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Theorem pgpfac1lem5 19949
Description: Lemma for pgpfac1 19950. (Contributed by Mario Carneiro, 27-Apr-2016.)
Hypotheses
Ref Expression
pgpfac1.k 𝐾 = (mrClsβ€˜(SubGrpβ€˜πΊ))
pgpfac1.s 𝑆 = (πΎβ€˜{𝐴})
pgpfac1.b 𝐡 = (Baseβ€˜πΊ)
pgpfac1.o 𝑂 = (odβ€˜πΊ)
pgpfac1.e 𝐸 = (gExβ€˜πΊ)
pgpfac1.z 0 = (0gβ€˜πΊ)
pgpfac1.l βŠ• = (LSSumβ€˜πΊ)
pgpfac1.p (πœ‘ β†’ 𝑃 pGrp 𝐺)
pgpfac1.g (πœ‘ β†’ 𝐺 ∈ Abel)
pgpfac1.n (πœ‘ β†’ 𝐡 ∈ Fin)
pgpfac1.oe (πœ‘ β†’ (π‘‚β€˜π΄) = 𝐸)
pgpfac1.u (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))
pgpfac1.au (πœ‘ β†’ 𝐴 ∈ π‘ˆ)
pgpfac1.3 (πœ‘ β†’ βˆ€π‘  ∈ (SubGrpβ€˜πΊ)((𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠) β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = 𝑠)))
Assertion
Ref Expression
pgpfac1lem5 (πœ‘ β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ))
Distinct variable groups:   𝑑,𝑠, 0   𝐴,𝑠,𝑑   βŠ• ,𝑠,𝑑   𝑃,𝑠,𝑑   𝐡,𝑠,𝑑   𝐺,𝑠,𝑑   π‘ˆ,𝑠,𝑑   𝑆,𝑠,𝑑   πœ‘,𝑠,𝑑   𝐾,𝑠,𝑑
Allowed substitution hints:   𝐸(𝑑,𝑠)   𝑂(𝑑,𝑠)

Proof of Theorem pgpfac1lem5
Dummy variables 𝑏 𝑒 𝑣 𝑦 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pgpfac1.n . . . . . . . . . 10 (πœ‘ β†’ 𝐡 ∈ Fin)
2 pwfi 9178 . . . . . . . . . 10 (𝐡 ∈ Fin ↔ 𝒫 𝐡 ∈ Fin)
31, 2sylib 217 . . . . . . . . 9 (πœ‘ β†’ 𝒫 𝐡 ∈ Fin)
43adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ 𝒫 𝐡 ∈ Fin)
5 pgpfac1.b . . . . . . . . . . . 12 𝐡 = (Baseβ€˜πΊ)
65subgss 19007 . . . . . . . . . . 11 (𝑣 ∈ (SubGrpβ€˜πΊ) β†’ 𝑣 βŠ† 𝐡)
763ad2ant2 1135 . . . . . . . . . 10 (((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ) ∧ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)) β†’ 𝑣 βŠ† 𝐡)
8 velpw 4608 . . . . . . . . . 10 (𝑣 ∈ 𝒫 𝐡 ↔ 𝑣 βŠ† 𝐡)
97, 8sylibr 233 . . . . . . . . 9 (((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ) ∧ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)) β†’ 𝑣 ∈ 𝒫 𝐡)
109rabssdv 4073 . . . . . . . 8 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} βŠ† 𝒫 𝐡)
114, 10ssfid 9267 . . . . . . 7 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ∈ Fin)
12 finnum 9943 . . . . . . 7 ({𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ∈ Fin β†’ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ∈ dom card)
1311, 12syl 17 . . . . . 6 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ∈ dom card)
14 pgpfac1.s . . . . . . . . . 10 𝑆 = (πΎβ€˜{𝐴})
15 pgpfac1.g . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐺 ∈ Abel)
16 ablgrp 19653 . . . . . . . . . . . . 13 (𝐺 ∈ Abel β†’ 𝐺 ∈ Grp)
1715, 16syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐺 ∈ Grp)
185subgacs 19041 . . . . . . . . . . . 12 (𝐺 ∈ Grp β†’ (SubGrpβ€˜πΊ) ∈ (ACSβ€˜π΅))
19 acsmre 17596 . . . . . . . . . . . 12 ((SubGrpβ€˜πΊ) ∈ (ACSβ€˜π΅) β†’ (SubGrpβ€˜πΊ) ∈ (Mooreβ€˜π΅))
2017, 18, 193syl 18 . . . . . . . . . . 11 (πœ‘ β†’ (SubGrpβ€˜πΊ) ∈ (Mooreβ€˜π΅))
21 pgpfac1.u . . . . . . . . . . . . 13 (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))
225subgss 19007 . . . . . . . . . . . . 13 (π‘ˆ ∈ (SubGrpβ€˜πΊ) β†’ π‘ˆ βŠ† 𝐡)
2321, 22syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ π‘ˆ βŠ† 𝐡)
24 pgpfac1.au . . . . . . . . . . . 12 (πœ‘ β†’ 𝐴 ∈ π‘ˆ)
2523, 24sseldd 3984 . . . . . . . . . . 11 (πœ‘ β†’ 𝐴 ∈ 𝐡)
26 pgpfac1.k . . . . . . . . . . . 12 𝐾 = (mrClsβ€˜(SubGrpβ€˜πΊ))
2726mrcsncl 17556 . . . . . . . . . . 11 (((SubGrpβ€˜πΊ) ∈ (Mooreβ€˜π΅) ∧ 𝐴 ∈ 𝐡) β†’ (πΎβ€˜{𝐴}) ∈ (SubGrpβ€˜πΊ))
2820, 25, 27syl2anc 585 . . . . . . . . . 10 (πœ‘ β†’ (πΎβ€˜{𝐴}) ∈ (SubGrpβ€˜πΊ))
2914, 28eqeltrid 2838 . . . . . . . . 9 (πœ‘ β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))
