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Theorem pgpssslw 19482
Description: Every 𝑃-subgroup is contained in a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
Hypotheses
Ref Expression
pgpssslw.1 𝑋 = (Baseβ€˜πΊ)
pgpssslw.2 𝑆 = (𝐺 β†Ύs 𝐻)
pgpssslw.3 𝐹 = (π‘₯ ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)} ↦ (β™―β€˜π‘₯))
Assertion
Ref Expression
pgpssslw ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ βˆƒπ‘˜ ∈ (𝑃 pSyl 𝐺)𝐻 βŠ† π‘˜)
Distinct variable groups:   π‘₯,π‘˜,𝑦,𝐺   π‘˜,𝐻,π‘₯,𝑦   𝑃,π‘˜,π‘₯,𝑦   π‘˜,𝑋,π‘₯   π‘˜,𝐹   𝑆,π‘˜,π‘₯,𝑦
Allowed substitution hints:   𝐹(π‘₯,𝑦)   𝑋(𝑦)
Proof of Theorem pgpssslw
Dummy variables π‘š 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 1138 . . . . . . . . . 10 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ 𝑋 ∈ Fin)
2 elrabi 3678 . . . . . . . . . . 11 (π‘₯ ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)} β†’ π‘₯ ∈ (SubGrpβ€˜πΊ))
3 pgpssslw.1 . . . . . . . . . . . 12 𝑋 = (Baseβ€˜πΊ)
43subgss 19007 . . . . . . . . . . 11 (π‘₯ ∈ (SubGrpβ€˜πΊ) β†’ π‘₯ βŠ† 𝑋)
52, 4syl 17 . . . . . . . . . 10 (π‘₯ ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)} β†’ π‘₯ βŠ† 𝑋)
6 ssfi 9173 . . . . . . . . . 10 ((𝑋 ∈ Fin ∧ π‘₯ βŠ† 𝑋) β†’ π‘₯ ∈ Fin)
71, 5, 6syl2an 597 . . . . . . . . 9 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ π‘₯ ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)}) β†’ π‘₯ ∈ Fin)
8 hashcl 14316 . . . . . . . . 9 (π‘₯ ∈ Fin β†’ (β™―β€˜π‘₯) ∈ β„•0)
97, 8syl 17 . . . . . . . 8 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ π‘₯ ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)}) β†’ (β™―β€˜π‘₯) ∈ β„•0)
109nn0zd 12584 . . . . . . 7 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ π‘₯ ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)}) β†’ (β™―β€˜π‘₯) ∈ β„€)
11 pgpssslw.3 . . . . . . 7 𝐹 = (π‘₯ ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)} ↦ (β™―β€˜π‘₯))
1210, 11fmptd 7114 . . . . . 6 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ 𝐹:{𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)}βŸΆβ„€)
1312frnd 6726 . . . . 5 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ ran 𝐹 βŠ† β„€)
14 fvex 6905 . . . . . . . 8 (β™―β€˜π‘₯) ∈ V
1514, 11fnmpti 6694 . . . . . . 7 𝐹 Fn {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)}
16 eqimss2 4042 . . . . . . . . . 10 (𝑦 = 𝐻 β†’ 𝐻 βŠ† 𝑦)
1716biantrud 533 . . . . . . . . 9 (𝑦 = 𝐻 β†’ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ↔ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)))
18 oveq2 7417 . . . . . . . . . . 11 (𝑦 = 𝐻 β†’ (𝐺 β†Ύs 𝑦) = (𝐺 β†Ύs 𝐻))
19 pgpssslw.2 . . . . . . . . . . 11 𝑆 = (𝐺 β†Ύs 𝐻)
2018, 19eqtr4di 2791 . . . . . . . . . 10 (𝑦 = 𝐻 β†’ (𝐺 β†Ύs 𝑦) = 𝑆)
2120breq2d 5161 . . . . . . . . 9 (𝑦 = 𝐻 β†’ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ↔ 𝑃 pGrp 𝑆))
2217, 21bitr3d 281 . . . . . . . 8 (𝑦 = 𝐻 β†’ ((𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦) ↔ 𝑃 pGrp 𝑆))
23 simp1 1137 . . . . . . . 8 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ 𝐻 ∈ (SubGrpβ€˜πΊ))
24 simp3 1139 . . . . . . . 8 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ 𝑃 pGrp 𝑆)
