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

Theorem pgpssslw 17957
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 1060 . . . . . . . . . 10 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝑋 ∈ Fin)
2 elrabi 3346 . . . . . . . . . . 11 (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} → 𝑥 ∈ (SubGrp‘𝐺))
3 pgpssslw.1 . . . . . . . . . . . 12 𝑋 = (Base‘𝐺)
43subgss 17523 . . . . . . . . . . 11 (𝑥 ∈ (SubGrp‘𝐺) → 𝑥𝑋)
52, 4syl 17 . . . . . . . . . 10 (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} → 𝑥𝑋)
6 ssfi 8131 . . . . . . . . . 10 ((𝑋 ∈ Fin ∧ 𝑥𝑋) → 𝑥 ∈ Fin)
71, 5, 6syl2an 494 . . . . . . . . 9 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → 𝑥 ∈ Fin)
8 hashcl 13094 . . . . . . . . 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 11431 . . . . . . 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 6346 . . . . . 6 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐹:{𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}⟶ℤ)
13 frn 6015 . . . . . 6 (𝐹:{𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}⟶ℤ → ran 𝐹 ⊆ ℤ)
1412, 13syl 17 . . . . 5 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ⊆ ℤ)
15 fvex 6163 . . . . . . . 8 (#‘𝑥) ∈ V
1615, 11fnmpti 5984 . . . . . . 7 𝐹 Fn {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}
17 simp1 1059 . . . . . . . 8 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐻 ∈ (SubGrp‘𝐺))
18 simp3 1061 . . . . . . . 8 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝑃 pGrp 𝑆)
19 eqimss2 3642 . . . . . . . . . . 11 (𝑦 = 𝐻𝐻𝑦)
2019biantrud 528 . . . . . . . . . 10 (𝑦 = 𝐻 → (𝑃 pGrp (𝐺s 𝑦) ↔ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)))
21 oveq2 6618 . . . . . . . . . . . 12 (𝑦 = 𝐻 → (𝐺s 𝑦) = (𝐺s 𝐻))
22 pgpssslw.2 . . . . . . . . . . . 12 𝑆 = (𝐺s 𝐻)
2321, 22syl6eqr 2673 . . . . . . . . . . 11 (𝑦 = 𝐻 → (𝐺s 𝑦) = 𝑆)
2423breq2d 4630 . . . . . . . . . 10 (𝑦 = 𝐻 → (𝑃 pGrp (𝐺s 𝑦) ↔ 𝑃 pGrp 𝑆))
2520, 24bitr3d 270 . . . . . . . . 9 (𝑦 = 𝐻 → ((𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦) ↔ 𝑃 pGrp 𝑆))
2625elrab 3350 . . . . . . . 8 (𝐻 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} ↔ (𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑃 pGrp 𝑆))
2717, 18, 26sylanbrc 697 . . . . . . 7 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐻 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)})
28 fnfvelrn 6317 . . . . . . 7 ((𝐹 Fn {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} ∧ 𝐻 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (𝐹𝐻) ∈ ran 𝐹)
2916, 27, 28sylancr 694 . . . . . 6 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (𝐹𝐻) ∈ ran 𝐹)
30 ne0i 3902 . . . . . 6 ((𝐹𝐻) ∈ ran 𝐹 → ran 𝐹 ≠ ∅)
3129, 30syl 17 . . . . 5 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ≠ ∅)
32 hashcl 13094 . . . . . . . 8 (𝑋 ∈ Fin → (#‘𝑋) ∈ ℕ0)
331, 32syl 17 . . . . . . 7 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (#‘𝑋) ∈ ℕ0)
3433nn0red 11303 . . . . . 6 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (#‘𝑋) ∈ ℝ)
35 fveq2 6153 . . . . . . . . . . 11 (𝑥 = 𝑚 → (#‘𝑥) = (#‘𝑚))
36 fvex 6163 . . . . . . . . . . 11 (#‘𝑚) ∈ V
3735, 11, 36fvmpt 6244 . . . . . . . . . 10 (𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} → (𝐹𝑚) = (#‘𝑚))
