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Theorem pgpssslw 19316
Description: Every 𝑃-subgroup is contained in a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
Hypotheses
Ref Expression
pgpssslw.1 𝑋 = (Base‘𝐺)
pgpssslw.2 𝑆 = (𝐺s 𝐻)
pgpssslw.3 𝐹 = (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} ↦ (♯‘𝑥))
Assertion
Ref Expression
pgpssslw ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑘 ∈ (𝑃 pSyl 𝐺)𝐻𝑘)
Distinct variable groups:   𝑥,𝑘,𝑦,𝐺   𝑘,𝐻,𝑥,𝑦   𝑃,𝑘,𝑥,𝑦   𝑘,𝑋,𝑥   𝑘,𝐹   𝑆,𝑘,𝑥,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)   𝑋(𝑦)

Proof of Theorem pgpssslw
Dummy variables 𝑚 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 1137 . . . . . . . . . 10 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝑋 ∈ Fin)
2 elrabi 3632 . . . . . . . . . . 11 (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} → 𝑥 ∈ (SubGrp‘𝐺))
3 pgpssslw.1 . . . . . . . . . . . 12 𝑋 = (Base‘𝐺)
43subgss 18853 . . . . . . . . . . 11 (𝑥 ∈ (SubGrp‘𝐺) → 𝑥𝑋)
52, 4syl 17 . . . . . . . . . 10 (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} → 𝑥𝑋)
6 ssfi 9043 . . . . . . . . . 10 ((𝑋 ∈ Fin ∧ 𝑥𝑋) → 𝑥 ∈ Fin)
71, 5, 6syl2an 597 . . . . . . . . 9 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → 𝑥 ∈ Fin)
8 hashcl 14176 . . . . . . . . 9 (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ0)
97, 8syl 17 . . . . . . . 8 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (♯‘𝑥) ∈ ℕ0)
109nn0zd 12530 . . . . . . 7 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (♯‘𝑥) ∈ ℤ)
11 pgpssslw.3 . . . . . . 7 𝐹 = (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} ↦ (♯‘𝑥))
1210, 11fmptd 7049 . . . . . 6 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐹:{𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}⟶ℤ)
1312frnd 6664 . . . . 5 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ⊆ ℤ)
14 fvex 6843 . . . . . . . 8 (♯‘𝑥) ∈ V
1514, 11fnmpti 6632 . . . . . . 7 𝐹 Fn {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}
16 eqimss2 3993 . . . . . . . . . 10 (𝑦 = 𝐻𝐻𝑦)
1716biantrud 533 . . . . . . . . 9 (𝑦 = 𝐻 → (𝑃 pGrp (𝐺s 𝑦) ↔ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)))
18 oveq2 7350 . . . . . . . . . . 11 (𝑦 = 𝐻 → (𝐺s 𝑦) = (𝐺s 𝐻))
19 pgpssslw.2 . . . . . . . . . . 11 𝑆 = (𝐺s 𝐻)
2018, 19eqtr4di 2795 . . . . . . . . . 10 (𝑦 = 𝐻 → (𝐺s 𝑦) = 𝑆)
2120breq2d 5109 . . . . . . . . 9 (𝑦 = 𝐻 → (𝑃 pGrp (𝐺s 𝑦) ↔ 𝑃 pGrp 𝑆))
2217, 21bitr3d 281 . . . . . . . 8 (𝑦 = 𝐻 → ((𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦) ↔ 𝑃 pGrp 𝑆))
23 simp1 1136 . . . . . . . 8 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐻 ∈ (SubGrp‘𝐺))
24 simp3 1138 . . . . . . . 8 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝑃 pGrp 𝑆)
2522, 23, 24elrabd 3640 . . . . . . 7 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐻 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)})
26 fnfvelrn 7019 . . . . . . 7 ((𝐹 Fn {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} ∧ 𝐻 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (𝐹𝐻) ∈ ran 𝐹)
2715, 25, 26sylancr 588 . . . . . 6 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (𝐹𝐻) ∈ ran 𝐹)
2827ne0d 4287 . . . . 5 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ≠ ∅)
29 hashcl 14176 . . . . . . . 8 (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ0)
301, 29syl 17 . . . . . . 7 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (♯‘𝑋) ∈ ℕ0)