3029adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))
31 simpr 486 . . . . . . . 8 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ 𝑆 ⊊ π‘ˆ)
3224snssd 4813 . . . . . . . . . . . . 13 (πœ‘ β†’ {𝐴} βŠ† π‘ˆ)
3332, 23sstrd 3993 . . . . . . . . . . . 12 (πœ‘ β†’ {𝐴} βŠ† 𝐡)
3420, 26, 33mrcssidd 17569 . . . . . . . . . . 11 (πœ‘ β†’ {𝐴} βŠ† (πΎβ€˜{𝐴}))
3534, 14sseqtrrdi 4034 . . . . . . . . . 10 (πœ‘ β†’ {𝐴} βŠ† 𝑆)
36 snssg 4788 . . . . . . . . . . 11 (𝐴 ∈ 𝐡 β†’ (𝐴 ∈ 𝑆 ↔ {𝐴} βŠ† 𝑆))
3725, 36syl 17 . . . . . . . . . 10 (πœ‘ β†’ (𝐴 ∈ 𝑆 ↔ {𝐴} βŠ† 𝑆))
3835, 37mpbird 257 . . . . . . . . 9 (πœ‘ β†’ 𝐴 ∈ 𝑆)
3938adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ 𝐴 ∈ 𝑆)
40 psseq1 4088 . . . . . . . . . 10 (𝑣 = 𝑆 β†’ (𝑣 ⊊ π‘ˆ ↔ 𝑆 ⊊ π‘ˆ))
41 eleq2 2823 . . . . . . . . . 10 (𝑣 = 𝑆 β†’ (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑆))
4240, 41anbi12d 632 . . . . . . . . 9 (𝑣 = 𝑆 β†’ ((𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣) ↔ (𝑆 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑆)))
4342rspcev 3613 . . . . . . . 8 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑆 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑆)) β†’ βˆƒπ‘£ ∈ (SubGrpβ€˜πΊ)(𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣))
4430, 31, 39, 43syl12anc 836 . . . . . . 7 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ βˆƒπ‘£ ∈ (SubGrpβ€˜πΊ)(𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣))
45 rabn0 4386 . . . . . . 7 ({𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} β‰  βˆ… ↔ βˆƒπ‘£ ∈ (SubGrpβ€˜πΊ)(𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣))
4644, 45sylibr 233 . . . . . 6 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} β‰  βˆ…)
47 simpr1 1195 . . . . . . . . 9 (((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) ∧ (𝑒 βŠ† {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ∧ 𝑒 β‰  βˆ… ∧ [⊊] Or 𝑒)) β†’ 𝑒 βŠ† {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)})
48 simpr2 1196 . . . . . . . . . 10 (((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) ∧ (𝑒 βŠ† {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ∧ 𝑒 β‰  βˆ… ∧ [⊊] Or 𝑒)) β†’ 𝑒 β‰  βˆ…)
4911adantr 482 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) ∧ (𝑒 βŠ† {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ∧ 𝑒 β‰  βˆ… ∧ [⊊] Or 𝑒)) β†’ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ∈ Fin)
5049, 47ssfid 9267 . . . . . . . . . 10 (((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) ∧ (𝑒 βŠ† {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ∧ 𝑒 β‰  βˆ… ∧ [⊊] Or 𝑒)) β†’ 𝑒 ∈ Fin)
51 simpr3 1197 . . . . . . . . . 10 (((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) ∧ (𝑒 βŠ† {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ∧ 𝑒 β‰  βˆ… ∧ [⊊] Or 𝑒)) β†’ [⊊] Or 𝑒)
52 fin1a2lem10 10404 . . . . . . . . . 10 ((𝑒 β‰  βˆ… ∧ 𝑒 ∈ Fin ∧ [⊊] Or 𝑒) β†’ βˆͺ 𝑒 ∈ 𝑒)
5348, 50, 51, 52syl3anc 1372 . . . . . . . . 9 (((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) ∧ (𝑒 βŠ† {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ∧ 𝑒 β‰  βˆ… ∧ [⊊] Or 𝑒)) β†’ βˆͺ 𝑒 ∈ 𝑒)
5447, 53sseldd 3984 . . . . . . . 8 (((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) ∧ (𝑒 βŠ† {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ∧ 𝑒 β‰  βˆ… ∧ [⊊] Or 𝑒)) β†’ βˆͺ 𝑒 ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)})
5554ex 414 . . . . . . 7 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ ((𝑒 βŠ† {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ∧ 𝑒 β‰  βˆ… ∧ [⊊] Or 𝑒) β†’ βˆͺ 𝑒 ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)}))
