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Theorem pgpssslw 19477
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 3677 . . . . . . . . . . 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 19002 . . . . . . . . . . 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 9170 . . . . . . . . . 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 14313 . . . . . . . . 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 12581 . . . . . . 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 7111 . . . . . 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 6723 . . . . 5 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ ran 𝐹 βŠ† β„€)
14 fvex 6902 . . . . . . . 8 (β™―β€˜π‘₯) ∈ V
1514, 11fnmpti 6691 . . . . . . 7 𝐹 Fn {𝑦 ∈ (SubGrpβ€˜πΊ) ∣ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ∧ 𝐻 βŠ† 𝑦)}
16 eqimss2 4041 . . . . . . . . . 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 7414 . . . . . . . . . . 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 5160 . . . . . . . . 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 3685 . . . . . . 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 7080 . . . . . . 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 4335 . . . . 5 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ ran 𝐹 β‰  βˆ…)
29 hashcl 14313 . . . . . . . 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 12530 . . . . . 6 ((𝐻 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) β†’ (β™―β€˜π‘‹) ∈ ℝ)
32 fveq2 6889 . . . . . . . . . . 11 (π‘₯ = π‘š β†’ (β™―β€˜π‘₯) = (β™―β€˜π‘š))
33 fvex 6902 . . . . . . . . . . 11 (β™―β€˜π‘š) ∈ V
3432, 11, 33fvmpt 6996 . . . . . . . . . 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 7414 . . . . . . . . . . . . 13 (𝑦 = π‘š β†’ (𝐺 β†Ύs 𝑦) = (𝐺 β†Ύs π‘š))
3736breq2d 5160 . . . . . . . . . . . 12 (𝑦 = π‘š β†’ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ↔ 𝑃 pGrp (𝐺 β†Ύs π‘š)))
38 sseq2 4008 . . . . . . . . . . . 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 3683 . . . . . . . . . 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 19002 . . . . . . . . . . . . 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 8993 . . . . . . . . . . . 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 9264 . . . . . . . . . . . 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 14336 . . . . . . . . . . . 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 5170 . . . . . . . 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 5151 . . . . . . . . 9 (𝑀 = (πΉβ€˜π‘š) β†’ (𝑀 ≀ (β™―β€˜π‘‹) ↔ (πΉβ€˜π‘š) ≀ (β™―β€˜π‘‹)))
5453ralrn 7087 . . . . . . . 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 5208 . . . . . 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 12639 . . . . 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 6950 . . . . 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 7414 . . . . . 6 (𝑦 = π‘˜ β†’ (𝐺 β†Ύs 𝑦) = (𝐺 β†Ύs π‘˜))
6564breq2d 5160 . . . . 5 (𝑦 = π‘˜ β†’ (𝑃 pGrp (𝐺 β†Ύs 𝑦) ↔ 𝑃 pGrp (𝐺 β†Ύs π‘˜)))
66 sseq2 4008 . . . . 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 3692 . . 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 19456 . . . 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 12562 . . . . . . . . . . . . 13 β„€ βŠ† ℝ
7513, 74sstrdi 3994 . . . . . . . . . . . 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 3992 . . . . . . . . . . . . . . 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 3685 . . . . . . . . . . . . 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 7080 . . . . . . . . . . . . 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 12173 . . . . . . . . . 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 3685 . . . . . . . . . . . 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 6889 . . . . . . . . . . . . 13 (π‘₯ = π‘˜ β†’ (β™―β€˜π‘₯) = (β™―β€˜π‘˜))
98 fvex 6902 . . . . . . . . . . . . 13 (β™―β€˜π‘˜) ∈ V
9997, 11, 98fvmpt 6996 . . . . . . . . . . . 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 5174 . . . . . . . . 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 9264 . . . . . . . . . 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 9264 . . . . . . . . . 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 14313 . . . . . . . . . . 11 (π‘š ∈ Fin β†’ (β™―β€˜π‘š) ∈ β„•0)
108 hashcl 14313 . . . . . . . . . . 11 (π‘˜ ∈ Fin β†’ (β™―β€˜π‘˜) ∈ β„•0)
109 nn0re 12478 . . . . . . . . . . . 12 ((β™―β€˜π‘š) ∈ β„•0 β†’ (β™―β€˜π‘š) ∈ ℝ)
110 nn0re 12478 . . . . . . . . . . . 12 ((β™―β€˜π‘˜) ∈ β„•0 β†’ (β™―β€˜π‘˜) ∈ ℝ)
111 lenlt 11289 . . . . . . . . . . . 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 9209 . . . . . . . . . . 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 14338 . . . . . . . . . 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 4099 . . . . . . . . 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 7414 . . . . . . . 8 (π‘˜ = π‘š β†’ (𝐺 β†Ύs π‘˜) = (𝐺 β†Ύs π‘š))
131130breq2d 5160 . . . . . . 7 (π‘˜ = π‘š β†’ (𝑃 pGrp (𝐺 β†Ύs π‘˜) ↔ 𝑃 pGrp (𝐺 β†Ύs π‘š)))
132 eqimss 4040 . . . . . . . 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 19471 . . 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 3948   ⊊ wpss 3949  βˆ…c0 4322   class class class wbr 5148   ↦ cmpt 5231  ran crn 5677   Fn wfn 6536  β€˜cfv 6541  (class class class)co 7406   β‰Ό cdom 8934   β‰Ί csdm 8935  Fincfn 8936  supcsup 9432  β„cr 11106   < clt 11245   ≀ cle 11246  β„•0cn0 12469  β„€cz 12555  β™―chash 14287  β„™cprime 16605  Basecbs 17141   β†Ύs cress 17170  SubGrpcsubg 18995   pGrp cpgp 19389   pSyl cslw 19390
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 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185
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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-oadd 8467  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-sup 9434  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-n0 12470  df-xnn0 12542  df-z 12556  df-uz 12820  df-fz 13482  df-hash 14288  df-subg 18998  df-pgp 19393  df-slw 19394
This theorem is referenced by:  slwn0  19478
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