2522, 23, 24elrabd 3686 . . . . . . 7 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ 𝐻 ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)})
26 fnfvelrn 7083 . . . . . . 7 ((𝐹 Fn {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)} ∧ 𝐻 ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)}) β†’ (πΉβ€˜π») ∈ ran 𝐹)
2715, 25, 26sylancr 588 . . . . . 6 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ (πΉβ€˜π») ∈ ran 𝐹)
2827ne0d 4336 . . . . 5 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ ran 𝐹 β‰  βˆ…)
29 hashcl 14316 . . . . . . . 8 (𝑋 ∈ Fin β†’ (β™―β€˜π‘‹) ∈ β„•0)
301, 29syl 17 . . . . . . 7 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ (β™―β€˜π‘‹) ∈ β„•0)
3130nn0red 12533 . . . . . 6 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ (β™―β€˜π‘‹) ∈ ℝ)
32 fveq2 6892 . . . . . . . . . . 11 (π‘₯ = π‘š β†’ (β™―β€˜π‘₯) = (β™―β€˜π‘š))
33 fvex 6905 . . . . . . . . . . 11 (β™―β€˜π‘š) ∈ V
3432, 11, 33fvmpt 6999 . . . . . . . . . 10 (π‘š ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)} β†’ (πΉβ€˜π‘š) = (β™―β€˜π‘š))
3534adantl 483 . . . . . . . . 9 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ π‘š ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)}) β†’ (πΉβ€˜π‘š) = (β™―β€˜π‘š))
36 oveq2 7417 . . . . . . . . . . . . 13 (𝑦 = π‘š β†’ (𝐺 β†Ύs 𝑦) = (𝐺 β†Ύs π‘š))
3736breq2d 5161 . . . . . . . . . . . 12 (𝑦 = π‘š β†’ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ↔ 𝑃 pGrp (𝐺 β†Ύs π‘š)))
38 sseq2 4009 . . . . . . . . . . . 12 (𝑦 = π‘š β†’ (𝐻 βŠ† 𝑦 ↔ 𝐻 βŠ† π‘š))
3937, 38anbi12d 632 . . . . . . . . . . 11 (𝑦 = π‘š β†’ ((𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦) ↔ (𝑃 pGrp (𝐺 β†Ύs π‘š) ∧ 𝐻 βŠ† π‘š)))
4039elrab 3684 . . . . . . . . . 10 (π‘š ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)} ↔ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (𝑃 pGrp (𝐺 β†Ύs π‘š) ∧ 𝐻 βŠ† π‘š)))
411adantr 482 . . . . . . . . . . . 12 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (𝑃 pGrp (𝐺 β†Ύs π‘š) ∧ 𝐻 βŠ† π‘š))) β†’ 𝑋 ∈ Fin)
423subgss 19007 . . . . . . . . . . . . 13 (π‘š ∈ (SubGrpβ€˜πΊ) β†’ π‘š βŠ† 𝑋)
4342ad2antrl 727 . . . . . . . . . . . 12 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (𝑃 pGrp (𝐺 β†Ύs π‘š) ∧ 𝐻 βŠ† π‘š))) β†’ π‘š βŠ† 𝑋)
44 ssdomg 8996 . . . . . . . . . . . 12 (𝑋 ∈ Fin β†’ (π‘š βŠ† 𝑋 β†’ π‘š β‰Ό 𝑋))
4541, 43, 44sylc 65 . . . . . . . . . . 11 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (𝑃 pGrp (𝐺 β†Ύs π‘š) ∧ 𝐻 βŠ† π‘š))) β†’ π‘š β‰Ό 𝑋)
4641, 43ssfid 9267 . . . . . . . . . . . 12 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (𝑃 pGrp (𝐺 β†Ύs π‘š) ∧ 𝐻 βŠ† π‘š))) β†’ π‘š ∈ Fin)
47 hashdom 14339 . . . . . . . . . . . 12 ((π‘š ∈ Fin ∧ 𝑋 ∈ Fin) β†’ ((β™―β€˜π‘š) ≀ (β™―β€˜π‘‹) ↔ π‘š β‰Ό 𝑋))
4846, 41, 47syl2anc 585 . . . . . . . . . . 11 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (𝑃 pGrp (𝐺 β†Ύs π‘š) ∧ 𝐻 βŠ† π‘š))) β†’ ((β™―β€˜π‘š) ≀ (β™―β€˜π‘‹) ↔ π‘š β‰Ό 𝑋))
4945, 48mpbird 257 . . . . . . . . . 10 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (𝑃 pGrp (𝐺 β†Ύs π‘š) ∧ 𝐻 βŠ† π‘š))) β†’ (β™―β€˜π‘š) ≀ (β™―β€˜π‘‹))