3130nn0red 12400 . . . . . 6 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (♯‘𝑋) ∈ ℝ)
32 fveq2 6830 . . . . . . . . . . 11 (𝑥 = 𝑚 → (♯‘𝑥) = (♯‘𝑚))
33 fvex 6843 . . . . . . . . . . 11 (♯‘𝑚) ∈ V
3432, 11, 33fvmpt 6936 . . . . . . . . . 10 (𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} → (𝐹𝑚) = (♯‘𝑚))
3534adantl 483 . . . . . . . . 9 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (𝐹𝑚) = (♯‘𝑚))
36 oveq2 7350 . . . . . . . . . . . . 13 (𝑦 = 𝑚 → (𝐺s 𝑦) = (𝐺s 𝑚))
3736breq2d 5109 . . . . . . . . . . . 12 (𝑦 = 𝑚 → (𝑃 pGrp (𝐺s 𝑦) ↔ 𝑃 pGrp (𝐺s 𝑚)))
38 sseq2 3962 . . . . . . . . . . . 12 (𝑦 = 𝑚 → (𝐻𝑦𝐻𝑚))
3937, 38anbi12d 632 . . . . . . . . . . 11 (𝑦 = 𝑚 → ((𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦) ↔ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚)))
4039elrab 3638 . . . . . . . . . 10 (𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} ↔ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚)))
411adantr 482 . . . . . . . . . . . 12 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → 𝑋 ∈ Fin)
423subgss 18853 . . . . . . . . . . . . 13 (𝑚 ∈ (SubGrp‘𝐺) → 𝑚𝑋)
4342ad2antrl 726 . . . . . . . . . . . 12 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → 𝑚𝑋)
44 ssdomg 8866 . . . . . . . . . . . 12 (𝑋 ∈ Fin → (𝑚𝑋𝑚𝑋))
4541, 43, 44sylc 65 . . . . . . . . . . 11 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → 𝑚𝑋)
4641, 43ssfid 9137 . . . . . . . . . . . 12 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → 𝑚 ∈ Fin)
47 hashdom 14199 . . . . . . . . . . . 12 ((𝑚 ∈ Fin ∧ 𝑋 ∈ Fin) → ((♯‘𝑚) ≤ (♯‘𝑋) ↔ 𝑚𝑋))
4846, 41, 47syl2anc 585 . . . . . . . . . . 11 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → ((♯‘𝑚) ≤ (♯‘𝑋) ↔ 𝑚𝑋))
4945, 48mpbird 257 . . . . . . . . . 10 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → (♯‘𝑚) ≤ (♯‘𝑋))
5040, 49sylan2b 595 . . . . . . . . 9 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (♯‘𝑚) ≤ (♯‘𝑋))
5135, 50eqbrtrd 5119 . . . . . . . 8 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (𝐹𝑚) ≤ (♯‘𝑋))
5251ralrimiva 3140 . . . . . . 7 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∀𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} (𝐹𝑚) ≤ (♯‘𝑋))
53 breq1 5100 . . . . . . . . 9 (𝑤 = (𝐹𝑚) → (𝑤 ≤ (♯‘𝑋) ↔ (𝐹𝑚) ≤ (♯‘𝑋)))
5453ralrn 7025 . . . . . . . 8 (𝐹 Fn {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} → (∀𝑤 ∈ ran 𝐹 𝑤 ≤ (♯‘𝑋) ↔ ∀𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} (𝐹𝑚) ≤ (♯‘𝑋)))
5515, 54ax-mp 5 . . . . . . 7 (∀𝑤 ∈ ran 𝐹 𝑤 ≤ (♯‘𝑋) ↔ ∀𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} (𝐹𝑚) ≤ (♯‘𝑋))
5652, 55sylibr 233 . . . . . 6 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∀𝑤 ∈ ran 𝐹 𝑤 ≤ (♯‘𝑋))
57 brralrspcev 5157 . . . . . 6 (((♯‘𝑋) ∈ ℝ ∧ ∀𝑤 ∈ ran 𝐹 𝑤 ≤ (♯‘𝑋)) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤𝑧)
5831, 56, 57syl2anc 585 . . . . 5 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤𝑧)
59 suprzcl 12506 . . . . 5 ((ran 𝐹 ⊆ ℤ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤𝑧) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
6013, 28, 58, 59syl3anc 1371 . . . 4 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
61 fvelrnb 6891 . . . . 5 (𝐹 Fn {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ↔ ∃𝑘 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))
6215, 61ax-mp 5 . . . 4 (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ↔ ∃𝑘 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} (𝐹𝑘) = sup(ran 𝐹, ℝ, < ))
