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Theorem pgpfaclem3 19791
Description: Lemma for pgpfac 19792. (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 14355 . . 3 ∅ ∈ Word 𝐶
2 pgpfac.g . . . . . 6 (𝜑𝐺 ∈ Abel)
3 ablgrp 19496 . . . . . 6 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
4 eqid 2738 . . . . . . 7 (0g𝐺) = (0g𝐺)
54dprd0 19739 . . . . . 6 (𝐺 ∈ Grp → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g𝐺)}))
62, 3, 53syl 18 . . . . 5 (𝜑 → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g𝐺)}))
76adantr 482 . . . 4 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = {(0g𝐺)}))
8 pgpfac.u . . . . . . . . 9 (𝜑𝑈 ∈ (SubGrp‘𝐺))
94subg0cl 18869 . . . . . . . . 9 (𝑈 ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ 𝑈)
108, 9syl 17 . . . . . . . 8 (𝜑 → (0g𝐺) ∈ 𝑈)
1110adantr 482 . . . . . . 7 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → (0g𝐺) ∈ 𝑈)
12 eqid 2738 . . . . . . . . . . 11 (𝐺s 𝑈) = (𝐺s 𝑈)
1312subgbas 18865 . . . . . . . . . 10 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 = (Base‘(𝐺s 𝑈)))
148, 13syl 17 . . . . . . . . 9 (𝜑𝑈 = (Base‘(𝐺s 𝑈)))
1514adantr 482 . . . . . . . 8 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → 𝑈 = (Base‘(𝐺s 𝑈)))
1612subggrp 18864 . . . . . . . . . . 11 (𝑈 ∈ (SubGrp‘𝐺) → (𝐺s 𝑈) ∈ Grp)
178, 16syl 17 . . . . . . . . . 10 (𝜑 → (𝐺s 𝑈) ∈ Grp)
18 grpmnd 18690 . . . . . . . . . 10 ((𝐺s 𝑈) ∈ Grp → (𝐺s 𝑈) ∈ Mnd)
19 eqid 2738 . . . . . . . . . . 11 (Base‘(𝐺s 𝑈)) = (Base‘(𝐺s 𝑈))
20 eqid 2738 . . . . . . . . . . 11 (gEx‘(𝐺s 𝑈)) = (gEx‘(𝐺s 𝑈))
2119, 20gex1 19302 . . . . . . . . . 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 478 . . . . . . . 8 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → (Base‘(𝐺s 𝑈)) ≈ 1o)
2415, 23eqbrtrd 5126 . . . . . . 7 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → 𝑈 ≈ 1o)
25 en1eqsn 9152 . . . . . . 7 (((0g𝐺) ∈ 𝑈𝑈 ≈ 1o) → 𝑈 = {(0g𝐺)})
2611, 24, 25syl2anc 585 . . . . . 6 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → 𝑈 = {(0g𝐺)})
2726eqeq2d 2749 . . . . 5 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → ((𝐺 DProd ∅) = 𝑈 ↔ (𝐺 DProd ∅) = {(0g𝐺)}))
2827anbi2d 630 . . . 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 5108 . . . . 5 (𝑠 = ∅ → (𝐺dom DProd 𝑠𝐺dom DProd ∅))
31 oveq2 7358 . . . . . 6 (𝑠 = ∅ → (𝐺 DProd 𝑠) = (𝐺 DProd ∅))
3231eqeq1d 2740 . . . . 5 (𝑠 = ∅ → ((𝐺 DProd 𝑠) = 𝑈 ↔ (𝐺 DProd ∅) = 𝑈))
3330, 32anbi12d 632 . . . 4 (𝑠 = ∅ → ((𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈) ↔ (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈)))
3433rspcev 3580 . . 3 ((∅ ∈ Word 𝐶 ∧ (𝐺dom DProd ∅ ∧ (𝐺 DProd ∅) = 𝑈)) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
351, 29, 34sylancr 588 . 2 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) = 1) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
3612subgabl 19543 . . . . . 6 ((𝐺 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝐺s 𝑈) ∈ Abel)
372, 8, 36syl2anc 585 . . . . 5 (𝜑 → (𝐺s 𝑈) ∈ Abel)
38 pgpfac.f . . . . . . . 8 (𝜑𝐵 ∈ Fin)
39 pgpfac.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
4039subgss 18862 . . . . . . . . 9 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈𝐵)
418, 40syl 17 . . . . . . . 8 (𝜑𝑈𝐵)
4238, 41ssfid 9145 . . . . . . 7 (𝜑𝑈 ∈ Fin)
4314, 42eqeltrrd 2840 . . . . . 6 (𝜑 → (Base‘(𝐺s 𝑈)) ∈ Fin)
4419, 20gexcl2 19300 . . . . . 6 (((𝐺s 𝑈) ∈ Grp ∧ (Base‘(𝐺s 𝑈)) ∈ Fin) → (gEx‘(𝐺s 𝑈)) ∈ ℕ)
4517, 43, 44syl2anc 585 . . . . 5 (𝜑 → (gEx‘(𝐺s 𝑈)) ∈ ℕ)
46 eqid 2738 . . . . . 6 (od‘(𝐺s 𝑈)) = (od‘(𝐺s 𝑈))
4719, 20, 46gexex 19560 . . . . 5 (((𝐺s 𝑈) ∈ Abel ∧ (gEx‘(𝐺s 𝑈)) ∈ ℕ) → ∃𝑥 ∈ (Base‘(𝐺s 𝑈))((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))
4837, 45, 47syl2anc 585 . . . 4 (𝜑 → ∃𝑥 ∈ (Base‘(𝐺s 𝑈))((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))
4948adantr 482 . . 3 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) → ∃𝑥 ∈ (Base‘(𝐺s 𝑈))((od‘(𝐺s 𝑈))‘𝑥) = (gEx‘(𝐺s 𝑈)))