3837adantl 482 . . . . . . . . 9 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (𝐹𝑚) = (#‘𝑚))
39 oveq2 6618 . . . . . . . . . . . . 13 (𝑦 = 𝑚 → (𝐺s 𝑦) = (𝐺s 𝑚))
4039breq2d 4630 . . . . . . . . . . . 12 (𝑦 = 𝑚 → (𝑃 pGrp (𝐺s 𝑦) ↔ 𝑃 pGrp (𝐺s 𝑚)))
41 sseq2 3611 . . . . . . . . . . . 12 (𝑦 = 𝑚 → (𝐻𝑦𝐻𝑚))
4240, 41anbi12d 746 . . . . . . . . . . 11 (𝑦 = 𝑚 → ((𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦) ↔ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚)))
4342elrab 3350 . . . . . . . . . 10 (𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} ↔ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚)))
441adantr 481 . . . . . . . . . . . 12 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → 𝑋 ∈ Fin)
453subgss 17523 . . . . . . . . . . . . 13 (𝑚 ∈ (SubGrp‘𝐺) → 𝑚𝑋)
4645ad2antrl 763 . . . . . . . . . . . 12 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → 𝑚𝑋)
47 ssdomg 7952 . . . . . . . . . . . 12 (𝑋 ∈ Fin → (𝑚𝑋𝑚𝑋))
4844, 46, 47sylc 65 . . . . . . . . . . 11 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → 𝑚𝑋)
49 ssfi 8131 . . . . . . . . . . . . 13 ((𝑋 ∈ Fin ∧ 𝑚𝑋) → 𝑚 ∈ Fin)
5044, 46, 49syl2anc 692 . . . . . . . . . . . 12 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → 𝑚 ∈ Fin)
51 hashdom 13115 . . . . . . . . . . . 12 ((𝑚 ∈ Fin ∧ 𝑋 ∈ Fin) → ((#‘𝑚) ≤ (#‘𝑋) ↔ 𝑚𝑋))
5250, 44, 51syl2anc 692 . . . . . . . . . . 11 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → ((#‘𝑚) ≤ (#‘𝑋) ↔ 𝑚𝑋))
5348, 52mpbird 247 . . . . . . . . . 10 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑚) ∧ 𝐻𝑚))) → (#‘𝑚) ≤ (#‘𝑋))
5443, 53sylan2b 492 . . . . . . . . 9 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (#‘𝑚) ≤ (#‘𝑋))
5538, 54eqbrtrd 4640 . . . . . . . 8 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (𝐹𝑚) ≤ (#‘𝑋))
5655ralrimiva 2961 . . . . . . 7 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∀𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} (𝐹𝑚) ≤ (#‘𝑋))
57 breq1 4621 . . . . . . . . 9 (𝑤 = (𝐹𝑚) → (𝑤 ≤ (#‘𝑋) ↔ (𝐹𝑚) ≤ (#‘𝑋)))
5857ralrn 6323 . . . . . . . 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)))
5916, 58ax-mp 5 . . . . . . 7 (∀𝑤 ∈ ran 𝐹 𝑤 ≤ (#‘𝑋) ↔ ∀𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} (𝐹𝑚) ≤ (#‘𝑋))
6056, 59sylibr 224 . . . . . 6 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∀𝑤 ∈ ran 𝐹 𝑤 ≤ (#‘𝑋))
61 breq2 4622 . . . . . . . 8 (𝑧 = (#‘𝑋) → (𝑤𝑧𝑤 ≤ (#‘𝑋)))
6261ralbidv 2981 . . . . . . 7 (𝑧 = (#‘𝑋) → (∀𝑤 ∈ ran 𝐹 𝑤𝑧 ↔ ∀𝑤 ∈ ran 𝐹 𝑤 ≤ (#‘𝑋)))
6362rspcev 3298 . . . . . 6 (((#‘𝑋) ∈ ℝ ∧ ∀𝑤 ∈ ran 𝐹 𝑤 ≤ (#‘𝑋)) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤𝑧)
6434, 60, 63syl2anc 692 . . . . 5 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤𝑧)
65 suprzcl 11408 . . . . 5 ((ran 𝐹 ⊆ ℤ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤𝑧) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
6614, 31, 64, 65syl3anc 1323 . . . 4 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
67 fvelrnb 6205 . . . . 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, < )))
6816, 67ax-mp 5 . . . 4 (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ↔ ∃𝑘 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} (𝐹𝑘) = sup(ran 𝐹, ℝ, < ))