6360, 62sylib 217 . . 3 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑘 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} (𝐹𝑘) = sup(ran 𝐹, ℝ, < ))
64 oveq2 7350 . . . . . 6 (𝑦 = 𝑘 → (𝐺s 𝑦) = (𝐺s 𝑘))
6564breq2d 5109 . . . . 5 (𝑦 = 𝑘 → (𝑃 pGrp (𝐺s 𝑦) ↔ 𝑃 pGrp (𝐺s 𝑘)))
66 sseq2 3962 . . . . 5 (𝑦 = 𝑘 → (𝐻𝑦𝐻𝑘))
6765, 66anbi12d 632 . . . 4 (𝑦 = 𝑘 → ((𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦) ↔ (𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘)))
6867rexrab 3647 . . 3 (∃𝑘 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} (𝐹𝑘) = sup(ran 𝐹, ℝ, < ) ↔ ∃𝑘 ∈ (SubGrp‘𝐺)((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))
6963, 68sylib 217 . 2 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑘 ∈ (SubGrp‘𝐺)((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))
70 simpl3 1193 . . . 4 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) → 𝑃 pGrp 𝑆)
71 pgpprm 19295 . . . 4 (𝑃 pGrp 𝑆𝑃 ∈ ℙ)
7270, 71syl 17 . . 3 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) → 𝑃 ∈ ℙ)
73 simprl 769 . . 3 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) → 𝑘 ∈ (SubGrp‘𝐺))
74 zssre 12432 . . . . . . . . . . . . 13 ℤ ⊆ ℝ
7513, 74sstrdi 3948 . . . . . . . . . . . 12 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ⊆ ℝ)
7675ad2antrr 724 . . . . . . . . . . 11 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → ran 𝐹 ⊆ ℝ)
7728ad2antrr 724 . . . . . . . . . . 11 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → ran 𝐹 ≠ ∅)
7858ad2antrr 724 . . . . . . . . . . 11 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤𝑧)
79 simprl 769 . . . . . . . . . . . . . 14 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → 𝑚 ∈ (SubGrp‘𝐺))
80 simprrr 780 . . . . . . . . . . . . . . 15 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → 𝑃 pGrp (𝐺s 𝑚))
81 simprrl 779 . . . . . . . . . . . . . . . . . 18 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) → (𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘))
8281adantr 482 . . . . . . . . . . . . . . . . 17 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → (𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘))
8382simprd 497 . . . . . . . . . . . . . . . 16 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → 𝐻𝑘)
84 simprrl 779 . . . . . . . . . . . . . . . 16 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → 𝑘𝑚)
8583, 84sstrd 3946 . . . . . . . . . . . . . . 15 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → 𝐻𝑚)
8680, 85jca 513 . . . . . . . . . . . . . 14 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))
8739, 79, 86elrabd 3640 . . . . . . . . . . . . 13 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)})
8887, 34syl 17 . . . . . . . . . . . 12 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → (𝐹𝑚) = (♯‘𝑚))
89 fnfvelrn 7019 . . . . . . . . . . . . 13 ((𝐹 Fn {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (𝐹𝑚) ∈ ran 𝐹)
9015, 87, 89sylancr 588 . . . . . . . . . . . 12 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → (𝐹𝑚) ∈ ran 𝐹)
9188, 90eqeltrrd 2839 . . . . . . . . . . 11 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → (♯‘𝑚) ∈ ran 𝐹)
9276, 77, 78, 91suprubd 12043 . . . . . . . . . 10 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → (♯‘𝑚) ≤ sup(ran 𝐹, ℝ, < ))
93 simprrr 780 . . . . . . . . . . . 12 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) → (𝐹𝑘) = sup(ran 𝐹, ℝ, < ))