5655alrimiv 1931 . . . . . 6 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ βˆ€π‘’((𝑒 βŠ† {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ∧ 𝑒 β‰  βˆ… ∧ [⊊] Or 𝑒) β†’ βˆͺ 𝑒 ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)}))
57 zornn0g 10500 . . . . . 6 (({𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ∈ dom card ∧ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} β‰  βˆ… ∧ βˆ€π‘’((𝑒 βŠ† {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ∧ 𝑒 β‰  βˆ… ∧ [⊊] Or 𝑒) β†’ βˆͺ 𝑒 ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)})) β†’ βˆƒπ‘  ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)}βˆ€π‘€ ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} Β¬ 𝑠 ⊊ 𝑀)
5813, 46, 56, 57syl3anc 1372 . . . . 5 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ βˆƒπ‘  ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)}βˆ€π‘€ ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} Β¬ 𝑠 ⊊ 𝑀)
59 psseq1 4088 . . . . . . . 8 (𝑣 = 𝑀 β†’ (𝑣 ⊊ π‘ˆ ↔ 𝑀 ⊊ π‘ˆ))
60 eleq2 2823 . . . . . . . 8 (𝑣 = 𝑀 β†’ (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑀))
6159, 60anbi12d 632 . . . . . . 7 (𝑣 = 𝑀 β†’ ((𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣) ↔ (𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀)))
6261ralrab 3690 . . . . . 6 (βˆ€π‘€ ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} Β¬ 𝑠 ⊊ 𝑀 ↔ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀))
6362rexbii 3095 . . . . 5 (βˆƒπ‘  ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)}βˆ€π‘€ ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} Β¬ 𝑠 ⊊ 𝑀 ↔ βˆƒπ‘  ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)}βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀))
6458, 63sylib 217 . . . 4 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ βˆƒπ‘  ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)}βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀))
6564ex 414 . . 3 (πœ‘ β†’ (𝑆 ⊊ π‘ˆ β†’ βˆƒπ‘  ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)}βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)))
66 pgpfac1.3 . . . . 5 (πœ‘ β†’ βˆ€π‘  ∈ (SubGrpβ€˜πΊ)((𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠) β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = 𝑠)))
67 psseq1 4088 . . . . . . 7 (𝑣 = 𝑠 β†’ (𝑣 ⊊ π‘ˆ ↔ 𝑠 ⊊ π‘ˆ))
68 eleq2 2823 . . . . . . 7 (𝑣 = 𝑠 β†’ (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑠))
6967, 68anbi12d 632 . . . . . 6 (𝑣 = 𝑠 β†’ ((𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣) ↔ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠)))
7069ralrab 3690 . . . . 5 (βˆ€π‘  ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)}βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = 𝑠) ↔ βˆ€π‘  ∈ (SubGrpβ€˜πΊ)((𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠) β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = 𝑠)))
7166, 70sylibr 233 . . . 4 (πœ‘ β†’ βˆ€π‘  ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)}βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = 𝑠))
72 r19.29 3115 . . . . 5 ((βˆ€π‘  ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)}βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = 𝑠) ∧ βˆƒπ‘  ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)}βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) β†’ βˆƒπ‘  ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} (βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = 𝑠) ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)))