50 eqid 2738 . . . . 5 (mrCls‘(SubGrp‘(𝐺s 𝑈))) = (mrCls‘(SubGrp‘(𝐺s 𝑈)))
51 eqid 2738 . . . . 5 ((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥}) = ((mrCls‘(SubGrp‘(𝐺s 𝑈)))‘{𝑥})
52 eqid 2738 . . . . 5 (0g‘(𝐺s 𝑈)) = (0g‘(𝐺s 𝑈))
53 eqid 2738 . . . . 5 (LSSum‘(𝐺s 𝑈)) = (LSSum‘(𝐺s 𝑈))
54 pgpfac.p . . . . . . 7 (𝜑𝑃 pGrp 𝐺)
55 subgpgp 19308 . . . . . . 7 ((𝑃 pGrp 𝐺𝑈 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺s 𝑈))
5654, 8, 55syl2anc 585 . . . . . 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 19788 . . . 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 2842 . . . . 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 2781 . . . . 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 19790 . . . 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 3157 . . 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 3157 . 2 ((𝜑 ∧ (gEx‘(𝐺s 𝑈)) ≠ 1) → ∃𝑠 ∈ Word 𝐶(𝐺dom DProd 𝑠 ∧ (𝐺 DProd 𝑠) = 𝑈))
8235, 81pm2.61dane 3031 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 397   = wceq 1542  wcel 2107  wne 2942  wral 3063  wrex 3072  {crab 3406  cin 3908  wss 3909  wpss 3910  c0 4281  {csn 4585   class class class wbr 5104  dom cdm 5631  ran crn 5632  cfv 6492  (class class class)co 7350  1oc1o 8373  cen 8814  Fincfn 8817  1c1 10986  cn 12087  Word cword 14330  Basecbs 17018  s cress 17047  0gc0g 17256  mrClscmrc 17398  Mndcmnd 18491  Grpcgrp 18683  SubGrpcsubg 18855  odcod 19238  gExcgex 19239   pGrp cpgp 19240  LSSumclsm 19345  Abelcabl 19492  CycGrpccyg 19583   DProd cdprd 19701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663  ax-inf2 9511  ax-cnex 11041  ax-resscn 11042  ax-1cn 11043  ax-icn 11044  ax-addcl 11045  ax-addrcl 11046  ax-mulcl 11047  ax-mulrcl 11048  ax-mulcom 11049  ax-addass 11050  ax-mulass 11051  ax-distr 11052  ax-i2m1 11053  ax-1ne0 11054  ax-1rid 11055  ax-rnegex 11056  ax-rrecex 11057  ax-cnre 11058  ax-pre-lttri 11059  ax-pre-lttrn 11060  ax-pre-ltadd 11061  ax-pre-mulgt0 11062  ax-pre-sup 11063
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-int 4907  df-iun 4955  df-iin 4956  df-disj 5070  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-se 5587  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6250  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7306  df-ov 7353  df-oprab 7354  df-mpo 7355  df-of 7608  df-rpss 7651  df-om 7794  df-1st 7912  df-2nd 7913  df-supp 8061  df-tpos 8125  df-frecs 8180  df-wrecs 8211  df-recs 8285  df-rdg 8324  df-1o 8380  df-2o 8381  df-oadd 8384  df-omul 8385  df-er 8582  df-ec 8584  df-qs 8588  df-map 8701  df-ixp 8770  df-en 8818  df-dom 8819  df-sdom 8820  df-fin 8821  df-fsupp 9240  df-sup 9312  df-inf 9313  df-oi 9380  df-dju 9771  df-card 9809  df-acn 9812  df-pnf 11125  df-mnf 11126  df-xr 11127  df-ltxr 11128  df-le 11129  df-sub 11321  df-neg 11322  df-div 11747  df-nn 12088  df-2 12150  df-3 12151  df-n0 12348  df-xnn0 12420  df-z 12434  df-uz 12697  df-q 12803  df-rp 12845  df-fz 13354  df-fzo 13497  df-fl 13626  df-mod 13704  df-seq 13836  df-exp 13897  df-fac 14102  df-bc 14131  df-hash 14159  df-word 14331  df-concat 14387  df-s1 14412  df-cj 14918  df-re 14919  df-im 14920  df-sqrt 15054  df-abs 15055  df-clim 15305  df-sum 15506  df-dvds 16072  df-gcd 16310  df-prm 16483  df-pc 16644  df-sets 16971  df-slot 16989  df-ndx 17001  df-base 17019  df-ress 17048  df-plusg 17081  df-0g 17258  df-gsum 17259  df-mre 17401  df-mrc 17402  df-acs 17404  df-mgm 18432  df-sgrp 18481  df-mnd 18492  df-mhm 18536  df-submnd 18537  df-grp 18686  df-minusg 18687  df-sbg 18688  df-mulg 18807  df-subg 18858  df-eqg 18860  df-ghm 18938  df-gim 18981  df-ga 19002  df-cntz 19029  df-oppg 19056  df-od 19242  df-gex 19243  df-pgp 19244  df-lsm 19347  df-pj1 19348  df-cmn 19493  df-abl 19494  df-cyg 19584  df-dprd 19703
This theorem is referenced by:  pgpfac  19792
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