6966, 68sylib 208 . . 3 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑘 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} (𝐹𝑘) = sup(ran 𝐹, ℝ, < ))
70 oveq2 6618 . . . . . 6 (𝑦 = 𝑘 → (𝐺s 𝑦) = (𝐺s 𝑘))
7170breq2d 4630 . . . . 5 (𝑦 = 𝑘 → (𝑃 pGrp (𝐺s 𝑦) ↔ 𝑃 pGrp (𝐺s 𝑘)))
72 sseq2 3611 . . . . 5 (𝑦 = 𝑘 → (𝐻𝑦𝐻𝑘))
7371, 72anbi12d 746 . . . 4 (𝑦 = 𝑘 → ((𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦) ↔ (𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘)))
7473rexrab 3356 . . 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, < )))
7569, 74sylib 208 . 2 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑘 ∈ (SubGrp‘𝐺)((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))
76 simpl3 1064 . . . . . . 7 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) → 𝑃 pGrp 𝑆)
77 pgpprm 17936 . . . . . . 7 (𝑃 pGrp 𝑆𝑃 ∈ ℙ)
7876, 77syl 17 . . . . . 6 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) → 𝑃 ∈ ℙ)
79 simprl 793 . . . . . 6 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) → 𝑘 ∈ (SubGrp‘𝐺))
80 zssre 11335 . . . . . . . . . . . . . . . 16 ℤ ⊆ ℝ
8114, 80syl6ss 3599 . . . . . . . . . . . . . . 15 ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ⊆ ℝ)
8281ad2antrr 761 . . . . . . . . . . . . . 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)))) → ran 𝐹 ⊆ ℝ)
8331ad2antrr 761 . . . . . . . . . . . . . 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)))) → ran 𝐹 ≠ ∅)
8464ad2antrr 761 . . . . . . . . . . . . . 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)))) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤𝑧)
85 simprl 793 . . . . . . . . . . . . . . . . 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)))) → 𝑚 ∈ (SubGrp‘𝐺))
86 simprrr 804 . . . . . . . . . . . . . . . . . 18 ((((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))
87 simprrl 803 . . . . . . . . . . . . . . . . . . . . 21 (((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))
8887adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((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))
8988simprd 479 . . . . . . . . . . . . . . . . . . 19 ((((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)
90 simprrl 803 . . . . . . . . . . . . . . . . . . 19 ((((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)
9189, 90sstrd 3597 . . . . . . . . . . . . . . . . . 18 ((((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)
9286, 91jca 554 . . . . . . . . . . . . . . . . 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 𝑚) ∧ 𝐻𝑚))
9385, 92, 43sylanbrc 697 . . . . . . . . . . . . . . . 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)))) → 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)})
9493, 37syl 17 . . . . . . . . . . . . . . 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)))) → (𝐹𝑚) = (#‘𝑚))
95 fnfvelrn 6317 . . . . . . . . . . . . . . . 16 ((𝐹 Fn {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)}) → (𝐹𝑚) ∈ ran 𝐹)
9616, 93, 95sylancr 694 . . . . . . . . . . . . . . 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)))) → (𝐹𝑚) ∈ ran 𝐹)
9794, 96eqeltrrd 2699 . . . . . . . . . . . . . 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) ∈ ran 𝐹)
98 suprub 10935 . . . . . . . . . . . . . 14 (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤𝑧) ∧ (#‘𝑚) ∈ ran 𝐹) → (#‘𝑚) ≤ sup(ran 𝐹, ℝ, < ))
