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Theorem pgpssslw 19396
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 3639 . . . . . . . . . . 11 (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} → 𝑥 ∈ (SubGrp‘𝐺))
3 pgpssslw.1 . . . . . . . . . . . 12 𝑋 = (Base‘𝐺)
43subgss 18929 . . . . . . . . . . 11 (𝑥 ∈ (SubGrp‘𝐺) → 𝑥𝑋)
52, 4syl 17 . . . . . . . . . 10 (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} → 𝑥𝑋)
6 ssfi 9117 . . . . . . . . . 10 ((𝑋 ∈ Fin ∧ 𝑥𝑋) → 𝑥 ∈ Fin)
71, 5, 6syl2an 596 . . . . . . . . 9 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → 𝑥 ∈ Fin)
8 hashcl 14256 . . . . . . . . 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 12525 . . . . . . 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 7062 . . . . . 6 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐹:{𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}⟶ℤ)
1312frnd 6676 . . . . 5 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ⊆ ℤ)
14 fvex 6855 . . . . . . . 8 (♯‘𝑥) ∈ V
1514, 11fnmpti 6644 . . . . . . 7 𝐹 Fn {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}
16 eqimss2 4001 . . . . . . . . . 10 (𝑦 = 𝐻𝐻𝑦)
1716biantrud 532 . . . . . . . . 9 (𝑦 = 𝐻 → (𝑃 pGrp (𝐺s 𝑦) ↔ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)))
18 oveq2 7365 . . . . . . . . . . 11 (𝑦 = 𝐻 → (𝐺s 𝑦) = (𝐺s 𝐻))
19 pgpssslw.2 . . . . . . . . . . 11 𝑆 = (𝐺s 𝐻)
2018, 19eqtr4di 2794 . . . . . . . . . 10 (𝑦 = 𝐻 → (𝐺s 𝑦) = 𝑆)
2120breq2d 5117 . . . . . . . . 9 (𝑦 = 𝐻 → (𝑃 pGrp (𝐺s 𝑦) ↔ 𝑃 pGrp 𝑆))
2217, 21bitr3d 280 . . . . . . . 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 3647 . . . . . . 7 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐻 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)})
26 fnfvelrn 7031 . . . . . . 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 587 . . . . . 6 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (𝐹𝐻) ∈ ran 𝐹)
2827ne0d 4295 . . . . 5 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ≠ ∅)
29 hashcl 14256 . . . . . . . 8 (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ0)
301, 29syl 17 . . . . . . 7 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (♯‘𝑋) ∈ ℕ0)
3130nn0red 12474 . . . . . 6 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (♯‘𝑋) ∈ ℝ)
32 fveq2 6842 . . . . . . . . . . 11 (𝑥 = 𝑚 → (♯‘𝑥) = (♯‘𝑚))
33 fvex 6855 . . . . . . . . . . 11 (♯‘𝑚) ∈ V
3432, 11, 33fvmpt 6948 . . . . . . . . . 10 (𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} → (𝐹𝑚) = (♯‘𝑚))
3534adantl 482 . . . . . . . . 9 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (𝐹𝑚) = (♯‘𝑚))
36 oveq2 7365 . . . . . . . . . . . . 13 (𝑦 = 𝑚 → (𝐺s 𝑦) = (𝐺s 𝑚))
3736breq2d 5117 . . . . . . . . . . . 12 (𝑦 = 𝑚 → (𝑃 pGrp (𝐺s 𝑦) ↔ 𝑃 pGrp (𝐺s 𝑚)))
38 sseq2 3970 . . . . . . . . . . . 12 (𝑦 = 𝑚 → (𝐻𝑦𝐻𝑚))
3937, 38anbi12d 631 . . . . . . . . . . 11 (𝑦 = 𝑚 → ((𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦) ↔ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚)))
4039elrab 3645 . . . . . . . . . 10 (𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} ↔ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚)))
411adantr 481 . . . . . . . . . . . 12 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → 𝑋 ∈ Fin)
423subgss 18929 . . . . . . . . . . . . 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 8940 . . . . . . . . . . . 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 9211 . . . . . . . . . . . 12 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → 𝑚 ∈ Fin)
47 hashdom 14279 . . . . . . . . . . . 12 ((𝑚 ∈ Fin ∧ 𝑋 ∈ Fin) → ((♯‘𝑚) ≤ (♯‘𝑋) ↔ 𝑚𝑋))
4846, 41, 47syl2anc 584 . . . . . . . . . . 11 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → ((♯‘𝑚) ≤ (♯‘𝑋) ↔ 𝑚𝑋))
4945, 48mpbird 256 . . . . . . . . . 10 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → (♯‘𝑚) ≤ (♯‘𝑋))
5040, 49sylan2b 594 . . . . . . . . 9 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (♯‘𝑚) ≤ (♯‘𝑋))
5135, 50eqbrtrd 5127 . . . . . . . 8 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (𝐹𝑚) ≤ (♯‘𝑋))
5251ralrimiva 3143 . . . . . . 7 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∀𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} (𝐹𝑚) ≤ (♯‘𝑋))
53 breq1 5108 . . . . . . . . 9 (𝑤 = (𝐹𝑚) → (𝑤 ≤ (♯‘𝑋) ↔ (𝐹𝑚) ≤ (♯‘𝑋)))
5453ralrn 7038 . . . . . . . 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 5165 . . . . . 6 (((♯‘𝑋) ∈ ℝ ∧ ∀𝑤 ∈ ran 𝐹 𝑤 ≤ (♯‘𝑋)) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤𝑧)
5831, 56, 57syl2anc 584 . . . . 5 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤𝑧)
