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Theorem subgpgp 17835
 Description: A subgroup of a p-group is a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.)
Assertion
Ref Expression
subgpgp ((𝑃 pGrp 𝐺𝑆 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺s 𝑆))

Proof of Theorem subgpgp
Dummy variables 𝑥 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pgpprm 17831 . . 3 (𝑃 pGrp 𝐺𝑃 ∈ ℙ)
21adantr 480 . 2 ((𝑃 pGrp 𝐺𝑆 ∈ (SubGrp‘𝐺)) → 𝑃 ∈ ℙ)
3 eqid 2610 . . . 4 (𝐺s 𝑆) = (𝐺s 𝑆)
43subggrp 17420 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → (𝐺s 𝑆) ∈ Grp)
54adantl 481 . 2 ((𝑃 pGrp 𝐺𝑆 ∈ (SubGrp‘𝐺)) → (𝐺s 𝑆) ∈ Grp)
6 eqid 2610 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
7 eqid 2610 . . . . . . 7 (od‘𝐺) = (od‘𝐺)
86, 7ispgp 17830 . . . . . 6 (𝑃 pGrp 𝐺 ↔ (𝑃 ∈ ℙ ∧ 𝐺 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑛)))
98simp3bi 1071 . . . . 5 (𝑃 pGrp 𝐺 → ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑛))
109adantr 480 . . . 4 ((𝑃 pGrp 𝐺𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑛))
116subgss 17418 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
1211adantl 481 . . . . . 6 ((𝑃 pGrp 𝐺𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
13 ssralv 3629 . . . . . 6 (𝑆 ⊆ (Base‘𝐺) → (∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑛) → ∀𝑥𝑆𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑛)))
1412, 13syl 17 . . . . 5 ((𝑃 pGrp 𝐺𝑆 ∈ (SubGrp‘𝐺)) → (∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑛) → ∀𝑥𝑆𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑛)))
15 eqid 2610 . . . . . . . . . 10 (od‘(𝐺s 𝑆)) = (od‘(𝐺s 𝑆))
163, 7, 15subgod 17808 . . . . . . . . 9 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆) → ((od‘𝐺)‘𝑥) = ((od‘(𝐺s 𝑆))‘𝑥))
1716adantll 746 . . . . . . . 8 (((𝑃 pGrp 𝐺𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑆) → ((od‘𝐺)‘𝑥) = ((od‘(𝐺s 𝑆))‘𝑥))
1817eqeq1d 2612 . . . . . . 7 (((𝑃 pGrp 𝐺𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑆) → (((od‘𝐺)‘𝑥) = (𝑃𝑛) ↔ ((od‘(𝐺s 𝑆))‘𝑥) = (𝑃𝑛)))
1918rexbidv 3034 . . . . . 6 (((𝑃 pGrp 𝐺𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑆) → (∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑛) ↔ ∃𝑛 ∈ ℕ0 ((od‘(𝐺s 𝑆))‘𝑥) = (𝑃𝑛)))
2019ralbidva 2968 . . . . 5 ((𝑃 pGrp 𝐺𝑆 ∈ (SubGrp‘𝐺)) → (∀𝑥𝑆𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑛) ↔ ∀𝑥𝑆𝑛 ∈ ℕ0 ((od‘(𝐺s 𝑆))‘𝑥) = (𝑃𝑛)))
2114, 20sylibd 228 . . . 4 ((𝑃 pGrp 𝐺𝑆 ∈ (SubGrp‘𝐺)) → (∀𝑥 ∈ (Base‘𝐺)∃𝑛 ∈ ℕ0 ((od‘𝐺)‘𝑥) = (𝑃𝑛) → ∀𝑥𝑆𝑛 ∈ ℕ0 ((od‘(𝐺s 𝑆))‘𝑥) = (𝑃𝑛)))
2210, 21mpd 15 . . 3 ((𝑃 pGrp 𝐺𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑥𝑆𝑛 ∈ ℕ0 ((od‘(𝐺s 𝑆))‘𝑥) = (𝑃𝑛))
233subgbas 17421 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘(𝐺s 𝑆)))
2423adantl 481 . . . 4 ((𝑃 pGrp 𝐺𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 = (Base‘(𝐺s 𝑆)))
2524raleqdv 3121 . . 3 ((𝑃 pGrp 𝐺𝑆 ∈ (SubGrp‘𝐺)) → (∀𝑥𝑆𝑛 ∈ ℕ0 ((od‘(𝐺s 𝑆))‘𝑥) = (𝑃𝑛) ↔ ∀𝑥 ∈ (Base‘(𝐺s 𝑆))∃𝑛 ∈ ℕ0 ((od‘(𝐺s 𝑆))‘𝑥) = (𝑃𝑛)))
2622, 25mpbid 221 . 2 ((𝑃 pGrp 𝐺𝑆 ∈ (SubGrp‘𝐺)) → ∀𝑥 ∈ (Base‘(𝐺s 𝑆))∃𝑛 ∈ ℕ0 ((od‘(𝐺s 𝑆))‘𝑥) = (𝑃𝑛))
27 eqid 2610 . . 3 (Base‘(𝐺s 𝑆)) = (Base‘(𝐺s 𝑆))
2827, 15ispgp 17830 . 2 (𝑃 pGrp (𝐺s 𝑆) ↔ (𝑃 ∈ ℙ ∧ (𝐺s 𝑆) ∈ Grp ∧ ∀𝑥 ∈ (Base‘(𝐺s 𝑆))∃𝑛 ∈ ℕ0 ((od‘(𝐺s 𝑆))‘𝑥) = (𝑃𝑛)))
292, 5, 26, 28syl3anbrc 1239 1 ((𝑃 pGrp 𝐺𝑆 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺s 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℕ0cn0 11169  ↑cexp 12722  ℙcprime 15223  Basecbs 15695   ↾s cress 15696  Grpcgrp 17245  SubGrpcsubg 17411  odcod 17767   pGrp cpgp 17769 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-seq 12664  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-mulg 17364  df-subg 17414  df-od 17771  df-pgp 17773 This theorem is referenced by:  pgpfaclem1  18303  pgpfaclem3  18305
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