5040, 49sylan2b 595 . . . . . . . . 9 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ π‘š ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)}) β†’ (β™―β€˜π‘š) ≀ (β™―β€˜π‘‹))
5135, 50eqbrtrd 5171 . . . . . . . 8 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ π‘š ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)}) β†’ (πΉβ€˜π‘š) ≀ (β™―β€˜π‘‹))
5251ralrimiva 3147 . . . . . . 7 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ βˆ€π‘š ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)} (πΉβ€˜π‘š) ≀ (β™―β€˜π‘‹))
53 breq1 5152 . . . . . . . . 9 (𝑀 = (πΉβ€˜π‘š) β†’ (𝑀 ≀ (β™―β€˜π‘‹) ↔ (πΉβ€˜π‘š) ≀ (β™―β€˜π‘‹)))
5453ralrn 7090 . . . . . . . 8 (𝐹 Fn {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)} β†’ (βˆ€π‘€ ∈ ran 𝐹 𝑀 ≀ (β™―β€˜π‘‹) ↔ βˆ€π‘š ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)} (πΉβ€˜π‘š) ≀ (β™―β€˜π‘‹)))
5515, 54ax-mp 5 . . . . . . 7 (βˆ€π‘€ ∈ ran 𝐹 𝑀 ≀ (β™―β€˜π‘‹) ↔ βˆ€π‘š ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)} (πΉβ€˜π‘š) ≀ (β™―β€˜π‘‹))
5652, 55sylibr 233 . . . . . 6 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ βˆ€π‘€ ∈ ran 𝐹 𝑀 ≀ (β™―β€˜π‘‹))
57 brralrspcev 5209 . . . . . 6 (((β™―β€˜π‘‹) ∈ ℝ ∧ βˆ€π‘€ ∈ ran 𝐹 𝑀 ≀ (β™―β€˜π‘‹)) β†’ βˆƒπ‘§ ∈ ℝ βˆ€π‘€ ∈ ran 𝐹 𝑀 ≀ 𝑧)
5831, 56, 57syl2anc 585 . . . . 5 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ βˆƒπ‘§ ∈ ℝ βˆ€π‘€ ∈ ran 𝐹 𝑀 ≀ 𝑧)
59 suprzcl 12642 . . . . 5 ((ran 𝐹 βŠ† β„€ ∧ ran 𝐹 β‰  βˆ… ∧ βˆƒπ‘§ ∈ ℝ βˆ€π‘€ ∈ ran 𝐹 𝑀 ≀ 𝑧) β†’ sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
6013, 28, 58, 59syl3anc 1372 . . . 4 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
61 fvelrnb 6953 . . . . 5 (𝐹 Fn {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)} β†’ (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ↔ βˆƒπ‘˜ ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)} (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))
6215, 61ax-mp 5 . . . 4 (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ↔ βˆƒπ‘˜ ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)} (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < ))
6360, 62sylib 217 . . 3 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ βˆƒπ‘˜ ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)} (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < ))
64 oveq2 7417 . . . . . 6 (𝑦 = π‘˜ β†’ (𝐺 β†Ύs 𝑦) = (𝐺 β†Ύs π‘˜))
6564breq2d 5161 . . . . 5 (𝑦 = π‘˜ β†’ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ↔ 𝑃 pGrp (𝐺 β†Ύs π‘˜)))
66 sseq2 4009 . . . . 5 (𝑦 = π‘˜ β†’ (𝐻 βŠ† 𝑦 ↔ 𝐻 βŠ† π‘˜))
6765, 66anbi12d 632 . . . 4 (𝑦 = π‘˜ β†’ ((𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦) ↔ (𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜)))
6867rexrab 3693 . . 3 (βˆƒπ‘˜ ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)} (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < ) ↔ βˆƒπ‘˜ ∈ (SubGrpβ€˜πΊ)((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))
6963, 68sylib 217 . 2 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ βˆƒπ‘˜ ∈ (SubGrpβ€˜πΊ)((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))
70 simpl3 1194 . . . 4 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) β†’ 𝑃 pGrp 𝑆)