7369elrab 3684 . . . . . . 7 (𝑠 ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ↔ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠)))
74 ineq2 4207 . . . . . . . . . . . 12 (𝑑 = 𝑣 β†’ (𝑆 ∩ 𝑑) = (𝑆 ∩ 𝑣))
7574eqeq1d 2735 . . . . . . . . . . 11 (𝑑 = 𝑣 β†’ ((𝑆 ∩ 𝑑) = { 0 } ↔ (𝑆 ∩ 𝑣) = { 0 }))
76 oveq2 7417 . . . . . . . . . . . 12 (𝑑 = 𝑣 β†’ (𝑆 βŠ• 𝑑) = (𝑆 βŠ• 𝑣))
7776eqeq1d 2735 . . . . . . . . . . 11 (𝑑 = 𝑣 β†’ ((𝑆 βŠ• 𝑑) = 𝑠 ↔ (𝑆 βŠ• 𝑣) = 𝑠))
7875, 77anbi12d 632 . . . . . . . . . 10 (𝑑 = 𝑣 β†’ (((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = 𝑠) ↔ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠)))
7978cbvrexvw 3236 . . . . . . . . 9 (βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = 𝑠) ↔ βˆƒπ‘£ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠))
80 simprrl 780 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) β†’ 𝑠 ⊊ π‘ˆ)
8180ad2antrr 725 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀))) β†’ 𝑠 ⊊ π‘ˆ)
82 simpr2 1196 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀))) β†’ (𝑆 βŠ• 𝑣) = 𝑠)
8382psseq1d 4093 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀))) β†’ ((𝑆 βŠ• 𝑣) ⊊ π‘ˆ ↔ 𝑠 ⊊ π‘ˆ))
8481, 83mpbird 257 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀))) β†’ (𝑆 βŠ• 𝑣) ⊊ π‘ˆ)
85 pssdif 4367 . . . . . . . . . . . . . . 15 ((𝑆 βŠ• 𝑣) ⊊ π‘ˆ β†’ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)) β‰  βˆ…)
86 n0 4347 . . . . . . . . . . . . . . 15 ((π‘ˆ βˆ– (𝑆 βŠ• 𝑣)) β‰  βˆ… ↔ βˆƒπ‘ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))
8785, 86sylib 217 . . . . . . . . . . . . . 14 ((𝑆 βŠ• 𝑣) ⊊ π‘ˆ β†’ βˆƒπ‘ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))
8884, 87syl 17 . . . . . . . . . . . . 13 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀))) β†’ βˆƒπ‘ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))
89 pgpfac1.o . . . . . . . . . . . . . . . 16 𝑂 = (odβ€˜πΊ)
90 pgpfac1.e . . . . . . . . . . . . . . . 16 𝐸 = (gExβ€˜πΊ)
91 pgpfac1.z . . . . . . . . . . . . . . . 16 0 = (0gβ€˜πΊ)
92 pgpfac1.l . . . . . . . . . . . . . . . 16 βŠ• = (LSSumβ€˜πΊ)
93 pgpfac1.p . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝑃 pGrp 𝐺)
9493ad3antrrr 729 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) ∧ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))) β†’ 𝑃 pGrp 𝐺)
9515ad3antrrr 729 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) ∧ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))) β†’ 𝐺 ∈ Abel)
961ad3antrrr 729 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) ∧ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))) β†’ 𝐡 ∈ Fin)
97 pgpfac1.oe . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ (π‘‚β€˜π΄) = 𝐸)
9897ad3antrrr 729 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) ∧ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))) β†’ (π‘‚β€˜π΄) = 𝐸)
9921ad3antrrr 729 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) ∧ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))) β†’ π‘ˆ ∈ (SubGrpβ€˜πΊ))
10024ad3antrrr 729 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) ∧ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))) β†’ 𝐴 ∈ π‘ˆ)
101 simplr 768 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) ∧ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))) β†’ 𝑣 ∈ (SubGrpβ€˜πΊ))
102 simprl1 1219 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) ∧ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))) β†’ (𝑆 ∩ 𝑣) = { 0 })
10384adantrr 716 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) ∧ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))) β†’ (𝑆 βŠ• 𝑣) ⊊ π‘ˆ)
