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Theorem pgpssslw 18734
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 1134 . . . . . . . . . 10 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝑋 ∈ Fin)
2 elrabi 3626 . . . . . . . . . . 11 (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} → 𝑥 ∈ (SubGrp‘𝐺))
3 pgpssslw.1 . . . . . . . . . . . 12 𝑋 = (Base‘𝐺)
43subgss 18275 . . . . . . . . . . 11 (𝑥 ∈ (SubGrp‘𝐺) → 𝑥𝑋)
52, 4syl 17 . . . . . . . . . 10 (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} → 𝑥𝑋)
6 ssfi 8726 . . . . . . . . . 10 ((𝑋 ∈ Fin ∧ 𝑥𝑋) → 𝑥 ∈ Fin)
71, 5, 6syl2an 598 . . . . . . . . 9 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → 𝑥 ∈ Fin)
8 hashcl 13717 . . . . . . . . 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 12077 . . . . . . 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 6859 . . . . . 6 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐹:{𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}⟶ℤ)
1312frnd 6498 . . . . 5 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ⊆ ℤ)
14 fvex 6662 . . . . . . . 8 (♯‘𝑥) ∈ V
1514, 11fnmpti 6467 . . . . . . 7 𝐹 Fn {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}
16 eqimss2 3975 . . . . . . . . . 10 (𝑦 = 𝐻𝐻𝑦)
1716biantrud 535 . . . . . . . . 9 (𝑦 = 𝐻 → (𝑃 pGrp (𝐺s 𝑦) ↔ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)))
18 oveq2 7147 . . . . . . . . . . 11 (𝑦 = 𝐻 → (𝐺s 𝑦) = (𝐺s 𝐻))
19 pgpssslw.2 . . . . . . . . . . 11 𝑆 = (𝐺s 𝐻)
2018, 19eqtr4di 2854 . . . . . . . . . 10 (𝑦 = 𝐻 → (𝐺s 𝑦) = 𝑆)
2120breq2d 5045 . . . . . . . . 9 (𝑦 = 𝐻 → (𝑃 pGrp (𝐺s 𝑦) ↔ 𝑃 pGrp 𝑆))
2217, 21bitr3d 284 . . . . . . . 8 (𝑦 = 𝐻 → ((𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦) ↔ 𝑃 pGrp 𝑆))
23 simp1 1133 . . . . . . . 8 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐻 ∈ (SubGrp‘𝐺))
24 simp3 1135 . . . . . . . 8 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝑃 pGrp 𝑆)
2522, 23, 24elrabd 3633 . . . . . . 7 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐻 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)})
26 fnfvelrn 6829 . . . . . . 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 590 . . . . . 6 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (𝐹𝐻) ∈ ran 𝐹)
2827ne0d 4254 . . . . 5 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ≠ ∅)
29 hashcl 13717 . . . . . . . 8 (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ0)
301, 29syl 17 . . . . . . 7 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (♯‘𝑋) ∈ ℕ0)
3130nn0red 11948 . . . . . 6 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (♯‘𝑋) ∈ ℝ)
32 fveq2 6649 . . . . . . . . . . 11 (𝑥 = 𝑚 → (♯‘𝑥) = (♯‘𝑚))
33 fvex 6662 . . . . . . . . . . 11 (♯‘𝑚) ∈ V
3432, 11, 33fvmpt 6749 . . . . . . . . . 10 (𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} → (𝐹𝑚) = (♯‘𝑚))
3534adantl 485 . . . . . . . . 9 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (𝐹𝑚) = (♯‘𝑚))
36 oveq2 7147 . . . . . . . . . . . . 13 (𝑦 = 𝑚 → (𝐺s 𝑦) = (𝐺s 𝑚))
3736breq2d 5045 . . . . . . . . . . . 12 (𝑦 = 𝑚 → (𝑃 pGrp (𝐺s 𝑦) ↔ 𝑃 pGrp (𝐺s 𝑚)))
38 sseq2 3944 . . . . . . . . . . . 12 (𝑦 = 𝑚 → (𝐻𝑦𝐻𝑚))
3937, 38anbi12d 633 . . . . . . . . . . 11 (𝑦 = 𝑚 → ((𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦) ↔ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚)))
4039elrab 3631 . . . . . . . . . 10 (𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} ↔ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚)))
411adantr 484 . . . . . . . . . . . 12 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → 𝑋 ∈ Fin)
423subgss 18275 . . . . . . . . . . . . 13 (𝑚 ∈ (SubGrp‘𝐺) → 𝑚𝑋)
4342ad2antrl 727 . . . . . . . . . . . 12 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → 𝑚𝑋)
44 ssdomg 8542 . . . . . . . . . . . 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 8729 . . . . . . . . . . . 12 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → 𝑚 ∈ Fin)