71 pgpprm 19461 . . . 4 (𝑃 pGrp 𝑆 β†’ 𝑃 ∈ β„™)
7270, 71syl 17 . . 3 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) β†’ 𝑃 ∈ β„™)
73 simprl 770 . . 3 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) β†’ π‘˜ ∈ (SubGrpβ€˜πΊ))
74 zssre 12565 . . . . . . . . . . . . 13 β„€ βŠ† ℝ
7513, 74sstrdi 3995 . . . . . . . . . . . 12 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ ran 𝐹 βŠ† ℝ)
7675ad2antrr 725 . . . . . . . . . . 11 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ ran 𝐹 βŠ† ℝ)
7728ad2antrr 725 . . . . . . . . . . 11 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ ran 𝐹 β‰  βˆ…)
7858ad2antrr 725 . . . . . . . . . . 11 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ βˆƒπ‘§ ∈ ℝ βˆ€π‘€ ∈ ran 𝐹 𝑀 ≀ 𝑧)
79 simprl 770 . . . . . . . . . . . . . 14 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ π‘š ∈ (SubGrpβ€˜πΊ))
80 simprrr 781 . . . . . . . . . . . . . . 15 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ 𝑃 pGrp (𝐺 β†Ύs π‘š))
81 simprrl 780 . . . . . . . . . . . . . . . . . 18 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) β†’ (𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜))
8281adantr 482 . . . . . . . . . . . . . . . . 17 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ (𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜))
8382simprd 497 . . . . . . . . . . . . . . . 16 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ 𝐻 βŠ† π‘˜)
84 simprrl 780 . . . . . . . . . . . . . . . 16 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ π‘˜ βŠ† π‘š)
8583, 84sstrd 3993 . . . . . . . . . . . . . . 15 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ 𝐻 βŠ† π‘š)
8680, 85jca 513 . . . . . . . . . . . . . 14 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ (𝑃 pGrp (𝐺 β†Ύs π‘š) ∧ 𝐻 βŠ† π‘š))
8739, 79, 86elrabd 3686 . . . . . . . . . . . . 13 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ π‘š ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)})
8887, 34syl 17 . . . . . . . . . . . 12 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ (πΉβ€˜π‘š) = (β™―β€˜π‘š))
89 fnfvelrn 7083 . . . . . . . . . . . . 13 ((𝐹 Fn {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)} ∧ π‘š ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)}) β†’ (πΉβ€˜π‘š) ∈ ran 𝐹)
9015, 87, 89sylancr 588 . . . . . . . . . . . 12 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ (πΉβ€˜π‘š) ∈ ran 𝐹)
9188, 90eqeltrrd 2835 . . . . . . . . . . 11 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ (β™―β€˜π‘š) ∈ ran 𝐹)
9276, 77, 78, 91suprubd 12176 . . . . . . . . . 10 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ (β™―β€˜π‘š) ≀ sup(ran 𝐹, ℝ, < ))
93 simprrr 781 . . . . . . . . . . . 12 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) β†’ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < ))
9493adantr 482 . . . . . . . . . . 11 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < ))
9573adantr 482 . . . . . . . . . . . . 13 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ π‘˜ ∈ (SubGrpβ€˜πΊ))
