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Theorem pgpfac1lem5 19787
Description: Lemma for pgpfac1 19788. (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 9055 . . . . . . . . . 10 (𝐡 ∈ Fin ↔ 𝒫 𝐡 ∈ Fin)
31, 2sylib 217 . . . . . . . . 9 (πœ‘ β†’ 𝒫 𝐡 ∈ Fin)
43adantr 481 . . . . . . . 8 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ 𝒫 𝐡 ∈ Fin)
5 pgpfac1.b . . . . . . . . . . . 12 𝐡 = (Baseβ€˜πΊ)
65subgss 18861 . . . . . . . . . . 11 (𝑣 ∈ (SubGrpβ€˜πΊ) β†’ 𝑣 βŠ† 𝐡)
763ad2ant2 1134 . . . . . . . . . 10 (((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) ∧ 𝑣 ∈ (SubGrpβ€˜πΊ) ∧ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)) β†’ 𝑣 βŠ† 𝐡)
8 velpw 4563 . . . . . . . . . 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 4030 . . . . . . . 8 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} βŠ† 𝒫 𝐡)
114, 10ssfid 9144 . . . . . . 7 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ∈ Fin)
12 finnum 9817 . . . . . . 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 19496 . . . . . . . . . . . . 13 (𝐺 ∈ Abel β†’ 𝐺 ∈ Grp)
1715, 16syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐺 ∈ Grp)
185subgacs 18895 . . . . . . . . . . . 12 (𝐺 ∈ Grp β†’ (SubGrpβ€˜πΊ) ∈ (ACSβ€˜π΅))
19 acsmre 17466 . . . . . . . . . . . 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 18861 . . . . . . . . . . . . 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 3943 . . . . . . . . . . 11 (πœ‘ β†’ 𝐴 ∈ 𝐡)
26 pgpfac1.k . . . . . . . . . . . 12 𝐾 = (mrClsβ€˜(SubGrpβ€˜πΊ))
2726mrcsncl 17426 . . . . . . . . . . 11 (((SubGrpβ€˜πΊ) ∈ (Mooreβ€˜π΅) ∧ 𝐴 ∈ 𝐡) β†’ (πΎβ€˜{𝐴}) ∈ (SubGrpβ€˜πΊ))
2820, 25, 27syl2anc 584 . . . . . . . . . 10 (πœ‘ β†’ (πΎβ€˜{𝐴}) ∈ (SubGrpβ€˜πΊ))
2914, 28eqeltrid 2842 . . . . . . . . 9 (πœ‘ β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))
3029adantr 481 . . . . . . . 8 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))
31 simpr 485 . . . . . . . 8 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ 𝑆 ⊊ π‘ˆ)
3224snssd 4767 . . . . . . . . . . . . 13 (πœ‘ β†’ {𝐴} βŠ† π‘ˆ)
3332, 23sstrd 3952 . . . . . . . . . . . 12 (πœ‘ β†’ {𝐴} βŠ† 𝐡)
3420, 26, 33mrcssidd 17439 . . . . . . . . . . 11 (πœ‘ β†’ {𝐴} βŠ† (πΎβ€˜{𝐴}))
3534, 14sseqtrrdi 3993 . . . . . . . . . 10 (πœ‘ β†’ {𝐴} βŠ† 𝑆)
36 snssg 4742 . . . . . . . . . . 11 (𝐴 ∈ 𝐡 β†’ (𝐴 ∈ 𝑆 ↔ {𝐴} βŠ† 𝑆))
3725, 36syl 17 . . . . . . . . . 10 (πœ‘ β†’ (𝐴 ∈ 𝑆 ↔ {𝐴} βŠ† 𝑆))
3835, 37mpbird 256 . . . . . . . . 9 (πœ‘ β†’ 𝐴 ∈ 𝑆)
3938adantr 481 . . . . . . . 8 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ 𝐴 ∈ 𝑆)
40 psseq1 4045 . . . . . . . . . 10 (𝑣 = 𝑆 β†’ (𝑣 ⊊ π‘ˆ ↔ 𝑆 ⊊ π‘ˆ))
41 eleq2 2826 . . . . . . . . . 10 (𝑣 = 𝑆 β†’ (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑆))
4240, 41anbi12d 631 . . . . . . . . 9 (𝑣 = 𝑆 β†’ ((𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣) ↔ (𝑆 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑆)))
4342rspcev 3579 . . . . . . . 8 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ (𝑆 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑆)) β†’ βˆƒπ‘£ ∈ (SubGrpβ€˜πΊ)(𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣))
4430, 31, 39, 43syl12anc 835 . . . . . . 7 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ βˆƒπ‘£ ∈ (SubGrpβ€˜πΊ)(𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣))
