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Theorem submod 19611
Description: The order of an element is the same in a submonoid. (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 766 . . . . . 6 (((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) ∧ 𝑥 ∈ ℕ) → 𝑌 ∈ (SubMnd‘𝐺))
2 nnnn0 12560 . . . . . . 7 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
32adantl 481 . . . . . 6 (((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ0)
4 simplr 768 . . . . . 6 (((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) ∧ 𝑥 ∈ ℕ) → 𝐴𝑌)
5 eqid 2740 . . . . . . 7 (.g𝐺) = (.g𝐺)
6 submod.h . . . . . . 7 𝐻 = (𝐺s 𝑌)
7 eqid 2740 . . . . . . 7 (.g𝐻) = (.g𝐻)
85, 6, 7submmulg 19158 . . . . . 6 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝑥 ∈ ℕ0𝐴𝑌) → (𝑥(.g𝐺)𝐴) = (𝑥(.g𝐻)𝐴))
91, 3, 4, 8syl3anc 1371 . . . . 5 (((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) ∧ 𝑥 ∈ ℕ) → (𝑥(.g𝐺)𝐴) = (𝑥(.g𝐻)𝐴))
10 eqid 2740 . . . . . . 7 (0g𝐺) = (0g𝐺)
116, 10subm0 18850 . . . . . 6 (𝑌 ∈ (SubMnd‘𝐺) → (0g𝐺) = (0g𝐻))
1211ad2antrr 725 . . . . 5 (((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) ∧ 𝑥 ∈ ℕ) → (0g𝐺) = (0g𝐻))
139, 12eqeq12d 2756 . . . 4 (((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) ∧ 𝑥 ∈ ℕ) → ((𝑥(.g𝐺)𝐴) = (0g𝐺) ↔ (𝑥(.g𝐻)𝐴) = (0g𝐻)))
1413rabbidva 3450 . . 3 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → {𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)} = {𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)})
15 eqeq1 2744 . . . 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 9545 . . . 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 4574 . . 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 2740 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
2019submss 18844 . . . 4 (𝑌 ∈ (SubMnd‘𝐺) → 𝑌 ⊆ (Base‘𝐺))
2120sselda 4008 . . 3 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → 𝐴 ∈ (Base‘𝐺))
22 submod.o . . . 4 𝑂 = (od‘𝐺)
23 eqid 2740 . . . 4 {𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)} = {𝑥 ∈ ℕ ∣ (𝑥(.g𝐺)𝐴) = (0g𝐺)}
2419, 5, 10, 22, 23odval 19576 . . 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 484 . . . 4 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → 𝐴𝑌)
2720adantr 480 . . . . 5 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → 𝑌 ⊆ (Base‘𝐺))
286, 19ressbas2 17296 . . . . 5 (𝑌 ⊆ (Base‘𝐺) → 𝑌 = (Base‘𝐻))
2927, 28syl 17 . . . 4 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → 𝑌 = (Base‘𝐻))
3026, 29eleqtrd 2846 . . 3 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → 𝐴 ∈ (Base‘𝐻))
31 eqid 2740 . . . 4 (Base‘𝐻) = (Base‘𝐻)
32 eqid 2740 . . . 4 (0g𝐻) = (0g𝐻)
33 submod.p . . . 4 𝑃 = (od‘𝐻)
34 eqid 2740 . . . 4 {𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)} = {𝑥 ∈ ℕ ∣ (𝑥(.g𝐻)𝐴) = (0g𝐻)}
3531, 7, 32, 33, 34odval 19576 . . 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 2790 1 ((𝑌 ∈ (SubMnd‘𝐺) ∧ 𝐴𝑌) → (𝑂𝐴) = (𝑃𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  {crab 3443  wss 3976  c0 4352  ifcif 4548  cfv 6573  (class class class)co 7448  infcinf 9510  cr 11183  0cc0 11184   < clt 11324  cn 12293  0cn0 12553  Basecbs 17258  s cress 17287  0gc0g 17499  SubMndcsubmnd 18817  .gcmg 19107  odcod 19566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-sup 9511  df-inf 9512  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-z 12640  df-uz 12904  df-seq 14053  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-submnd 18819  df-mulg 19108  df-od 19570
This theorem is referenced by:  subgod  19612  unitscyglem5  42156
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