9493adantr 482 . . . . . . . . . . 11 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → (𝐹𝑘) = sup(ran 𝐹, ℝ, < ))
9573adantr 482 . . . . . . . . . . . . 13 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → 𝑘 ∈ (SubGrp‘𝐺))
9667, 95, 82elrabd 3640 . . . . . . . . . . . 12 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → 𝑘 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)})
97 fveq2 6830 . . . . . . . . . . . . 13 (𝑥 = 𝑘 → (♯‘𝑥) = (♯‘𝑘))
98 fvex 6843 . . . . . . . . . . . . 13 (♯‘𝑘) ∈ V
9997, 11, 98fvmpt 6936 . . . . . . . . . . . 12 (𝑘 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} → (𝐹𝑘) = (♯‘𝑘))
10096, 99syl 17 . . . . . . . . . . 11 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → (𝐹𝑘) = (♯‘𝑘))
10194, 100eqtr3d 2779 . . . . . . . . . 10 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → sup(ran 𝐹, ℝ, < ) = (♯‘𝑘))
10292, 101breqtrd 5123 . . . . . . . . 9 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → (♯‘𝑚) ≤ (♯‘𝑘))
103 simpll2 1213 . . . . . . . . . . 11 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → 𝑋 ∈ Fin)
10442ad2antrl 726 . . . . . . . . . . 11 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → 𝑚𝑋)
105103, 104ssfid 9137 . . . . . . . . . 10 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → 𝑚 ∈ Fin)
106105, 84ssfid 9137 . . . . . . . . . 10 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → 𝑘 ∈ Fin)
107 hashcl 14176 . . . . . . . . . . 11 (𝑚 ∈ Fin → (♯‘𝑚) ∈ ℕ0)
108 hashcl 14176 . . . . . . . . . . 11 (𝑘 ∈ Fin → (♯‘𝑘) ∈ ℕ0)
109 nn0re 12348 . . . . . . . . . . . 12 ((♯‘𝑚) ∈ ℕ0 → (♯‘𝑚) ∈ ℝ)
110 nn0re 12348 . . . . . . . . . . . 12 ((♯‘𝑘) ∈ ℕ0 → (♯‘𝑘) ∈ ℝ)
111 lenlt 11159 . . . . . . . . . . . 12 (((♯‘𝑚) ∈ ℝ ∧ (♯‘𝑘) ∈ ℝ) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
112109, 110, 111syl2an 597 . . . . . . . . . . 11 (((♯‘𝑚) ∈ ℕ0 ∧ (♯‘𝑘) ∈ ℕ0) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
113107, 108, 112syl2an 597 . . . . . . . . . 10 ((𝑚 ∈ Fin ∧ 𝑘 ∈ Fin) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
114105, 106, 113syl2anc 585 . . . . . . . . 9 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
115102, 114mpbid 231 . . . . . . . 8 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → ¬ (♯‘𝑘) < (♯‘𝑚))
116 php3 9082 . . . . . . . . . . 11 ((𝑚 ∈ Fin ∧ 𝑘𝑚) → 𝑘𝑚)
117116ex 414 . . . . . . . . . 10 (𝑚 ∈ Fin → (𝑘𝑚𝑘𝑚))
118105, 117syl 17 . . . . . . . . 9 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → (𝑘𝑚𝑘𝑚))
119 hashsdom 14201 . . . . . . . . . 10 ((𝑘 ∈ Fin ∧ 𝑚 ∈ Fin) → ((♯‘𝑘) < (♯‘𝑚) ↔ 𝑘𝑚))
120106, 105, 119syl2anc 585 . . . . . . . . 9 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → ((♯‘𝑘) < (♯‘𝑚) ↔ 𝑘𝑚))
121118, 120sylibrd 259 . . . . . . . 8 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → (𝑘𝑚 → (♯‘𝑘) < (♯‘𝑚)))
122115, 121mtod 197 . . . . . . 7 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → ¬ 𝑘𝑚)
123 sspss 4051 . . . . . . . . 9 (𝑘𝑚 ↔ (𝑘𝑚𝑘 = 𝑚))
12484, 123sylib 217 . . . . . . . 8 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → (𝑘𝑚𝑘 = 𝑚))
125124ord 862 . . . . . . 7 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → (¬ 𝑘𝑚𝑘 = 𝑚))
126122, 125mpd 15 . . . . . 6 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚)))) → 𝑘 = 𝑚)
127126expr 458 . . . . 5 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ 𝑚 ∈ (SubGrp‘𝐺)) → ((𝑘𝑚𝑃 pGrp (𝐺s 𝑚)) → 𝑘 = 𝑚))
12881simpld 496 . . . . . . 7 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) → 𝑃 pGrp (𝐺s 𝑘))