45 rabn0 4343 . . . . . . 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 1194 . . . . . . . . 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 1195 . . . . . . . . . 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 481 . . . . . . . . . . 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 9144 . . . . . . . . . 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 1196 . . . . . . . . . 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 10278 . . . . . . . . . 10 ((𝑒 β‰  βˆ… ∧ 𝑒 ∈ Fin ∧ [⊊] Or 𝑒) β†’ βˆͺ 𝑒 ∈ 𝑒)
5348, 50, 51, 52syl3anc 1371 . . . . . . . . 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 3943 . . . . . . . 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 413 . . . . . . 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 1930 . . . . . 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 10374 . . . . . 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 1371 . . . . 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 4045 . . . . . . . 8 (𝑣 = 𝑀 β†’ (𝑣 ⊊ π‘ˆ ↔ 𝑀 ⊊ π‘ˆ))
60 eleq2 2826 . . . . . . . 8 (𝑣 = 𝑀 β†’ (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑀))
6159, 60anbi12d 631 . . . . . . 7 (𝑣 = 𝑀 β†’ ((𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣) ↔ (𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀)))
6261ralrab 3649 . . . . . 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 413 . . 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 4045 . . . . . . 7 (𝑣 = 𝑠 β†’ (𝑣 ⊊ π‘ˆ ↔ 𝑠 ⊊ π‘ˆ))
68 eleq2 2826 . . . . . . 7 (𝑣 = 𝑠 β†’ (𝐴 ∈ 𝑣 ↔ 𝐴 ∈ 𝑠))
6967, 68anbi12d 631 . . . . . 6 (𝑣 = 𝑠 β†’ ((𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣) ↔ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠)))
7069ralrab 3649 . . . . 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 3643 . . . . . . 7 (𝑠 ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ↔ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠)))
74 ineq2 4164 . . . . . . . . . . . 12 (𝑑 = 𝑣 β†’ (𝑆 ∩ 𝑑) = (𝑆 ∩ 𝑣))
7574eqeq1d 2739 . . . . . . . . . . 11 (𝑑 = 𝑣 β†’ ((𝑆 ∩ 𝑑) = { 0 } ↔ (𝑆 ∩ 𝑣) = { 0 }))
76 oveq2 7357 . . . . . . . . . . . 12 (𝑑 = 𝑣 β†’ (𝑆 βŠ• 𝑑) = (𝑆 βŠ• 𝑣))
7776eqeq1d 2739 . . . . . . . . . . 11 (𝑑 = 𝑣 β†’ ((𝑆 βŠ• 𝑑) = 𝑠 ↔ (𝑆 βŠ• 𝑣) = 𝑠))
7875, 77anbi12d 631 . . . . . . . . . 10 (𝑑 = 𝑣 β†’ (((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = 𝑠) ↔ ((𝑆 ∩ 𝑣) = { 0 } ∧ (𝑆 βŠ• 𝑣) = 𝑠)))
7978cbvrexvw 3224 . . . . . . . . 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 779 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠))) β†’ 𝑠 ⊊ π‘ˆ)
8180ad2antrr 724 . . . . . . . . . . . . . . 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 1195 . . . . . . . . . . . . . . . 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 4050 . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . 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 4324 . . . . . . . . . . . . . . 15 ((𝑆 βŠ• 𝑣) ⊊ π‘ˆ β†’ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)) β‰  βˆ…)
86 n0 4304 . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . . 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 4055 . . . . . . . . . . . . . . . 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 1220 . . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . . . 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 4045 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 βŠ• 𝑣) = 𝑠 β†’ ((𝑆 βŠ• 𝑣) ⊊ 𝑦 ↔ 𝑠 ⊊ 𝑦))
108107notbid 317 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆 βŠ• 𝑣) = 𝑠 β†’ (Β¬ (𝑆 βŠ• 𝑣) ⊊ 𝑦 ↔ Β¬ 𝑠 ⊊ 𝑦))