9982, 83, 84, 97, 98syl31anc 1326 . . . . . . . . . . . . 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) ≤ sup(ran 𝐹, ℝ, < ))
100 simprrr 804 . . . . . . . . . . . . . . 15 (((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, < ))
101100adantr 481 . . . . . . . . . . . . . 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)))) → (𝐹𝑘) = sup(ran 𝐹, ℝ, < ))
10279adantr 481 . . . . . . . . . . . . . . . 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)))) → 𝑘 ∈ (SubGrp‘𝐺))
10373elrab 3350 . . . . . . . . . . . . . . . 16 (𝑘 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} ↔ (𝑘 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘)))
104102, 88, 103sylanbrc 697 . . . . . . . . . . . . . . 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)))) → 𝑘 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)})
105 fveq2 6153 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑘 → (#‘𝑥) = (#‘𝑘))
106 fvex 6163 . . . . . . . . . . . . . . . 16 (#‘𝑘) ∈ V
107105, 11, 106fvmpt 6244 . . . . . . . . . . . . . . 15 (𝑘 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺s 𝑦) ∧ 𝐻𝑦)} → (𝐹𝑘) = (#‘𝑘))
108104, 107syl 17 . . . . . . . . . . . . . 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)))) → (𝐹𝑘) = (#‘𝑘))
109101, 108eqtr3d 2657 . . . . . . . . . . . . 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)))) → sup(ran 𝐹, ℝ, < ) = (#‘𝑘))
11099, 109breqtrd 4644 . . . . . . . . . . . 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)))) → (#‘𝑚) ≤ (#‘𝑘))
111 simpll2 1099 . . . . . . . . . . . . . 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)))) → 𝑋 ∈ Fin)
11245ad2antrl 763 . . . . . . . . . . . . . 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)))) → 𝑚𝑋)
113111, 112, 49syl2anc 692 . . . . . . . . . . . . 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 ∈ Fin)
114 ssfi 8131 . . . . . . . . . . . . . 14 ((𝑚 ∈ Fin ∧ 𝑘𝑚) → 𝑘 ∈ Fin)
115113, 90, 114syl2anc 692 . . . . . . . . . . . . 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 ∈ Fin)
116 hashcl 13094 . . . . . . . . . . . . . 14 (𝑚 ∈ Fin → (#‘𝑚) ∈ ℕ0)
117 hashcl 13094 . . . . . . . . . . . . . 14 (𝑘 ∈ Fin → (#‘𝑘) ∈ ℕ0)
118 nn0re 11252 . . . . . . . . . . . . . . 15 ((#‘𝑚) ∈ ℕ0 → (#‘𝑚) ∈ ℝ)
119 nn0re 11252 . . . . . . . . . . . . . . 15 ((#‘𝑘) ∈ ℕ0 → (#‘𝑘) ∈ ℝ)
120 lenlt 10067 . . . . . . . . . . . . . . 15 (((#‘𝑚) ∈ ℝ ∧ (#‘𝑘) ∈ ℝ) → ((#‘𝑚) ≤ (#‘𝑘) ↔ ¬ (#‘𝑘) < (#‘𝑚)))
121118, 119, 120syl2an 494 . . . . . . . . . . . . . 14 (((#‘𝑚) ∈ ℕ0 ∧ (#‘𝑘) ∈ ℕ0) → ((#‘𝑚) ≤ (#‘𝑘) ↔ ¬ (#‘𝑘) < (#‘𝑚)))
122116, 117, 121syl2an 494 . . . . . . . . . . . . 13 ((𝑚 ∈ Fin ∧ 𝑘 ∈ Fin) → ((#‘𝑚) ≤ (#‘𝑘) ↔ ¬ (#‘𝑘) < (#‘𝑚)))
123113, 115, 122syl2anc 692 . . . . . . . . . . . 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)))) → ((#‘𝑚) ≤ (#‘𝑘) ↔ ¬ (#‘𝑘) < (#‘𝑚)))
124110, 123mpbid 222 . . . . . . . . . . 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)))) → ¬ (#‘𝑘) < (#‘𝑚))
125 php3 8097 . . . . . . . . . . . . . 14 ((𝑚 ∈ Fin ∧ 𝑘𝑚) → 𝑘𝑚)
126125ex 450 . . . . . . . . . . . . 13 (𝑚 ∈ Fin → (𝑘𝑚𝑘𝑚))
127113, 126syl 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)))) → (𝑘𝑚𝑘𝑚))