59 suprzcl 12583 . . . . 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 6903 . . . . 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 7365 . . . . . 6 (𝑦 = 𝑘 → (𝐺s 𝑦) = (𝐺s 𝑘))
6564breq2d 5117 . . . . 5 (𝑦 = 𝑘 → (𝑃 pGrp (𝐺s 𝑦) ↔ 𝑃 pGrp (𝐺s 𝑘)))
66 sseq2 3970 . . . . 5 (𝑦 = 𝑘 → (𝐻𝑦𝐻𝑘))
6765, 66anbi12d 631 . . . 4 (𝑦 = 𝑘 → ((𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦) ↔ (𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘)))
6867rexrab 3654 . . 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 19375 . . . 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 12506 . . . . . . . . . . . . 13 ℤ ⊆ ℝ
7513, 74sstrdi 3956 . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . 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 3954 . . . . . . . . . . . . . . 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 512 . . . . . . . . . . . . . 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 3647 . . . . . . . . . . . . 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 7031 . . . . . . . . . . . . 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 587 . . . . . . . . . . . 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 12117 . . . . . . . . . 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 481 . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 3647 . . . . . . . . . . . 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 6842 . . . . . . . . . . . . 13 (𝑥 = 𝑘 → (♯‘𝑥) = (♯‘𝑘))
98 fvex 6855 . . . . . . . . . . . . 13 (♯‘𝑘) ∈ V
9997, 11, 98fvmpt 6948 . . . . . . . . . . . 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 2778 . . . . . . . . . 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 5131 . . . . . . . . 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 9211 . . . . . . . . . 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 9211 . . . . . . . . . 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 14256 . . . . . . . . . . 11 (𝑚 ∈ Fin → (♯‘𝑚) ∈ ℕ0)
108 hashcl 14256 . . . . . . . . . . 11 (𝑘 ∈ Fin → (♯‘𝑘) ∈ ℕ0)
109 nn0re 12422 . . . . . . . . . . . 12 ((♯‘𝑚) ∈ ℕ0 → (♯‘𝑚) ∈ ℝ)
110 nn0re 12422 . . . . . . . . . . . 12 ((♯‘𝑘) ∈ ℕ0 → (♯‘𝑘) ∈ ℝ)
111 lenlt 11233 . . . . . . . . . . . 12 (((♯‘𝑚) ∈ ℝ ∧ (♯‘𝑘) ∈ ℝ) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
112109, 110, 111syl2an 596 . . . . . . . . . . 11 (((♯‘𝑚) ∈ ℕ0 ∧ (♯‘𝑘) ∈ ℕ0) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
113107, 108, 112syl2an 596 . . . . . . . . . 10 ((𝑚 ∈ Fin ∧ 𝑘 ∈ Fin) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
114105, 106, 113syl2anc 584 . . . . . . . . 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 9156 . . . . . . . . . . 11 ((𝑚 ∈ Fin ∧ 𝑘𝑚) → 𝑘𝑚)
117116ex 413 . . . . . . . . . 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 14281 . . . . . . . . . 10 ((𝑘 ∈ Fin ∧ 𝑚 ∈ Fin) → ((♯‘𝑘) < (♯‘𝑚) ↔ 𝑘𝑚))
120106, 105, 119syl2anc 584 . . . . . . . . 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 258 . . . . . . . 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 4059 . . . . . . . . 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 457 . . . . 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 495 . . . . . . 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 481 . . . . . 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 7365 . . . . . . . 8 (𝑘 = 𝑚 → (𝐺s 𝑘) = (𝐺s 𝑚))
131130breq2d 5117 . . . . . . 7 (𝑘 = 𝑚 → (𝑃 pGrp (𝐺s 𝑘) ↔ 𝑃 pGrp (𝐺s 𝑚)))
132 eqimss 4000 . . . . . . . 8 (𝑘 = 𝑚𝑘𝑚)
133132biantrurd 533 . . . . . . 7 (𝑘 = 𝑚 → (𝑃 pGrp (𝐺s 𝑚) ↔ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚))))
134131, 133bitrd 278 . . . . . 6 (𝑘 = 𝑚 → (𝑃 pGrp (𝐺s 𝑘) ↔ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚))))
135129, 134syl5ibcom 244 . . . . 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 3143 . . 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 19390 . . 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 496 . 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 3169 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 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  {crab 3407  wss 3910  wpss 3911  c0 4282   class class class wbr 5105  cmpt 5188  ran crn 5634   Fn wfn 6491  cfv 6496  (class class class)co 7357  cdom 8881  csdm 8882  Fincfn 8883  supcsup 9376  cr 11050   < clt 11189  cle 11190  0cn0 12413  cz 12499  chash 14230  cprime 16547  Basecbs 17083  s cress 17112  SubGrpcsubg 18922   pGrp cpgp 19308   pSyl cslw 19309
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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231  df-subg 18925  df-pgp 19312  df-slw 19313
This theorem is referenced by:  slwn0  19397
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