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Theorem submod 19270
Description: The order of an element is the same in a subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by AV, 5-Oct-2020.)
Hypotheses
Ref Expression
submod.h 𝐻 = (𝐺s 𝑌)
submod.o 𝑂 = (od‘𝐺)
submod.p 𝑃 = (od‘𝐻)
Assertion
Ref Expression
submod ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → (𝑂𝐴) = (𝑃𝐴))

Proof of Theorem submod
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpll 764 . . . . . 6 (((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) ∧ 𝑥 ∈ ℕ) → 𝑌 ∈ (SubMnd‘𝐺))
2 nnnn0 12341 . . . . . . 7 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
32adantl 482 . . . . . 6 (((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ0)
4 simplr 766 . . . . . 6 (((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) ∧ 𝑥 ∈ ℕ) → 𝐴𝑌)
5 eqid 2736 . . . . . . 7 (.g𝐺) = (.g𝐺)
6 submod.h . . . . . . 7 𝐻 = (𝐺s 𝑌)
7 eqid 2736 . . . . . . 7 (.g𝐻) = (.g𝐻)
85, 6, 7submmulg 18843 . . . . . 6 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝑥 ∈ ℕ0𝐴𝑌) → (𝑥(.g𝐺)𝐴) = (𝑥(.g𝐻)𝐴))
91, 3, 4, 8syl3anc 1370 . . . . 5 (((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) ∧ 𝑥 ∈ ℕ) → (𝑥(.g𝐺)𝐴) = (𝑥(.g𝐻)𝐴))
10 eqid 2736 . . . . . . 7 (0g𝐺) = (0g𝐺)
116, 10subm0 18551 . . . . . 6 (𝑌 ∈ (SubMnd‘𝐺) → (0g𝐺) = (0g𝐻))
1211ad2antrr 723 . . . . 5 (((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) ∧ 𝑥 ∈ ℕ) → (0g𝐺) = (0g𝐻))
139, 12eqeq12d 2752 . . . 4 (((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) ∧ 𝑥 ∈ ℕ) → ((𝑥(.g𝐺)𝐴) = (0g𝐺) ↔ (𝑥(.g𝐻)𝐴) = (0g𝐻)))
1413rabbidva 3410 . . 3 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → {𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)} = {𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)})
15 eqeq1 2740 . . . 4 ({𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)} = {𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)} → ({𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)} = ∅ ↔ {𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)} = ∅))
16 infeq1 9333 . . . 4 ({𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)} = {𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)} → inf({𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)}, ℝ, < ) = inf({𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)}, ℝ, < ))
1715, 16ifbieq2d 4499 . . 3 ({𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)} = {𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)} → if({𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)} = ∅, 0, inf({𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)}, ℝ, < )) = if({𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)} = ∅, 0, inf({𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)}, ℝ, < )))
1814, 17syl 17 . 2 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → if({𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)} = ∅, 0, inf({𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)}, ℝ, < )) = if({𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)} = ∅, 0, inf({𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)}, ℝ, < )))
19 eqid 2736 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
2019submss 18545 . . . 4 (𝑌 ∈ (SubMnd‘𝐺) → 𝑌 ⊆ (Base‘𝐺))
2120sselda 3932 . . 3 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → 𝐴 ∈ (Base‘𝐺))
22 submod.o . . . 4 𝑂 = (od‘𝐺)
23 eqid 2736 . . . 4 {𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)} = {𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)}
2419, 5, 10, 22, 23odval 19238 . . 3 (𝐴 ∈ (Base‘𝐺) → (𝑂𝐴) = if({𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)} = ∅, 0, inf({𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)}, ℝ, < )))
2521, 24syl 17 . 2 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → (𝑂𝐴) = if({𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)} = ∅, 0, inf({𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)}, ℝ, < )))
26 simpr 485 . . . 4 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → 𝐴𝑌)
2720adantr 481 . . . . 5 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → 𝑌 ⊆ (Base‘𝐺))
286, 19ressbas2 17046 . . . . 5 (𝑌 ⊆ (Base‘𝐺) → 𝑌 = (Base‘𝐻))
2927, 28syl 17 . . . 4 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → 𝑌 = (Base‘𝐻))
3026, 29eleqtrd 2839 . . 3 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → 𝐴 ∈ (Base‘𝐻))
31 eqid 2736 . . . 4 (Base‘𝐻) = (Base‘𝐻)
32 eqid 2736 . . . 4 (0g𝐻) = (0g𝐻)
33 submod.p . . . 4 𝑃 = (od‘𝐻)
34 eqid 2736 . . . 4 {𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)} = {𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)}
3531, 7, 32, 33, 34odval 19238 . . 3 (𝐴 ∈ (Base‘𝐻) → (𝑃𝐴) = if({𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)} = ∅, 0, inf({𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)}, ℝ, < )))
3630, 35syl 17 . 2 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → (𝑃𝐴) = if({𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)} = ∅, 0, inf({𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)}, ℝ, < )))
3718, 25, 363eqtr4d 2786 1 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → (𝑂𝐴) = (𝑃𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  {crab 3403  wss 3898  c0 4269  ifcif 4473  cfv 6479  (class class class)co 7337  infcinf 9298  cr 10971  0cc0 10972   < clt 11110  cn 12074  0cn0 12334  Basecbs 17009  s cress 17038  0gc0g 17247  SubMndcsubmnd 18526  .gcmg 18796  odcod 19228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-1st 7899  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-er 8569  df-en 8805  df-dom 8806  df-sdom 8807  df-sup 9299  df-inf 9300  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-nn 12075  df-2 12137  df-n0 12335  df-z 12421  df-uz 12684  df-seq 13823  df-sets 16962  df-slot 16980  df-ndx 16992  df-base 17010  df-ress 17039  df-plusg 17072  df-0g 17249  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-submnd 18528  df-mulg 18797  df-od 19232
This theorem is referenced by:  subgod  19271
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