104103pssssd 4098 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) ∧ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))) β†’ (𝑆 βŠ• 𝑣) βŠ† π‘ˆ)
105 simprl3 1221 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) ∧ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))) β†’ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀))
10682adantrr 716 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) ∧ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))) β†’ (𝑆 βŠ• 𝑣) = 𝑠)
107 psseq1 4088 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 βŠ• 𝑣) = 𝑠 β†’ ((𝑆 βŠ• 𝑣) ⊊ 𝑦 ↔ 𝑠 ⊊ 𝑦))
108107notbid 318 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆 βŠ• 𝑣) = 𝑠 β†’ (Β¬ (𝑆 βŠ• 𝑣) ⊊ 𝑦 ↔ Β¬ 𝑠 ⊊ 𝑦))
109108imbi2d 341 . . . . . . . . . . . . . . . . . . . 20 ((𝑆 βŠ• 𝑣) = 𝑠 β†’ (((𝑦 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑦) β†’ Β¬ (𝑆 βŠ• 𝑣) ⊊ 𝑦) ↔ ((𝑦 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑦) β†’ Β¬ 𝑠 ⊊ 𝑦)))
110109ralbidv 3178 . . . . . . . . . . . . . . . . . . 19 ((𝑆 βŠ• 𝑣) = 𝑠 β†’ (βˆ€π‘¦ ∈ (SubGrpβ€˜πΊ)((𝑦 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑦) β†’ Β¬ (𝑆 βŠ• 𝑣) ⊊ 𝑦) ↔ βˆ€π‘¦ ∈ (SubGrpβ€˜πΊ)((𝑦 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑦) β†’ Β¬ 𝑠 ⊊ 𝑦)))
111 psseq1 4088 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑀 β†’ (𝑦 ⊊ π‘ˆ ↔ 𝑀 ⊊ π‘ˆ))
112 eleq2 2823 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑀 β†’ (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑀))
113111, 112anbi12d 632 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑀 β†’ ((𝑦 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑦) ↔ (𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀)))
114 psseq2 4089 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑀 β†’ (𝑠 ⊊ 𝑦 ↔ 𝑠 ⊊ 𝑀))
115114notbid 318 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑀 β†’ (Β¬ 𝑠 ⊊ 𝑦 ↔ Β¬ 𝑠 ⊊ 𝑀))
116113, 115imbi12d 345 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑀 β†’ (((𝑦 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑦) β†’ Β¬ 𝑠 ⊊ 𝑦) ↔ ((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)))
117116cbvralvw 3235 . . . . . . . . . . . . . . . . . . 19 (βˆ€π‘¦ ∈ (SubGrpβ€˜πΊ)((𝑦 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑦) β†’ Β¬ 𝑠 ⊊ 𝑦) ↔ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀))
118110, 117bitrdi 287 . . . . . . . . . . . . . . . . . 18 ((𝑆 βŠ• 𝑣) = 𝑠 β†’ (βˆ€π‘¦ ∈ (SubGrpβ€˜πΊ)((𝑦 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑦) β†’ Β¬ (𝑆 βŠ• 𝑣) ⊊ 𝑦) ↔ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)))
119106, 118syl 17 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) ∧ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))) β†’ (βˆ€π‘¦ ∈ (SubGrpβ€˜πΊ)((𝑦 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑦) β†’ Β¬ (𝑆 βŠ• 𝑣) ⊊ 𝑦) ↔ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)))
120105, 119mpbird 257 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) ∧ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))) β†’ βˆ€π‘¦ ∈ (SubGrpβ€˜πΊ)((𝑦 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑦) β†’ Β¬ (𝑆 βŠ• 𝑣) ⊊ 𝑦))
121 simprr 772 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) ∧ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))) β†’ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))
122 eqid 2733 . . . . . . . . . . . . . . . 16 (.gβ€˜πΊ) = (.gβ€˜πΊ)
12326, 14, 5, 89, 90, 91, 92, 94, 95, 96, 98, 99, 100, 101, 102, 104, 120, 121, 122pgpfac1lem4 19948 . . . . . . . . . . . . . . 15 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) ∧ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)))) β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ))
