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Theorem pgpssslw 19632
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 1138 . . . . . . . . . 10 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝑋 ∈ Fin)
2 elrabi 3687 . . . . . . . . . . 11 (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} → 𝑥 ∈ (SubGrp‘𝐺))
3 pgpssslw.1 . . . . . . . . . . . 12 𝑋 = (Base‘𝐺)
43subgss 19145 . . . . . . . . . . 11 (𝑥 ∈ (SubGrp‘𝐺) → 𝑥𝑋)
52, 4syl 17 . . . . . . . . . 10 (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} → 𝑥𝑋)
6 ssfi 9213 . . . . . . . . . 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 14395 . . . . . . . . 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 12639 . . . . . . 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 7134 . . . . . 6 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐹:{𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}⟶ℤ)
1312frnd 6744 . . . . 5 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ⊆ ℤ)
14 fvex 6919 . . . . . . . 8 (♯‘𝑥) ∈ V
1514, 11fnmpti 6711 . . . . . . 7 𝐹 Fn {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}
16 eqimss2 4043 . . . . . . . . . 10 (𝑦 = 𝐻𝐻𝑦)
1716biantrud 531 . . . . . . . . 9 (𝑦 = 𝐻 → (𝑃 pGrp (𝐺s 𝑦) ↔ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)))
18 oveq2 7439 . . . . . . . . . . 11 (𝑦 = 𝐻 → (𝐺s 𝑦) = (𝐺s 𝐻))
19 pgpssslw.2 . . . . . . . . . . 11 𝑆 = (𝐺s 𝐻)
2018, 19eqtr4di 2795 . . . . . . . . . 10 (𝑦 = 𝐻 → (𝐺s 𝑦) = 𝑆)
2120breq2d 5155 . . . . . . . . 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 1137 . . . . . . . 8 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐻 ∈ (SubGrp‘𝐺))
24 simp3 1139 . . . . . . . 8 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝑃 pGrp 𝑆)
2522, 23, 24elrabd 3694 . . . . . . 7 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐻 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)})
26 fnfvelrn 7100 . . . . . . 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 4342 . . . . 5 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ≠ ∅)
29 hashcl 14395 . . . . . . . 8 (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ0)
301, 29syl 17 . . . . . . 7 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (♯‘𝑋) ∈ ℕ0)
3130nn0red 12588 . . . . . 6 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (♯‘𝑋) ∈ ℝ)
32 fveq2 6906 . . . . . . . . . . 11 (𝑥 = 𝑚 → (♯‘𝑥) = (♯‘𝑚))
33 fvex 6919 . . . . . . . . . . 11 (♯‘𝑚) ∈ V
3432, 11, 33fvmpt 7016 . . . . . . . . . 10 (𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} → (𝐹𝑚) = (♯‘𝑚))
3534adantl 481 . . . . . . . . 9 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (𝐹𝑚) = (♯‘𝑚))
36 oveq2 7439 . . . . . . . . . . . . 13 (𝑦 = 𝑚 → (𝐺s 𝑦) = (𝐺s 𝑚))
3736breq2d 5155 . . . . . . . . . . . 12 (𝑦 = 𝑚 → (𝑃 pGrp (𝐺s 𝑦) ↔ 𝑃 pGrp (𝐺s 𝑚)))
38 sseq2 4010 . . . . . . . . . . . 12 (𝑦 = 𝑚 → (𝐻𝑦𝐻𝑚))
3937, 38anbi12d 632 . . . . . . . . . . 11 (𝑦 = 𝑚 → ((𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦) ↔ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚)))
4039elrab 3692 . . . . . . . . . 10 (𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} ↔ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚)))
411adantr 480 . . . . . . . . . . . 12 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → 𝑋 ∈ Fin)
423subgss 19145 . . . . . . . . . . . . 13 (𝑚 ∈ (SubGrp‘𝐺) → 𝑚𝑋)
4342ad2antrl 728 . . . . . . . . . . . 12 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → 𝑚𝑋)
44 ssdomg 9040 . . . . . . . . . . . 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 9301 . . . . . . . . . . . 12 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → 𝑚 ∈ Fin)
47 hashdom 14418 . . . . . . . . . . . 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 257 . . . . . . . . . 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 5165 . . . . . . . 8 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (𝐹𝑚) ≤ (♯‘𝑋))
5251ralrimiva 3146 . . . . . . 7 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∀𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} (𝐹𝑚) ≤ (♯‘𝑋))
53 breq1 5146 . . . . . . . . 9 (𝑤 = (𝐹𝑚) → (𝑤 ≤ (♯‘𝑋) ↔ (𝐹𝑚) ≤ (♯‘𝑋)))
5453ralrn 7108 . . . . . . . 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 234 . . . . . 6 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∀𝑤 ∈ ran 𝐹 𝑤 ≤ (♯‘𝑋))
57 brralrspcev 5203 . . . . . 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 12698 . . . . 5 ((ran 𝐹 ⊆ ℤ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤𝑧) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
6013, 28, 58, 59syl3anc 1373 . . . 4 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
61 fvelrnb 6969 . . . . 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 218 . . 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 7439 . . . . . 6 (𝑦 = 𝑘 → (𝐺s 𝑦) = (𝐺s 𝑘))
6564breq2d 5155 . . . . 5 (𝑦 = 𝑘 → (𝑃 pGrp (𝐺s 𝑦) ↔ 𝑃 pGrp (𝐺s 𝑘)))
66 sseq2 4010 . . . . 5 (𝑦 = 𝑘 → (𝐻𝑦𝐻𝑘))
