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Theorem submod 17807
 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 786 . . . . . 6 (((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) ∧ 𝑥 ∈ ℕ) → 𝑌 ∈ (SubMnd‘𝐺))
2 nnnn0 11176 . . . . . . 7 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
32adantl 481 . . . . . 6 (((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ0)
4 simplr 788 . . . . . 6 (((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) ∧ 𝑥 ∈ ℕ) → 𝐴𝑌)
5 eqid 2610 . . . . . . 7 (.g𝐺) = (.g𝐺)
6 submod.h . . . . . . 7 𝐻 = (𝐺s 𝑌)
7 eqid 2610 . . . . . . 7 (.g𝐻) = (.g𝐻)
85, 6, 7submmulg 17409 . . . . . 6 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝑥 ∈ ℕ0𝐴𝑌) → (𝑥(.g𝐺)𝐴) = (𝑥(.g𝐻)𝐴))
91, 3, 4, 8syl3anc 1318 . . . . 5 (((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) ∧ 𝑥 ∈ ℕ) → (𝑥(.g𝐺)𝐴) = (𝑥(.g𝐻)𝐴))
10 eqid 2610 . . . . . . 7 (0g𝐺) = (0g𝐺)
116, 10subm0 17179 . . . . . 6 (𝑌 ∈ (SubMnd‘𝐺) → (0g𝐺) = (0g𝐻))
1211ad2antrr 758 . . . . 5 (((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) ∧ 𝑥 ∈ ℕ) → (0g𝐺) = (0g𝐻))
139, 12eqeq12d 2625 . . . 4 (((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) ∧ 𝑥 ∈ ℕ) → ((𝑥(.g𝐺)𝐴) = (0g𝐺) ↔ (𝑥(.g𝐻)𝐴) = (0g𝐻)))
1413rabbidva 3163 . . 3 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → {𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)} = {𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)})
15 eqeq1 2614 . . . 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 8265 . . . 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 4061 . . 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 2610 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
2019submss 17173 . . . 4 (𝑌 ∈ (SubMnd‘𝐺) → 𝑌 ⊆ (Base‘𝐺))
2120sselda 3568 . . 3 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → 𝐴 ∈ (Base‘𝐺))
22 submod.o . . . 4 𝑂 = (od‘𝐺)
23 eqid 2610 . . . 4 {𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)} = {𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)}
2419, 5, 10, 22, 23odval 17776 . . 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 476 . . . 4 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → 𝐴𝑌)
2720adantr 480 . . . . 5 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → 𝑌 ⊆ (Base‘𝐺))
286, 19ressbas2 15758 . . . . 5 (𝑌 ⊆ (Base‘𝐺) → 𝑌 = (Base‘𝐻))
2927, 28syl 17 . . . 4 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → 𝑌 = (Base‘𝐻))
3026, 29eleqtrd 2690 . . 3 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → 𝐴 ∈ (Base‘𝐻))
31 eqid 2610 . . . 4 (Base‘𝐻) = (Base‘𝐻)
32 eqid 2610 . . . 4 (0g𝐻) = (0g𝐻)
33 submod.p . . . 4 𝑃 = (od‘𝐻)
34 eqid 2610 . . . 4 {𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)} = {𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)}
3531, 7, 32, 33, 34odval 17776 . . 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 2654 1 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → (𝑂𝐴) = (𝑃𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900   ⊆ wss 3540  ∅c0 3874  ifcif 4036  ‘cfv 5804  (class class class)co 6549  infcinf 8230  ℝcr 9814  0cc0 9815   < clt 9953  ℕcn 10897  ℕ0cn0 11169  Basecbs 15695   ↾s cress 15696  0gc0g 15923  SubMndcsubmnd 17157  .gcmg 17363  odcod 17767 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-mulg 17364  df-od 17771 This theorem is referenced by:  subgod  17808
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