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Theorem pgpfaclem3 19955
Description: Lemma for pgpfac 19956. (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 14491 . . 3 ∅ ∈ Word 𝐶
2 pgpfac.g . . . . . 6 (𝜑𝐺 ∈ Abel)
3 ablgrp 19655 . . . . . 6 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
4 eqid 2732 . . . . . . 7 (0g𝐺) = (0g𝐺)
54dprd0 19903 . . . . . 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 19016 . . . . . . . . 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 2732 . . . . . . . . . . 11 (𝐺s 𝑈) = (𝐺s 𝑈)
1312subgbas 19012 . . . . . . . . . 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 19011 . . . . . . . . . . 11 (𝑈 ∈ (SubGrp‘𝐺) → (𝐺s 𝑈) ∈ Grp)
178, 16syl 17 . . . . . . . . . 10 (𝜑 → (𝐺s 𝑈) ∈ Grp)
18 grpmnd 18828 . . . . . . . . . 10 ((𝐺s 𝑈) ∈ Grp → (𝐺s 𝑈) ∈ Mnd)
19 eqid 2732 . . . . . . . . . . 11 (Base‘(𝐺s 𝑈)) = (Base‘(𝐺s 𝑈))
20 eqid 2732 . . . . . . . . . . 11 (gEx‘(𝐺s 𝑈)) = (gEx‘(𝐺s 𝑈))
2119, 20gex1 19461 . . . . . . . . . 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 5170 . . . . . . 7 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → 𝑈 ≈ 1o)
25 en1eqsn 9276 . . . . . . 7 (((0g𝐺) ∈ 𝑈𝑈 ≈ 1o) → 𝑈 = {(0g𝐺)})
2611, 24, 25syl2anc 584 . . . . . 6 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → 𝑈 = {(0g𝐺)})
2726eqeq2d 2743 . . . . 5 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → ((𝐺 DProd ∅) = 𝑈 ↔ (𝐺 DProd ∅) = {(0g𝐺)}))
2827anbi2d 629 . . . 4 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → ((𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈) ↔ (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g𝐺)})))
297, 28mpbird 256 . . 3 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈))
30 breq2 5152 . . . . 5 (𝑠 = ∅ → (𝐺dom DProd 𝑠𝐺dom DProd ∅))
31 oveq2 7419 . . . . . 6 (𝑠 = ∅ → (𝐺 DProd 𝑠) = (𝐺 DProd ∅))
3231eqeq1d 2734 . . . . 5 (𝑠 = ∅ → ((𝐺 DProd 𝑠) = 𝑈 ↔ (𝐺 DProd ∅) = 𝑈))
3330, 32anbi12d 631 . . . 4 (𝑠 = ∅ → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈) ↔ (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈)))
3433rspcev 3612 . . 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 19706 . . . . . 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 19009 . . . . . . . . 9 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈𝐵)
418, 40syl 17 . . . . . . . 8 (𝜑𝑈𝐵)
4238, 41ssfid 9269 . . . . . . 7 (𝜑𝑈 ∈ Fin)
4314, 42eqeltrrd 2834 . . . . . 6 (𝜑 → (Base‘(𝐺s 𝑈)) ∈ Fin)
4419, 20gexcl2 19459 . . . . . 6 (((𝐺s 𝑈) ∈ Grp ∧ (Base‘(𝐺s 𝑈)) ∈ Fin) → (gEx‘(𝐺s 𝑈)) ∈ ℕ)
4517, 43, 44syl2anc 584 . . . . 5 (𝜑 → (gEx‘(𝐺s 𝑈)) ∈ ℕ)
46 eqid 2732 . . . . . 6 (od‘(𝐺s 𝑈)) = (od‘(𝐺s 𝑈))
4719, 20, 46gexex 19723 . . . . 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 2732 . . . . 5 (mrCls‘(SubGrp‘(𝐺s 𝑈))) = (mrCls‘(SubGrp‘(𝐺s 𝑈)))
51 eqid 2732 . . . . 5 ((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) = ((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})
52 eqid 2732 . . . . 5 (0g‘(𝐺s 𝑈)) = (0g‘(𝐺s 𝑈))
53 eqid 2732 . . . . 5 (LSSum‘(𝐺s 𝑈)) = (LSSum‘(𝐺s 𝑈))
54 pgpfac.p . . . . . . 7 (𝜑𝑃 pGrp 𝐺)
55 subgpgp 19467 . . . . . . 7 ((𝑃 pGrp 𝐺𝑈 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺s 𝑈))
5654, 8, 55syl2anc 584 . . . . . 6 (𝜑𝑃 pGrp (𝐺s 𝑈))
5756ad2antrr 724 . . . . 5 (((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) → 𝑃 pGrp (𝐺s 𝑈))
5837ad2antrr 724 . . . . 5 (((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) → (𝐺s 𝑈) ∈ Abel)
5943ad2antrr 724 . . . . 5 (((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) ∧ (𝑥 ∈ (Base‘(𝐺s 𝑈)) ∧ ((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))) → (Base‘(𝐺s 𝑈)) ∈ Fin)
60 simprr 771 . . . . 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 769 . . . . 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 19952 . . . 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 728 . . . . 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 728 . . . . 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 728 . . . . 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 728 . . . . 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 728 . . . . 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 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))))) → (gEx‘(𝐺s 𝑈)) ≠ 1)
71 simplrl 775 . . . . . 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 2836 . . . . 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 776 . . . . 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 769 . . . . 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 779 . . . . 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 780 . . . . . 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 2775 . . . . 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 19954 . . . 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 3161 . . 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 3161 . 2 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
8235, 81pm2.61dane 3029 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 396   = wceq 1541  wcel 2106  wne 2940  wral 3061  wrex 3070  {crab 3432  cin 3947  wss 3948  wpss 3949  c0 4322  {csn 4628   class class class wbr 5148  dom cdm 5676  ran crn 5677  cfv 6543  (class class class)co 7411  1oc1o 8461  cen 8938  Fincfn 8941  1c1 11113  cn 12214  Word cword 14466  Basecbs 17146  s cress 17175  0gc0g 17387  mrClscmrc 17529  Mndcmnd 18627  Grpcgrp 18821  SubGrpcsubg 19002  odcod 19394  gExcgex 19395   pGrp cpgp 19396  LSSumclsm 19504  Abelcabl 19651  CycGrpccyg 19747   DProd cdprd 19865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  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 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  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 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-n0 12475  df-xnn0 12547  df-z 12561  df-uz 12825  df-q 12935  df-rp 12977  df-fz 13487  df-fzo 13630  df-fl 13759  df-mod 13837  df-seq 13969  df-exp 14030  df-fac 14236  df-bc 14265  df-hash 14293  df-word 14467  df-concat 14523  df-s1 14548  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-clim 15434  df-sum 15635  df-dvds 16200  df-gcd 16438  df-prm 16611  df-pc 16772  df-sets 17099  df-slot 17117  df-ndx 17129  df-base 17147  df-ress 17176  df-plusg 17212  df-0g 17389  df-gsum 17390  df-mre 17532  df-mrc 17533  df-acs 17535  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-mhm 18673  df-submnd 18674  df-grp 18824  df-minusg 18825  df-sbg 18826  df-mulg 18953  df-subg 19005  df-eqg 19007  df-ghm 19092  df-gim 19135  df-ga 19156  df-cntz 19183  df-oppg 19212  df-od 19398  df-gex 19399  df-pgp 19400  df-lsm 19506  df-pj1 19507  df-cmn 19652  df-abl 19653  df-cyg 19748  df-dprd 19867
This theorem is referenced by:  pgpfac  19956
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