6765, 66anbi12d 632 . . . 4 (𝑦 = 𝑘 → ((𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦) ↔ (𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘)))
6867rexrab 3702 . . 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 218 . 2 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑘 ∈ (SubGrp‘𝐺)((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))
70 simpl3 1194 . . . 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 19611 . . . 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 771 . . 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 12620 . . . . . . . . . . . . 13 ℤ ⊆ ℝ
7513, 74sstrdi 3996 . . . . . . . . . . . 12 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ⊆ ℝ)
7675ad2antrr 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)))) → ran 𝐹 ⊆ ℝ)
7728ad2antrr 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)))) → ran 𝐹 ≠ ∅)
7858ad2antrr 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)))) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤𝑧)
79 simprl 771 . . . . . . . . . . . . . 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 782 . . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . . . 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 3994 . . . . . . . . . . . . . . 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 511 . . . . . . . . . . . . . 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 3694 . . . . . . . . . . . . 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 7100 . . . . . . . . . . . . 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 2842 . . . . . . . . . . 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 12230 . . . . . . . . . 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 782 . . . . . . . . . . . 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 480 . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 3694 . . . . . . . . . . . 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 6906 . . . . . . . . . . . . 13 (𝑥 = 𝑘 → (♯‘𝑥) = (♯‘𝑘))
98 fvex 6919 . . . . . . . . . . . . 13 (♯‘𝑘) ∈ V
9997, 11, 98fvmpt 7016 . . . . . . . . . . . 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 5169 . . . . . . . . 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 1214 . . . . . . . . . . 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 728 . . . . . . . . . . 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 9301 . . . . . . . . . 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 9301 . . . . . . . . . 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 14395 . . . . . . . . . . 11 (𝑚 ∈ Fin → (♯‘𝑚) ∈ ℕ0)
108 hashcl 14395 . . . . . . . . . . 11 (𝑘 ∈ Fin → (♯‘𝑘) ∈ ℕ0)
109 nn0re 12535 . . . . . . . . . . . 12 ((♯‘𝑚) ∈ ℕ0 → (♯‘𝑚) ∈ ℝ)
110 nn0re 12535 . . . . . . . . . . . 12 ((♯‘𝑘) ∈ ℕ0 → (♯‘𝑘) ∈ ℝ)
111 lenlt 11339 . . . . . . . . . . . 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 232 . . . . . . . 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 9249 . . . . . . . . . . 11 ((𝑚 ∈ Fin ∧ 𝑘𝑚) → 𝑘𝑚)
117116ex 412 . . . . . . . . . 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 14420 . . . . . . . . . 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 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 198 . . . . . . 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 4102 . . . . . . . . 9 (𝑘𝑚 ↔ (𝑘𝑚𝑘 = 𝑚))
12484, 123sylib 218 . . . . . . . 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 865 . . . . . . 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 456 . . . . 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 494 . . . . . . 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 480 . . . . . 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 7439 . . . . . . . 8 (𝑘 = 𝑚 → (𝐺s 𝑘) = (𝐺s 𝑚))
131130breq2d 5155 . . . . . . 7 (𝑘 = 𝑚 → (𝑃 pGrp (𝐺s 𝑘) ↔ 𝑃 pGrp (𝐺s 𝑚)))
132 eqimss 4042 . . . . . . . 8 (𝑘 = 𝑚𝑘𝑚)
133132biantrurd 532 . . . . . . 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 212 . . . 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 3146 . . 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 19626 . . 3 (𝑘 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝑘 ∈ (SubGrp‘𝐺) ∧ ∀𝑚 ∈ (SubGrp‘𝐺)((𝑘𝑚𝑃 pGrp (𝐺s 𝑚)) ↔ 𝑘 = 𝑚)))
13972, 73, 137, 138syl3anbrc 1344 . 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 495 . 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 3173 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 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  wss 3951  wpss 3952  c0 4333   class class class wbr 5143  cmpt 5225  ran crn 5686   Fn wfn 6556  cfv 6561  (class class class)co 7431  cdom 8983  csdm 8984  Fincfn 8985  supcsup 9480  cr 11154   < clt 11295  cle 11296  0cn0 12526  cz 12613  chash 14369  cprime 16708  Basecbs 17247  s cress 17274  SubGrpcsubg 19138   pGrp cpgp 19544   pSyl cslw 19545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-hash 14370  df-subg 19141  df-pgp 19548  df-slw 19549
This theorem is referenced by:  slwn0  19633
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