MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pgpfac1lem5 Structured version   Visualization version   GIF version

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 9056 . . . . . . . . . 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 18862 . . . . . . . . . . 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 4564 . . . . . . . . . 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 4031 . . . . . . . 8 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} βŠ† 𝒫 𝐡)
114, 10ssfid 9145 . . . . . . 7 ((πœ‘ ∧ 𝑆 ⊊ π‘ˆ) β†’ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ∈ Fin)
12 finnum 9818 . . . . . . 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 17467 . . . . . . . . . . . 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 18862 . . . . . . . . . . . . 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 3944 . . . . . . . . . . 11 (πœ‘ β†’ 𝐴 ∈ 𝐡)
26 pgpfac1.k . . . . . . . . . . . 12 𝐾 = (mrClsβ€˜(SubGrpβ€˜πΊ))
2726mrcsncl 17427 . . . . . . . . . . 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 2843 . . . . . . . . 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 4768 . . . . . . . . . . . . 13 (πœ‘ β†’ {𝐴} βŠ† π‘ˆ)
3332, 23sstrd 3953 . . . . . . . . . . . 12 (πœ‘ β†’ {𝐴} βŠ† 𝐡)
3420, 26, 33mrcssidd 17440 . . . . . . . . . . 11 (πœ‘ β†’ {𝐴} βŠ† (πΎβ€˜{𝐴}))
3534, 14sseqtrrdi 3994 . . . . . . . . . 10 (πœ‘ β†’ {𝐴} βŠ† 𝑆)
36 snssg 4743 . . . . . . . . . . 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 4046 . . . . . . . . . 10 (𝑣 = 𝑆 β†’ (𝑣 ⊊ π‘ˆ ↔ 𝑆 ⊊ π‘ˆ))
41 eleq2 2827 . . . . . . . . . 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 3580 . . . . . . . 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 4344 . . . . . . 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 9145 . . . . . . . . . 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 10279 . . . . . . . . . 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 3944 . . . . . . . 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 10375 . . . . . 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 4046 . . . . . . . 8 (𝑣 = 𝑀 β†’ (𝑣 ⊊ π‘ˆ ↔ 𝑀 ⊊ π‘ˆ))
60 eleq2 2827 . . . . . . . 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 3650 . . . . . 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 3096 . . . . 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 4046 . . . . . . 7 (𝑣 = 𝑠 β†’ (𝑣 ⊊ π‘ˆ ↔ 𝑠 ⊊ π‘ˆ))
68 eleq2 2827 . . . . . . 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 3650 . . . . 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 3116 . . . . 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 3644 . . . . . . 7 (𝑠 ∈ {𝑣 ∈ (SubGrpβ€˜πΊ) ∣ (𝑣 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑣)} ↔ (𝑠 ∈ (SubGrpβ€˜πΊ) ∧ (𝑠 ⊊ π‘ˆ ∧ 𝐴 ∈ 𝑠)))
74 ineq2 4165 . . . . . . . . . . . 12 (𝑑 = 𝑣 β†’ (𝑆 ∩ 𝑑) = (𝑆 ∩ 𝑣))
7574eqeq1d 2740 . . . . . . . . . . 11 (𝑑 = 𝑣 β†’ ((𝑆 ∩ 𝑑) = { 0 } ↔ (𝑆 ∩ 𝑣) = { 0 }))
76 oveq2 7358 . . . . . . . . . . . 12 (𝑑 = 𝑣 β†’ (𝑆 βŠ• 𝑑) = (𝑆 βŠ• 𝑣))
7776eqeq1d 2740 . . . . . . . . . . 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 3225 . . . . . . . . 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 4051 . . . . . . . . . . . . . . 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 4325 . . . . . . . . . . . . . . 15 ((𝑆 βŠ• 𝑣) ⊊ π‘ˆ β†’ (π‘ˆ βˆ– (𝑆 βŠ• 𝑣)) β‰  βˆ…)
86 n0 4305 . . . . . . . . . . . . . . 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 4056 . . . . . . . . . . . . . . . 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 4046 . . . . . . . . . . . . . . . . . . . . . 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 3173 . . . . . . . . . . . . . . . . . . 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 4046 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑀 β†’ (𝑦 ⊊ π‘ˆ ↔ 𝑀 ⊊ π‘ˆ))
112 eleq2 2827 . . . . . . . . . . . . . . . . . . . . . 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 4047 . . . . . . . . . . . . . . . . . . . . . 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 3224 . . . . . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . . 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 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 3151 . . . . . . . . 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 3151 . . . . 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 18886 . . . . . 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 18869 . . . . . . . 8 (𝑆 ∈ (SubGrpβ€˜πΊ) β†’ 0 ∈ 𝑆)
14129, 140syl 17 . . . . . . 7 (πœ‘ β†’ 0 ∈ 𝑆)
142141snssd 4768 . . . . . 6 (πœ‘ β†’ { 0 } βŠ† 𝑆)
143142adantr 482 . . . . 5 ((πœ‘ ∧ 𝑆 = π‘ˆ) β†’ { 0 } βŠ† 𝑆)
144 sseqin2 4174 . . . . 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 1372 . . . . . 6 (πœ‘ β†’ (𝑆 βŠ• { 0 }) = 𝑆)
148147eqeq1d 2740 . . . . 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 4165 . . . . . . 7 (𝑑 = { 0 } β†’ (𝑆 ∩ 𝑑) = (𝑆 ∩ { 0 }))
151150eqeq1d 2740 . . . . . 6 (𝑑 = { 0 } β†’ ((𝑆 ∩ 𝑑) = { 0 } ↔ (𝑆 ∩ { 0 }) = { 0 }))
152 oveq2 7358 . . . . . . 7 (𝑑 = { 0 } β†’ (𝑆 βŠ• 𝑑) = (𝑆 βŠ• { 0 }))
153152eqeq1d 2740 . . . . . 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 3580 . . . 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 17435 . . . . 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 3991 . . 3 (πœ‘ β†’ 𝑆 βŠ† π‘ˆ)
161 sspss 4058 . . 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 2942  βˆ€wral 3063  βˆƒwrex 3072  {crab 3406   βˆ– cdif 3906   ∩ cin 3908   βŠ† wss 3909   ⊊ wpss 3910  βˆ…c0 4281  π’« cpw 4559  {csn 4585  βˆͺ cuni 4864   class class class wbr 5104   Or wor 5542  dom cdm 5631  β€˜cfv 6492  (class class class)co 7350   [⊊] crpss 7650  Fincfn 8817  cardccrd 9805  Basecbs 17018  0gc0g 17256  Moorecmre 17397  mrClscmrc 17398  ACScacs 17400  Grpcgrp 18683  .gcmg 18806  SubGrpcsubg 18855  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 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-rpss 7651  df-om 7794  df-1st 7912  df-2nd 7913  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-en 8818  df-dom 8819  df-sdom 8820  df-fin 8821  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-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-mre 17401  df-mrc 17402  df-acs 17404  df-mgm 18432  df-sgrp 18481  df-mnd 18492  df-submnd 18537  df-grp 18686  df-minusg 18687  df-sbg 18688  df-mulg 18807  df-subg 18858  df-eqg 18860  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
  Copyright terms: Public domain W3C validator