109108imbi2d 340 . . . . . . . . . . . . . . . . . . . 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 3172 . . . . . . . . . . . . . . . . . . 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 4045 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑀 β†’ (𝑦 ⊊ π‘ˆ ↔ 𝑀 ⊊ π‘ˆ))
112 eleq2 2826 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑀 β†’ (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑀))
113111, 112anbi12d 631 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑀 β†’ ((𝑦 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑦) ↔ (𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀)))
114 psseq2 4046 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑀 β†’ (𝑠 ⊊ 𝑦 ↔ 𝑠 ⊊ 𝑀))
115114notbid 317 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑀 β†’ (Β¬ 𝑠 ⊊ 𝑦 ↔ Β¬ 𝑠 ⊊ 𝑀))
116113, 115imbi12d 344 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑀 β†’ (((𝑦 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑦) β†’ Β¬ 𝑠 ⊊ 𝑦) ↔ ((𝑀 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑀) β†’ Β¬ 𝑠 ⊊ 𝑀)))
117116cbvralvw 3223 . . . . . . . . . . . . . . . . . . 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 286 . . . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . 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 19786 . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . 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 1936 . . . . . . . . . . . . 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 1354 . . . . . . . . . . 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 411 . . . . . . . . . 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 3150 . . . . . . . . 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 411 . . . . . . 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 594 . . . . . 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 3150 . . . . 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 693 . . 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 18885 . . . . . 6 (𝐺 ∈ Grp β†’ { 0 } ∈ (SubGrpβ€˜πΊ))
13817, 137syl 17 . . . . 5 (πœ‘ β†’ { 0 } ∈ (SubGrpβ€˜πΊ))
139138adantr 481 . . . 4 ((πœ‘ ∧ 𝑆 = π‘ˆ) β†’ { 0 } ∈ (SubGrpβ€˜πΊ))
14091subg0cl 18868 . . . . . . . 8 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ 0 ∈ 𝑆)
14129, 140syl 17 . . . . . . 7 (πœ‘ β†’ 0 ∈ 𝑆)
142141snssd 4767 . . . . . 6 (πœ‘ β†’ { 0 } βŠ† 𝑆)
143142adantr 481 . . . . 5 ((πœ‘ ∧ 𝑆 = π‘ˆ) β†’ { 0 } βŠ† 𝑆)
144 sseqin2 4173 . . . . 5 ({ 0 } βŠ† 𝑆 ↔ (𝑆 ∩ { 0 }) = { 0 })
145143, 144sylib 217 . . . 4 ((πœ‘ ∧ 𝑆 = π‘ˆ) β†’ (𝑆 ∩ { 0 }) = { 0 })
14692lsmss2 19378 . . . . . . 7 ((𝑆 ∈ (SubGrpβ€˜πΊ) ∧ { 0 } ∈ (SubGrpβ€˜πΊ) ∧ { 0 } βŠ† 𝑆) β†’ (𝑆 βŠ• { 0 }) = 𝑆)
14729, 138, 142, 146syl3anc 1371 . . . . . 6 (πœ‘ β†’ (𝑆 βŠ• { 0 }) = 𝑆)
148147eqeq1d 2739 . . . . 5 (πœ‘ β†’ ((𝑆 βŠ• { 0 }) = π‘ˆ ↔ 𝑆 = π‘ˆ))
149148biimpar 478 . . . 4 ((πœ‘ ∧ 𝑆 = π‘ˆ) β†’ (𝑆 βŠ• { 0 }) = π‘ˆ)
150 ineq2 4164 . . . . . . 7 (𝑑 = { 0 } β†’ (𝑆 ∩ 𝑑) = (𝑆 ∩ { 0 }))
151150eqeq1d 2739 . . . . . 6 (𝑑 = { 0 } β†’ ((𝑆 ∩ 𝑑) = { 0 } ↔ (𝑆 ∩ { 0 }) = { 0 }))
152 oveq2 7357 . . . . . . 7 (𝑑 = { 0 } β†’ (𝑆 βŠ• 𝑑) = (𝑆 βŠ• { 0 }))
153152eqeq1d 2739 . . . . . 6 (𝑑 = { 0 } β†’ ((𝑆 βŠ• 𝑑) = π‘ˆ ↔ (𝑆 βŠ• { 0 }) = π‘ˆ))
154151, 153anbi12d 631 . . . . 5 (𝑑 = { 0 } β†’ (((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ) ↔ ((𝑆 ∩ { 0 }) = { 0 } ∧ (𝑆 βŠ• { 0 }) = π‘ˆ)))
155154rspcev 3579 . . . 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 835 . . 3 ((πœ‘ ∧ 𝑆 = π‘ˆ) β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ))