9667, 95, 82elrabd 3686 . . . . . . . . . . . 12 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ π‘˜ ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)})
97 fveq2 6892 . . . . . . . . . . . . 13 (π‘₯ = π‘˜ β†’ (β™―β€˜π‘₯) = (β™―β€˜π‘˜))
98 fvex 6905 . . . . . . . . . . . . 13 (β™―β€˜π‘˜) ∈ V
9997, 11, 98fvmpt 6999 . . . . . . . . . . . 12 (π‘˜ ∈ {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)} β†’ (πΉβ€˜π‘˜) = (β™―β€˜π‘˜))
10096, 99syl 17 . . . . . . . . . . 11 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ (πΉβ€˜π‘˜) = (β™―β€˜π‘˜))
10194, 100eqtr3d 2775 . . . . . . . . . 10 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ sup(ran 𝐹, ℝ, < ) = (β™―β€˜π‘˜))
10292, 101breqtrd 5175 . . . . . . . . 9 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ (β™―β€˜π‘š) ≀ (β™―β€˜π‘˜))
103 simpll2 1214 . . . . . . . . . . 11 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ 𝑋 ∈ Fin)
10442ad2antrl 727 . . . . . . . . . . 11 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ π‘š βŠ† 𝑋)
105103, 104ssfid 9267 . . . . . . . . . 10 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ π‘š ∈ Fin)
106105, 84ssfid 9267 . . . . . . . . . 10 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ π‘˜ ∈ Fin)
107 hashcl 14316 . . . . . . . . . . 11 (π‘š ∈ Fin β†’ (β™―β€˜π‘š) ∈ β„•0)
108 hashcl 14316 . . . . . . . . . . 11 (π‘˜ ∈ Fin β†’ (β™―β€˜π‘˜) ∈ β„•0)
109 nn0re 12481 . . . . . . . . . . . 12 ((β™―β€˜π‘š) ∈ β„•0 β†’ (β™―β€˜π‘š) ∈ ℝ)
110 nn0re 12481 . . . . . . . . . . . 12 ((β™―β€˜π‘˜) ∈ β„•0 β†’ (β™―β€˜π‘˜) ∈ ℝ)
111 lenlt 11292 . . . . . . . . . . . 12 (((β™―β€˜π‘š) ∈ ℝ ∧ (β™―β€˜π‘˜) ∈ ℝ) β†’ ((β™―β€˜π‘š) ≀ (β™―β€˜π‘˜) ↔ Β¬ (β™―β€˜π‘˜) < (β™―β€˜π‘š)))
112109, 110, 111syl2an 597 . . . . . . . . . . 11 (((β™―β€˜π‘š) ∈ β„•0 ∧ (β™―β€˜π‘˜) ∈ β„•0) β†’ ((β™―β€˜π‘š) ≀ (β™―β€˜π‘˜) ↔ Β¬ (β™―β€˜π‘˜) < (β™―β€˜π‘š)))
113107, 108, 112syl2an 597 . . . . . . . . . 10 ((π‘š ∈ Fin ∧ π‘˜ ∈ Fin) β†’ ((β™―β€˜π‘š) ≀ (β™―β€˜π‘˜) ↔ Β¬ (β™―β€˜π‘˜) < (β™―β€˜π‘š)))
114105, 106, 113syl2anc 585 . . . . . . . . 9 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ ((β™―β€˜π‘š) ≀ (β™―β€˜π‘˜) ↔ Β¬ (β™―β€˜π‘˜) < (β™―β€˜π‘š)))
115102, 114mpbid 231 . . . . . . . 8 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ Β¬ (β™―β€˜π‘˜) < (β™―β€˜π‘š))
116 php3 9212 . . . . . . . . . . 11 ((π‘š ∈ Fin ∧ π‘˜ ⊊ π‘š) β†’ π‘˜ β‰Ί π‘š)
117116ex 414 . . . . . . . . . 10 (π‘š ∈ Fin β†’ (π‘˜ ⊊ π‘š β†’ π‘˜ β‰Ί π‘š))
118105, 117syl 17 . . . . . . . . 9 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ (π‘˜ ⊊ π‘š β†’ π‘˜ β‰Ί π‘š))
119 hashsdom 14341 . . . . . . . . . 10 ((π‘˜ ∈ Fin ∧ π‘š ∈ Fin) β†’ ((β™―β€˜π‘˜) < (β™―β€˜π‘š) ↔ π‘˜ β‰Ί π‘š))
120106, 105, 119syl2anc 585 . . . . . . . . 9 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ ((β™―β€˜π‘˜) < (β™―β€˜π‘š) ↔ π‘˜ β‰Ί π‘š))
121118, 120sylibrd 259 . . . . . . . 8 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ (π‘˜ ⊊ π‘š β†’ (β™―β€˜π‘˜) < (β™―β€˜π‘š)))
122115, 121mtod 197 . . . . . . 7 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ Β¬ π‘˜ ⊊ π‘š)
123 sspss 4100 . . . . . . . . 9 (π‘˜ βŠ† π‘š ↔ (π‘˜ ⊊ π‘š ∨ π‘˜ = π‘š))
12484, 123sylib 217 . . . . . . . 8 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ (π‘˜ ⊊ π‘š ∨ π‘˜ = π‘š))