47 hashdom 13740 . . . . . . . . . . . 12 ((𝑚 ∈ Fin ∧ 𝑋 ∈ Fin) → ((♯‘𝑚) ≤ (♯‘𝑋) ↔ 𝑚𝑋))
4846, 41, 47syl2anc 587 . . . . . . . . . . 11 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → ((♯‘𝑚) ≤ (♯‘𝑋) ↔ 𝑚𝑋))
4945, 48mpbird 260 . . . . . . . . . 10 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → (♯‘𝑚) ≤ (♯‘𝑋))
5040, 49sylan2b 596 . . . . . . . . 9 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (♯‘𝑚) ≤ (♯‘𝑋))
5135, 50eqbrtrd 5055 . . . . . . . 8 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (𝐹𝑚) ≤ (♯‘𝑋))
5251ralrimiva 3152 . . . . . . 7 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∀𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} (𝐹𝑚) ≤ (♯‘𝑋))
53 breq1 5036 . . . . . . . . 9 (𝑤 = (𝐹𝑚) → (𝑤 ≤ (♯‘𝑋) ↔ (𝐹𝑚) ≤ (♯‘𝑋)))
5453ralrn 6835 . . . . . . . 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 237 . . . . . 6 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∀𝑤 ∈ ran 𝐹 𝑤 ≤ (♯‘𝑋))
57 brralrspcev 5093 . . . . . 6 (((♯‘𝑋) ∈ ℝ ∧ ∀𝑤 ∈ ran 𝐹 𝑤 ≤ (♯‘𝑋)) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤𝑧)
5831, 56, 57syl2anc 587 . . . . 5 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤𝑧)
59 suprzcl 12054 . . . . 5 ((ran 𝐹 ⊆ ℤ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤𝑧) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
6013, 28, 58, 59syl3anc 1368 . . . 4 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
61 fvelrnb 6705 . . . . 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 221 . . 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 7147 . . . . . 6 (𝑦 = 𝑘 → (𝐺s 𝑦) = (𝐺s 𝑘))
6564breq2d 5045 . . . . 5 (𝑦 = 𝑘 → (𝑃 pGrp (𝐺s 𝑦) ↔ 𝑃 pGrp (𝐺s 𝑘)))
66 sseq2 3944 . . . . 5 (𝑦 = 𝑘 → (𝐻𝑦𝐻𝑘))
6765, 66anbi12d 633 . . . 4 (𝑦 = 𝑘 → ((𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦) ↔ (𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘)))
6867rexrab 3638 . . 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 221 . 2 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑘 ∈ (SubGrp‘𝐺)((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))
70 simpl3 1190 . . . 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 18713 . . . 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 770 . . 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 11980 . . . . . . . . . . . . 13 ℤ ⊆ ℝ
7513, 74sstrdi 3930 . . . . . . . . . . . 12 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ⊆ ℝ)
7675ad2antrr 725 . . . . . . . . . . 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 725 . . . . . . . . . . 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 725 . . . . . . . . . . 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 770 . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . 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 499 . . . . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . . . 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 3928 . . . . . . . . . . . . . . 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 515 . . . . . . . . . . . . . 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 3633 . . . . . . . . . . . . 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 6829 . . . . . . . . . . . . 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 590 . . . . . . . . . . . 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 2894 . . . . . . . . . . 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 11594 . . . . . . . . . 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 781 . . . . . . . . . . . 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 484 . . . . . . . . . . 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 484 . . . . . . . . . . . . 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 3633 . . . . . . . . . . . 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 6649 . . . . . . . . . . . . 13 (𝑥 = 𝑘 → (♯‘𝑥) = (♯‘𝑘))