124123expr 458 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀))) β†’ (𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)) β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ)))
125124exlimdv 1937 . . . . . . . . . . . . 13 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀))) β†’ (βˆƒπ‘ 𝑏 ∈ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)) β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ)))
12688, 125mpd 15 . . . . . . . . . . . 12 ((((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) ∧ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠 ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀))) β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ))
1271263exp2 1355 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) β†’ ((𝑆 ∩ 𝑣) = { 0 } β†’ ((𝑆 βŠ• 𝑣) = 𝑠 β†’ (βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀) β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ)))))
128127impd 412 . . . . . . . . . 10 (((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ)) β†’ (((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠) β†’ (βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀) β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ))))
129128rexlimdva 3156 . . . . . . . . 9 ((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) β†’ (βˆƒπ‘£ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠) β†’ (βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀) β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ))))
13079, 129biimtrid 241 . . . . . . . 8 ((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) β†’ (βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = 𝑠) β†’ (βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀) β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ))))
131130impd 412 . . . . . . 7 ((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) β†’ ((βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = 𝑠) ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ)))
13273, 131sylan2b 595 . . . . . 6 ((πœ‘ ∧ 𝑠 ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)}) β†’ ((βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = 𝑠) ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ)))
133132rexlimdva 3156 . . . . 5 (πœ‘ β†’ (βˆƒπ‘  ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} (βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = 𝑠) ∧ βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ)))
13472, 133syl5 34 . . . 4 (πœ‘ β†’ ((βˆ€π‘  ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)}βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = 𝑠) ∧ βˆƒπ‘  ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)}βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)) β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ)))
13571, 134mpand 694 . . 3 (πœ‘ β†’ (βˆƒπ‘  ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)}βˆ€π‘€ ∈ (SubGrpβ€˜πΊ)((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀) β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ)))
13665, 135syld 47 . 2 (πœ‘ β†’ (𝑆 ⊊ π‘ˆ β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ)))
137910subg 19031 . . . . . 6 (𝐺 ∈ Grp β†’ { 0 } ∈ (SubGrpβ€˜πΊ))
13817, 137syl 17 . . . . 5 (πœ‘ β†’ { 0 } ∈ (SubGrpβ€˜πΊ))
139138adantr 482 . . . 4 ((πœ‘ ∧ 𝑆 = π‘ˆ) β†’ { 0 } ∈ (SubGrpβ€˜πΊ))
14091subg0cl 19014 . . . . . . . 8 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ 0 ∈ 𝑆)
14129, 140syl 17 . . . . . . 7 (πœ‘ β†’ 0 ∈ 𝑆)