125124ord 863 . . . . . . 7 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ (Β¬ π‘˜ ⊊ π‘š β†’ π‘˜ = π‘š))
126122, 125mpd 15 . . . . . 6 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ (π‘š ∈ (SubGrpβ€˜πΊ) ∧ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)))) β†’ π‘˜ = π‘š)
127126expr 458 . . . . 5 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ π‘š ∈ (SubGrpβ€˜πΊ)) β†’ ((π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)) β†’ π‘˜ = π‘š))
12881simpld 496 . . . . . . 7 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) β†’ 𝑃 pGrp (𝐺 β†Ύs π‘˜))
129128adantr 482 . . . . . 6 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ π‘š ∈ (SubGrpβ€˜πΊ)) β†’ 𝑃 pGrp (𝐺 β†Ύs π‘˜))
130 oveq2 7417 . . . . . . . 8 (π‘˜ = π‘š β†’ (𝐺 β†Ύs π‘˜) = (𝐺 β†Ύs π‘š))
131130breq2d 5161 . . . . . . 7 (π‘˜ = π‘š β†’ (𝑃 pGrp (𝐺 β†Ύs π‘˜) ↔ 𝑃 pGrp (𝐺 β†Ύs π‘š)))
132 eqimss 4041 . . . . . . . 8 (π‘˜ = π‘š β†’ π‘˜ βŠ† π‘š)
133132biantrurd 534 . . . . . . 7 (π‘˜ = π‘š β†’ (𝑃 pGrp (𝐺 β†Ύs π‘š) ↔ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š))))
134131, 133bitrd 279 . . . . . 6 (π‘˜ = π‘š β†’ (𝑃 pGrp (𝐺 β†Ύs π‘˜) ↔ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š))))
135129, 134syl5ibcom 244 . . . . 5 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ π‘š ∈ (SubGrpβ€˜πΊ)) β†’ (π‘˜ = π‘š β†’ (π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š))))
136127, 135impbid 211 . . . 4 ((((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) ∧ π‘š ∈ (SubGrpβ€˜πΊ)) β†’ ((π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)) ↔ π‘˜ = π‘š))
137136ralrimiva 3147 . . 3 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) β†’ βˆ€π‘š ∈ (SubGrpβ€˜πΊ)((π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)) ↔ π‘˜ = π‘š))
138 isslw 19476 . . 3 (π‘˜ ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ β„™ ∧ π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ βˆ€π‘š ∈ (SubGrpβ€˜πΊ)((π‘˜ βŠ† π‘š ∧ 𝑃 pGrp (𝐺 β†Ύs π‘š)) ↔ π‘˜ = π‘š)))
13972, 73, 137, 138syl3anbrc 1344 . 2 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) β†’ π‘˜ ∈ (𝑃 pSyl 𝐺))
14081simprd 497 . 2 (((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (π‘˜ ∈ (SubGrpβ€˜πΊ) ∧ ((𝑃 pGrp (𝐺 β†Ύs π‘˜) ∧ 𝐻 βŠ† π‘˜) ∧ (πΉβ€˜π‘˜) = sup(ran 𝐹, ℝ, < )))) β†’ 𝐻 βŠ† π‘˜)
14169, 139, 140reximssdv 3173 1 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ βˆƒπ‘˜ ∈ (𝑃 pSyl 𝐺)𝐻 βŠ† π‘˜)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  {crab 3433   βŠ† wss 3949   ⊊ wpss 3950  βˆ…c0 4323   class class class wbr 5149   ↦ cmpt 5232  ran crn 5678   Fn wfn 6539  β€˜cfv 6544  (class class class)co 7409   β‰Ό cdom 8937   β‰Ί csdm 8938  Fincfn 8939  supcsup 9435  β„cr 11109   < clt 11248   ≀ cle 11249  β„•0cn0 12472  β„€cz 12558  β™―chash 14290  β„™cprime 16608  Basecbs 17144   β†Ύs cress 17173  SubGrpcsubg 19000   pGrp cpgp 19394   pSyl cslw 19395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-fz 13485  df-hash 14291  df-subg 19003  df-pgp 19398  df-slw 19399
This theorem is referenced by:  slwn0  19483
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