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Theorem symgmatr01lem 21257
Description: Lemma for symgmatr01 21258. (Contributed by AV, 3-Jan-2019.)
Hypothesis
Ref Expression
symgmatr01.p 𝑃 = (Base‘(SymGrp‘𝑁))
Assertion
Ref Expression
symgmatr01lem ((𝐾𝑁𝐿𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → ∃𝑘𝑁 if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 𝐴, 𝐵), (𝑘𝑀(𝑄𝑘))) = 𝐵))
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘   𝑘,𝑞,𝐿   𝑘,𝐾,𝑞   𝑘,𝑀   𝑘,𝑁   𝑃,𝑘,𝑞   𝑄,𝑘,𝑞
Allowed substitution hints:   𝐴(𝑞)   𝐵(𝑞)   𝑀(𝑞)   𝑁(𝑞)

Proof of Theorem symgmatr01lem
StepHypRef Expression
1 simpll 766 . . 3 (((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) → 𝐾𝑁)
2 eqeq1 2828 . . . . . 6 (𝑘 = 𝐾 → (𝑘 = 𝐾𝐾 = 𝐾))
3 fveq2 6659 . . . . . . . 8 (𝑘 = 𝐾 → (𝑄𝑘) = (𝑄𝐾))
43eqeq1d 2826 . . . . . . 7 (𝑘 = 𝐾 → ((𝑄𝑘) = 𝐿 ↔ (𝑄𝐾) = 𝐿))
54ifbid 4472 . . . . . 6 (𝑘 = 𝐾 → if((𝑄𝑘) = 𝐿, 𝐴, 𝐵) = if((𝑄𝐾) = 𝐿, 𝐴, 𝐵))
6 id 22 . . . . . . 7 (𝑘 = 𝐾𝑘 = 𝐾)
76, 3oveq12d 7164 . . . . . 6 (𝑘 = 𝐾 → (𝑘𝑀(𝑄𝑘)) = (𝐾𝑀(𝑄𝐾)))
82, 5, 7ifbieq12d 4477 . . . . 5 (𝑘 = 𝐾 → if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 𝐴, 𝐵), (𝑘𝑀(𝑄𝑘))) = if(𝐾 = 𝐾, if((𝑄𝐾) = 𝐿, 𝐴, 𝐵), (𝐾𝑀(𝑄𝐾))))
98eqeq1d 2826 . . . 4 (𝑘 = 𝐾 → (if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 𝐴, 𝐵), (𝑘𝑀(𝑄𝑘))) = 𝐵 ↔ if(𝐾 = 𝐾, if((𝑄𝐾) = 𝐿, 𝐴, 𝐵), (𝐾𝑀(𝑄𝐾))) = 𝐵))
109adantl 485 . . 3 ((((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) ∧ 𝑘 = 𝐾) → (if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 𝐴, 𝐵), (𝑘𝑀(𝑄𝑘))) = 𝐵 ↔ if(𝐾 = 𝐾, if((𝑄𝐾) = 𝐿, 𝐴, 𝐵), (𝐾𝑀(𝑄𝐾))) = 𝐵))
11 eqidd 2825 . . . . 5 (((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) → 𝐾 = 𝐾)
1211iftrued 4458 . . . 4 (((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) → if(𝐾 = 𝐾, if((𝑄𝐾) = 𝐿, 𝐴, 𝐵), (𝐾𝑀(𝑄𝐾))) = if((𝑄𝐾) = 𝐿, 𝐴, 𝐵))
13 eldif 3929 . . . . . . 7 (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) ↔ (𝑄𝑃 ∧ ¬ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}))
14 ianor 979 . . . . . . . . . 10 (¬ (𝑄𝑃 ∧ (𝑄𝐾) = 𝐿) ↔ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿))
15 fveq1 6658 . . . . . . . . . . . 12 (𝑞 = 𝑄 → (𝑞𝐾) = (𝑄𝐾))
1615eqeq1d 2826 . . . . . . . . . . 11 (𝑞 = 𝑄 → ((𝑞𝐾) = 𝐿 ↔ (𝑄𝐾) = 𝐿))
1716elrab 3666 . . . . . . . . . 10 (𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿} ↔ (𝑄𝑃 ∧ (𝑄𝐾) = 𝐿))
1814, 17xchnxbir 336 . . . . . . . . 9 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿} ↔ (¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿))
19 pm2.21 123 . . . . . . . . . 10 𝑄𝑃 → (𝑄𝑃 → ¬ (𝑄𝐾) = 𝐿))
20 ax-1 6 . . . . . . . . . 10 (¬ (𝑄𝐾) = 𝐿 → (𝑄𝑃 → ¬ (𝑄𝐾) = 𝐿))
2119, 20jaoi 854 . . . . . . . . 9 ((¬ 𝑄𝑃 ∨ ¬ (𝑄𝐾) = 𝐿) → (𝑄𝑃 → ¬ (𝑄𝐾) = 𝐿))
2218, 21sylbi 220 . . . . . . . 8 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿} → (𝑄𝑃 → ¬ (𝑄𝐾) = 𝐿))
2322impcom 411 . . . . . . 7 ((𝑄𝑃 ∧ ¬ 𝑄 ∈ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → ¬ (𝑄𝐾) = 𝐿)
2413, 23sylbi 220 . . . . . 6 (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → ¬ (𝑄𝐾) = 𝐿)
2524adantl 485 . . . . 5 (((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) → ¬ (𝑄𝐾) = 𝐿)
2625iffalsed 4461 . . . 4 (((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) → if((𝑄𝐾) = 𝐿, 𝐴, 𝐵) = 𝐵)
2712, 26eqtrd 2859 . . 3 (((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) → if(𝐾 = 𝐾, if((𝑄𝐾) = 𝐿, 𝐴, 𝐵), (𝐾𝑀(𝑄𝐾))) = 𝐵)
281, 10, 27rspcedvd 3612 . 2 (((𝐾𝑁𝐿𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿})) → ∃𝑘𝑁 if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 𝐴, 𝐵), (𝑘𝑀(𝑄𝑘))) = 𝐵)
2928ex 416 1 ((𝐾𝑁𝐿𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → ∃𝑘𝑁 if(𝑘 = 𝐾, if((𝑄𝑘) = 𝐿, 𝐴, 𝐵), (𝑘𝑀(𝑄𝑘))) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2115  wrex 3134  {crab 3137  cdif 3916  ifcif 4450  cfv 6344  (class class class)co 7146  Basecbs 16481  SymGrpcsymg 18493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-iota 6303  df-fv 6352  df-ov 7149
This theorem is referenced by:  symgmatr01  21258
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