142141snssd 4813 . . . . . 6 (πœ‘ β†’ { 0 } βŠ† 𝑆)
143142adantr 482 . . . . 5 ((πœ‘ ∧ 𝑆 = π‘ˆ) β†’ { 0 } βŠ† 𝑆)
144 sseqin2 4216 . . . . 5 ({ 0 } βŠ† 𝑆 ↔ (𝑆 ∩ { 0 }) = { 0 })
145143, 144sylib 217 . . . 4 ((πœ‘ ∧ 𝑆 = π‘ˆ) β†’ (𝑆 ∩ { 0 }) = { 0 })
14692lsmss2 19535 . . . . . . 7 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ { 0 } ∈ (SubGrpβ€˜πΊ) ∧ { 0 } βŠ† 𝑆) β†’ (𝑆 βŠ• { 0 }) = 𝑆)
14729, 138, 142, 146syl3anc 1372 . . . . . 6 (πœ‘ β†’ (𝑆 βŠ• { 0 }) = 𝑆)
148147eqeq1d 2735 . . . . 5 (πœ‘ β†’ ((𝑆 βŠ• { 0 }) = π‘ˆ ↔ 𝑆 = π‘ˆ))
149148biimpar 479 . . . 4 ((πœ‘ ∧ 𝑆 = π‘ˆ) β†’ (𝑆 βŠ• { 0 }) = π‘ˆ)
150 ineq2 4207 . . . . . . 7 (𝑑 = { 0 } β†’ (𝑆 ∩ 𝑑) = (𝑆 ∩ { 0 }))
151150eqeq1d 2735 . . . . . 6 (𝑑 = { 0 } β†’ ((𝑆 ∩ 𝑑) = { 0 } ↔ (𝑆 ∩ { 0 }) = { 0 }))
152 oveq2 7417 . . . . . . 7 (𝑑 = { 0 } β†’ (𝑆 βŠ• 𝑑) = (𝑆 βŠ• { 0 }))
153152eqeq1d 2735 . . . . . 6 (𝑑 = { 0 } β†’ ((𝑆 βŠ• 𝑑) = π‘ˆ ↔ (𝑆 βŠ• { 0 }) = π‘ˆ))
154151, 153anbi12d 632 . . . . 5 (𝑑 = { 0 } β†’ (((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ) ↔ ((𝑆 ∩ { 0 }) = { 0 } ∧ (𝑆 βŠ• { 0 }) = π‘ˆ)))
155154rspcev 3613 . . . 4 (({ 0 } ∈ (SubGrpβ€˜πΊ) ∧ ((𝑆 ∩ { 0 }) = { 0 } ∧ (𝑆 βŠ• { 0 }) = π‘ˆ)) β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ))
156139, 145, 149, 155syl12anc 836 . . 3 ((πœ‘ ∧ 𝑆 = π‘ˆ) β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ))
157156ex 414 . 2 (πœ‘ β†’ (𝑆 = π‘ˆ β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ)))
15826mrcsscl 17564 . . . . 5 (((SubGrpβ€˜πΊ) ∈ (Mooreβ€˜π΅) ∧ {𝐴} βŠ† π‘ˆ ∧ π‘ˆ ∈ (SubGrpβ€˜πΊ)) β†’ (πΎβ€˜{𝐴}) βŠ† π‘ˆ)
15920, 32, 21, 158syl3anc 1372 . . . 4 (πœ‘ β†’ (πΎβ€˜{𝐴}) βŠ† π‘ˆ)
16014, 159eqsstrid 4031 . . 3 (πœ‘ β†’ 𝑆 βŠ† π‘ˆ)
161 sspss 4100 . . 3 (𝑆 βŠ† π‘ˆ ↔ (𝑆 ⊊ π‘ˆ ∨ 𝑆 = π‘ˆ))
162160, 161sylib 217 . 2 (πœ‘ β†’ (𝑆 ⊊ π‘ˆ ∨ 𝑆 = π‘ˆ))
163136, 157, 162mpjaod 859 1 (πœ‘ β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088  βˆ€wal 1540   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  {crab 3433   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949   ⊊ wpss 3950  βˆ…c0 4323  π’« cpw 4603  {csn 4629  βˆͺ cuni 4909   class class class wbr 5149   Or wor 5588  dom cdm 5677  β€˜cfv 6544  (class class class)co 7409   [⊊] crpss 7712  Fincfn 8939  cardccrd 9930  Basecbs 17144  0gc0g 17385  Moorecmre 17526  mrClscmrc 17527  ACScacs 17529  Grpcgrp 18819  .gcmg 18950  SubGrpcsubg 19000  odcod 19392  gExcgex 19393   pGrp cpgp 19394  LSSumclsm 19502  Abelcabl 19649
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-disj 5115  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-rpss 7713  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-oadd 8470  df-omul 8471  df-er 8703  df-ec 8705  df-qs 8709  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-inf 9438  df-oi 9505  df-dju 9896  df-card 9934  df-acn 9937  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-fz 13485  df-fzo 13628  df-fl 13757  df-mod 13835  df-seq 13967  df-exp 14028  df-fac 14234  df-bc 14263  df-hash 14291  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-sum 15633  df-dvds 16198  df-gcd 16436  df-prm 16609  df-pc 16770  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-0g 17387  df-mre 17530  df-mrc 17531  df-acs 17533  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-submnd 18672  df-grp 18822  df-minusg 18823  df-sbg 18824  df-mulg 18951  df-subg 19003  df-eqg 19005  df-ga 19154  df-cntz 19181  df-od 19396  df-gex 19397  df-pgp 19398  df-lsm 19504  df-cmn 19650  df-abl 19651
This theorem is referenced by:  pgpfac1  19950
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