . . . . 5 (LSSum‘(𝐺s 𝑈)) = (LSSum‘(𝐺s 𝑈))
54 pgpfac.p . . . . . . 7 (𝜑𝑃 pGrp 𝐺)
55 subgpgp 19543 . . . . . . 7 ((𝑃 pGrp 𝐺𝑈 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺s 𝑈))
5654, 8, 55syl2anc 583 . . . . . 6 (𝜑𝑃 pGrp (𝐺s 𝑈))
5756ad2antrr 725 . . . . 5 (((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) → 𝑃 pGrp (𝐺s 𝑈))
5837ad2antrr 725 . . . . 5 (((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) → (𝐺s 𝑈) ∈ Abel)
5943ad2antrr 725 . . . . 5 (((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) → (Base‘(𝐺s 𝑈)) ∈ Fin)
60 simprr 772 . . . . 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 770 . . . . 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 20028 . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 729 . . . . 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 775 . . . . 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 776 . . . . . 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 2831 . . . . 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 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))))) → ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))
75 simprl 770 . . . . 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 780 . . . . 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 781 . . . . . 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 2770 . . . . 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 20030 . . . 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 3156 . . 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 3156 . 2 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
8235, 81pm2.61dane 3024 1 (𝜑 → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  wne 2935  wral 3056  wrex 3065  {crab 3427  cin 3943  wss 3944  wpss 3945  c0 4318  {csn 4624   class class class wbr 5142  dom cdm 5672  ran crn 5673  cfv 6542  (class class class)co 7414  1oc1o 8473  cen 8952  Fincfn 8955  1c1 11131  cn 12234  Word cword 14488  Basecbs 17171  s cress 17200  0gc0g 17412  mrClscmrc 17554  Mndcmnd 18685  Grpcgrp 18881  SubGrpcsubg 19066  odcod 19470  gExcgex 19471   pGrp cpgp 19472  LSSumclsm 19580  Abelcabl 19727  CycGrpccyg 19823   DProd cdprd 19941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-disj 5108  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679  df-rpss 7722  df-om 7865  df-1st 7987  df-2nd 7988  df-supp 8160  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-omul 8485  df-er 8718  df-ec 8720  df-qs 8724  df-map 8838  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9378  df-sup 9457  df-inf 9458  df-oi 9525  df-dju 9916  df-card 9954  df-acn 9957  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-q 12955  df-rp 12999  df-fz 13509  df-fzo 13652  df-fl 13781  df-mod 13859  df-seq 13991  df-exp 14051  df-fac 14257  df-bc 14286  df-hash 14314  df-word 14489  df-concat 14545  df-s1 14570  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-clim 15456  df-sum 15657  df-dvds 16223  df-gcd 16461  df-prm 16634  df-pc 16797  df-sets 17124  df-slot 17142  df-ndx 17154  df-base 17172  df-ress 17201  df-plusg 17237  df-0g 17414  df-gsum 17415  df-mre 17557  df-mrc 17558  df-acs 17560  df-mgm 18591  df-sgrp 18670  df-mnd 18686  df-mhm 18731  df-submnd 18732  df-grp 18884  df-minusg 18885  df-sbg 18886  df-mulg 19015  df-subg 19069  df-eqg 19071  df-ghm 19159  df-gim 19204  df-ga 19232  df-cntz 19259  df-oppg 19288  df-od 19474  df-gex 19475  df-pgp 19476  df-lsm 19582  df-pj1 19583  df-cmn 19728  df-abl 19729  df-cyg 19824  df-dprd 19943
This theorem is referenced by:  pgpfac  20032
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