128 hashsdom 13117 . . . . . . . . . . . . 13 ((𝑘 ∈ Fin ∧ 𝑚 ∈ Fin) → ((#‘𝑘) < (#‘𝑚) ↔ 𝑘𝑚))
129115, 113, 128syl2anc 692 . . . . . . . . . . . 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) < (#‘𝑚) ↔ 𝑘𝑚))
130127, 129sylibrd 249 . . . . . . . . . . 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)))) → (𝑘𝑚 → (#‘𝑘) < (#‘𝑚)))
131124, 130mtod 189 . . . . . . . . . 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)))) → ¬ 𝑘𝑚)
132 sspss 3689 . . . . . . . . . . . 12 (𝑘𝑚 ↔ (𝑘𝑚𝑘 = 𝑚))
13390, 132sylib 208 . . . . . . . . . . 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)))) → (𝑘𝑚𝑘 = 𝑚))
134133ord 392 . . . . . . . . . 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)))) → (¬ 𝑘𝑚𝑘 = 𝑚))
135131, 134mpd 15 . . . . . . . . 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)
136135expr 642 . . . . . . . 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))
13787simpld 475 . . . . . . . . . 10 (((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))
138137adantr 481 . . . . . . . . 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)) → 𝑃 pGrp (𝐺s 𝑘))
139 oveq2 6618 . . . . . . . . . . 11 (𝑘 = 𝑚 → (𝐺s 𝑘) = (𝐺s 𝑚))
140139breq2d 4630 . . . . . . . . . 10 (𝑘 = 𝑚 → (𝑃 pGrp (𝐺s 𝑘) ↔ 𝑃 pGrp (𝐺s 𝑚)))
141 eqimss 3641 . . . . . . . . . . 11 (𝑘 = 𝑚𝑘𝑚)
142141biantrurd 529 . . . . . . . . . 10 (𝑘 = 𝑚 → (𝑃 pGrp (𝐺s 𝑚) ↔ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚))))
143140, 142bitrd 268 . . . . . . . . 9 (𝑘 = 𝑚 → (𝑃 pGrp (𝐺s 𝑘) ↔ (𝑘𝑚𝑃 pGrp (𝐺s 𝑚))))
144138, 143syl5ibcom 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)) → (𝑘 = 𝑚 → (𝑘𝑚𝑃 pGrp (𝐺s 𝑚))))
145136, 144impbid 202 . . . . . . 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)) ↔ 𝑘 = 𝑚))
146145ralrimiva 2961 . . . . . 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))
147 isslw 17951 . . . . . 6 (𝑘 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝑘 ∈ (SubGrp‘𝐺) ∧ ∀𝑚 ∈ (SubGrp‘𝐺)((𝑘𝑚𝑃 pGrp (𝐺s 𝑚)) ↔ 𝑘 = 𝑚)))
14878, 79, 146, 147syl3anbrc 1244 . . . . 5 (((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))
14987simprd 479 . . . . 5 (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺s 𝑘) ∧ 𝐻𝑘) ∧ (𝐹𝑘) = sup(ran 𝐹, ℝ, < )))) → 𝐻𝑘)
150148, 149jca 554 . . . 4 (((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) ∧ 𝐻𝑘))
151150ex 450 . . 3 ((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) ∧ 𝐻𝑘)))
152151reximdv2 3009 . 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)𝐻𝑘))
15375, 152mpd 15 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 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  wss 3559  wpss 3560  c0 3896   class class class wbr 4618  cmpt 4678  ran crn 5080   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  cdom 7904  csdm 7905  Fincfn 7906  supcsup 8297  cr 9886   < clt 10025  cle 10026  0cn0 11243  cz 11328  #chash 13064  cprime 15316  Basecbs 15788  s cress 15789  SubGrpcsubg 17516   pGrp cpgp 17874   pSyl cslw 17875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-sup 8299  df-card 8716  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-n0 11244  df-xnn0 11315  df-z 11329  df-uz 11639  df-fz 12276  df-hash 13065  df-subg 17519  df-pgp 17878  df-slw 17879
This theorem is referenced by:  slwn0  17958
  Copyright terms: Public domain W3C validator