98 fvex 6662 . . . . . . . . . . . . 13 (♯‘𝑘) ∈ V
9997, 11, 98fvmpt 6749 . . . . . . . . . . . 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 2838 . . . . . . . . . 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 5059 . . . . . . . . 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 1210 . . . . . . . . . . 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 727 . . . . . . . . . . 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 8729 . . . . . . . . . 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 8729 . . . . . . . . . 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 13717 . . . . . . . . . . 11 (𝑚 ∈ Fin → (♯‘𝑚) ∈ ℕ0)
108 hashcl 13717 . . . . . . . . . . 11 (𝑘 ∈ Fin → (♯‘𝑘) ∈ ℕ0)
109 nn0re 11898 . . . . . . . . . . . 12 ((♯‘𝑚) ∈ ℕ0 → (♯‘𝑚) ∈ ℝ)
110 nn0re 11898 . . . . . . . . . . . 12 ((♯‘𝑘) ∈ ℕ0 → (♯‘𝑘) ∈ ℝ)
111 lenlt 10712 . . . . . . . . . . . 12 (((♯‘𝑚) ∈ ℝ ∧ (♯‘𝑘) ∈ ℝ) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
112109, 110, 111syl2an 598 . . . . . . . . . . 11 (((♯‘𝑚) ∈ ℕ0 ∧ (♯‘𝑘) ∈ ℕ0) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
113107, 108, 112syl2an 598 . . . . . . . . . 10 ((𝑚 ∈ Fin ∧ 𝑘 ∈ Fin) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
114105, 106, 113syl2anc 587 . . . . . . . . 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 235 . . . . . . . 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 8691 . . . . . . . . . . 11 ((𝑚 ∈ Fin ∧ 𝑘𝑚) → 𝑘𝑚)
117116ex 416 . . . . . . . . . 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 13742 . . . . . . . . . 10 ((𝑘 ∈ Fin ∧ 𝑚 ∈ Fin) → ((♯‘𝑘) < (♯‘𝑚) ↔ 𝑘𝑚))
120106, 105, 119syl2anc 587 . . . . . . . . 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 262 . . . . . . . 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 201 . . . . . . 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 4030 . . . . . . . . 9 (𝑘𝑚 ↔ (𝑘𝑚𝑘 = 𝑚))
12484, 123sylib 221 . . . . . . . 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 861 . . . . . . 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 460 . . . . 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 498 . . . . . . 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 484 . . . . . 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 7147 . . . . . . . 8 (𝑘 = 𝑚 → (𝐺s 𝑘) = (𝐺s 𝑚))
131130breq2d 5045 . . . . . . 7 (𝑘 = 𝑚 → (𝑃 pGrp (𝐺s 𝑘) ↔ 𝑃 pGrp (𝐺s 𝑚)))
132 eqimss 3974 . . . . . . . 8 (𝑘 = 𝑚𝑘𝑚)
133132biantrurd 536 . . . . . . 7 (𝑘 = 𝑚 → (𝑃 pGrp (𝐺s 𝑚) ↔ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚))))
134131, 133bitrd 282 . . . . . 6 (𝑘 = 𝑚 → (𝑃 pGrp (𝐺s 𝑘) ↔ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚))))
135129, 134syl5ibcom 248 . . . . 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 215 . . . 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 3152 . . 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 18728 . . 3 (𝑘 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝑘 ∈ (SubGrp‘𝐺) ∧ ∀𝑚 ∈ (SubGrp‘𝐺)((𝑘𝑚𝑃 pGrp (𝐺s 𝑚)) ↔ 𝑘 = 𝑚)))
13972, 73, 137, 138syl3anbrc 1340 . 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 499 . 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 3238 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 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2112  wne 2990  wral 3109  wrex 3110  {crab 3113  wss 3884  wpss 3885  c0 4246   class class class wbr 5033  cmpt 5113  ran crn 5524   Fn wfn 6323  cfv 6328  (class class class)co 7139  cdom 8494  csdm 8495  Fincfn 8496  supcsup 8892  cr 10529   < clt 10668  cle 10669  0cn0 11889  cz 11973  chash 13690  cprime 16008  Basecbs 16478  s cress 16479  SubGrpcsubg 18268   pGrp cpgp 18649   pSyl cslw 18650
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-fz 12890  df-hash 13691  df-subg 18271  df-pgp 18653  df-slw 18654
This theorem is referenced by:  slwn0  18735
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