129128adantr 482 . . . . . 6 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ 𝑚 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺s 𝑘))
130 oveq2 7350 . . . . . . . 8 (𝑘 = 𝑚 → (𝐺s 𝑘) = (𝐺s 𝑚))
131130breq2d 5109 . . . . . . 7 (𝑘 = 𝑚 → (𝑃 pGrp (𝐺s 𝑘) ↔ 𝑃 pGrp (𝐺s 𝑚)))
132 eqimss 3992 . . . . . . . 8 (𝑘 = 𝑚𝑘𝑚)
133132biantrurd 534 . . . . . . 7 (𝑘 = 𝑚 → (𝑃 pGrp (𝐺s 𝑚) ↔ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚))))
134131, 133bitrd 279 . . . . . 6 (𝑘 = 𝑚 → (𝑃 pGrp (𝐺s 𝑘) ↔ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚))))
135129, 134syl5ibcom 245 . . . . 5 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ 𝑚 ∈ (SubGrp‘𝐺)) → (𝑘 = 𝑚 → (𝑘𝑚𝑃 pGrp (𝐺s 𝑚))))
136127, 135impbid 211 . . . 4 ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ 𝑚 ∈ (SubGrp‘𝐺)) → ((𝑘𝑚𝑃 pGrp (𝐺s 𝑚)) ↔ 𝑘 = 𝑚))
137136ralrimiva 3140 . . 3 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) → ∀𝑚 ∈ (SubGrp‘𝐺)((𝑘𝑚𝑃 pGrp (𝐺s 𝑚)) ↔ 𝑘 = 𝑚))
138 isslw 19310 . . 3 (𝑘 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝑘 ∈ (SubGrp‘𝐺) ∧ ∀𝑚 ∈ (SubGrp‘𝐺)((𝑘𝑚𝑃 pGrp (𝐺s 𝑚)) ↔ 𝑘 = 𝑚)))
13972, 73, 137, 138syl3anbrc 1343 . 2 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) → 𝑘 ∈ (𝑃 pSyl 𝐺))
14081simprd 497 . 2 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) → 𝐻𝑘)
14169, 139, 140reximssdv 3166 1 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑘 ∈ (𝑃 pSyl 𝐺)𝐻𝑘)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2941  wral 3062  wrex 3071  {crab 3404  wss 3902  wpss 3903  c0 4274   class class class wbr 5097  cmpt 5180  ran crn 5626   Fn wfn 6479  cfv 6484  (class class class)co 7342  cdom 8807  csdm 8808  Fincfn 8809  supcsup 9302  cr 10976   < clt 11115  cle 11116  0cn0 12339  cz 12425  chash 14150  cprime 16474  Basecbs 17010  s cress 17039  SubGrpcsubg 18846   pGrp cpgp 19231   pSyl cslw 19232
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-sep 5248  ax-nul 5255  ax-pow 5313  ax-pr 5377  ax-un 7655  ax-cnex 11033  ax-resscn 11034  ax-1cn 11035  ax-icn 11036  ax-addcl 11037  ax-addrcl 11038  ax-mulcl 11039  ax-mulrcl 11040  ax-mulcom 11041  ax-addass 11042  ax-mulass 11043  ax-distr 11044  ax-i2m1 11045  ax-1ne0 11046  ax-1rid 11047  ax-rnegex 11048  ax-rrecex 11049  ax-cnre 11050  ax-pre-lttri 11051  ax-pre-lttrn 11052  ax-pre-ltadd 11053  ax-pre-mulgt0 11054  ax-pre-sup 11055
This theorem depends on definitions:  df-bi 206  df-an 398  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 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3921  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-int 4900  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5181  df-tr 5215  df-id 5523  df-eprel 5529  df-po 5537  df-so 5538  df-fr 5580  df-we 5582  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6243  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-riota 7298  df-ov 7345  df-oprab 7346  df-mpo 7347  df-om 7786  df-1st 7904  df-2nd 7905  df-frecs 8172  df-wrecs 8203  df-recs 8277  df-rdg 8316  df-1o 8372  df-oadd 8376  df-er 8574  df-en 8810  df-dom 8811  df-sdom 8812  df-fin 8813  df-sup 9304  df-card 9801  df-pnf 11117  df-mnf 11118  df-xr 11119  df-ltxr 11120  df-le 11121  df-sub 11313  df-neg 11314  df-nn 12080  df-n0 12340  df-xnn0 12412  df-z 12426  df-uz 12689  df-fz 13346  df-hash 14151  df-subg 18849  df-pgp 19235  df-slw 19236
This theorem is referenced by:  slwn0  19317
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