157156ex 413 . 2 (πœ‘ β†’ (𝑆 = π‘ˆ β†’ βˆƒπ‘‘ ∈ (SubGrpβ€˜πΊ)((𝑆 ∩ 𝑑) = { 0 } ∧ (𝑆 βŠ• 𝑑) = π‘ˆ)))
15826mrcsscl 17434 . . . . 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 1371 . . . 4 (πœ‘ β†’ (πΎβ€˜{𝐴}) βŠ† π‘ˆ)
16014, 159eqsstrid 3990 . . 3 (πœ‘ β†’ 𝑆 βŠ† π‘ˆ)
161 sspss 4057 . . 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 858 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 396   ∨ wo 845   ∧ w3a 1087  βˆ€wal 1539   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  {crab 3405   βˆ– cdif 3905   ∩ cin 3907   βŠ† wss 3908   ⊊ wpss 3909  βˆ…c0 4280  π’« cpw 4558  {csn 4584  βˆͺ cuni 4863   class class class wbr 5103   Or wor 5541  dom cdm 5630  β€˜cfv 6491  (class class class)co 7349   [⊊] crpss 7649  Fincfn 8816  cardccrd 9804  Basecbs 17017  0gc0g 17255  Moorecmre 17396  mrClscmrc 17397  ACScacs 17399  Grpcgrp 18682  .gcmg 18805  SubGrpcsubg 18854  odcod 19238  gExcgex 19239   pGrp cpgp 19240  LSSumclsm 19345  Abelcabl 19492
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 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7662  ax-inf2 9510  ax-cnex 11040  ax-resscn 11041  ax-1cn 11042  ax-icn 11043  ax-addcl 11044  ax-addrcl 11045  ax-mulcl 11046  ax-mulrcl 11047  ax-mulcom 11048  ax-addass 11049  ax-mulass 11050  ax-distr 11051  ax-i2m1 11052  ax-1ne0 11053  ax-1rid 11054  ax-rnegex 11055  ax-rrecex 11056  ax-cnre 11057  ax-pre-lttri 11058  ax-pre-lttrn 11059  ax-pre-ltadd 11060  ax-pre-mulgt0 11061  ax-pre-sup 11062
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-iin 4955  df-disj 5069  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5528  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5585  df-se 5586  df-we 5587  df-xp 5636  df-rel 5637  df-cnv 5638  df-co 5639  df-dm 5640  df-rn 5641  df-res 5642  df-ima 5643  df-pred 6249  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6443  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-isom 6500  df-riota 7305  df-ov 7352  df-oprab 7353  df-mpo 7354  df-rpss 7650  df-om 7793  df-1st 7911  df-2nd 7912  df-frecs 8179  df-wrecs 8210  df-recs 8284  df-rdg 8323  df-1o 8379  df-2o 8380  df-oadd 8383  df-omul 8384  df-er 8581  df-ec 8583  df-qs 8587  df-map 8700  df-en 8817  df-dom 8818  df-sdom 8819  df-fin 8820  df-sup 9311  df-inf 9312  df-oi 9379  df-dju 9770  df-card 9808  df-acn 9811  df-pnf 11124  df-mnf 11125  df-xr 11126  df-ltxr 11127  df-le 11128  df-sub 11320  df-neg 11321  df-div 11746  df-nn 12087  df-2 12149  df-3 12150  df-n0 12347  df-xnn0 12419  df-z 12433  df-uz 12696  df-q 12802  df-rp 12844  df-fz 13353  df-fzo 13496  df-fl 13625  df-mod 13703  df-seq 13835  df-exp 13896  df-fac 14101  df-bc 14130  df-hash 14158  df-cj 14917  df-re 14918  df-im 14919  df-sqrt 15053  df-abs 15054  df-clim 15304  df-sum 15505  df-dvds 16071  df-gcd 16309  df-prm 16482  df-pc 16643  df-sets 16970  df-slot 16988  df-ndx 17000  df-base 17018  df-ress 17047  df-plusg 17080  df-0g 17257  df-mre 17400  df-mrc 17401  df-acs 17403  df-mgm 18431  df-sgrp 18480  df-mnd 18491  df-submnd 18536  df-grp 18685  df-minusg 18686  df-sbg 18687  df-mulg 18806  df-subg 18857  df-eqg 18859  df-ga 19002  df-cntz 19029  df-od 19242  df-gex 19243  df-pgp 19244  df-lsm 19347  df-cmn 19493  df-abl 19494
This theorem is referenced by:  pgpfac1  19788
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