MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mgm2nsgrplem2 Structured version   Visualization version   GIF version

Theorem mgm2nsgrplem2 18945
Description: Lemma 2 for mgm2nsgrp 18948. (Contributed by AV, 27-Jan-2020.)
Hypotheses
Ref Expression
mgm2nsgrp.s 𝑆 = {𝐴, 𝐵}
mgm2nsgrp.b (Base‘𝑀) = 𝑆
mgm2nsgrp.o (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))
mgm2nsgrp.p = (+g𝑀)
Assertion
Ref Expression
mgm2nsgrplem2 ((𝐴𝑉𝐵𝑊) → ((𝐴 𝐴) 𝐵) = 𝐴)
Distinct variable groups:   𝑥,𝑆,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑀   𝑥, ,𝑦
Allowed substitution hints:   𝑀(𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem mgm2nsgrplem2
StepHypRef Expression
1 prid1g 4765 . . 3 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
2 mgm2nsgrp.s . . 3 𝑆 = {𝐴, 𝐵}
31, 2eleqtrrdi 2850 . 2 (𝐴𝑉𝐴𝑆)
4 prid2g 4766 . . 3 (𝐵𝑊𝐵 ∈ {𝐴, 𝐵})
54, 2eleqtrrdi 2850 . 2 (𝐵𝑊𝐵𝑆)
6 mgm2nsgrp.p . . . . 5 = (+g𝑀)
7 mgm2nsgrp.o . . . . 5 (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))
86, 7eqtri 2763 . . . 4 = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))
98a1i 11 . . 3 ((𝐴𝑆𝐵𝑆) → = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴)))
10 ifeq1 4535 . . . . . . 7 (𝐵 = 𝐴 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = if((𝑥 = 𝐴𝑦 = 𝐴), 𝐴, 𝐴))
11 ifid 4571 . . . . . . 7 if((𝑥 = 𝐴𝑦 = 𝐴), 𝐴, 𝐴) = 𝐴
1210, 11eqtrdi 2791 . . . . . 6 (𝐵 = 𝐴 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
1312a1d 25 . . . . 5 (𝐵 = 𝐴 → (𝑦 = 𝐵 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴))
14 eqeq1 2739 . . . . . . . . . . 11 (𝑦 = 𝐵 → (𝑦 = 𝐴𝐵 = 𝐴))
1514bicomd 223 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝐵 = 𝐴𝑦 = 𝐴))
1615notbid 318 . . . . . . . . 9 (𝑦 = 𝐵 → (¬ 𝐵 = 𝐴 ↔ ¬ 𝑦 = 𝐴))
1716biimpac 478 . . . . . . . 8 ((¬ 𝐵 = 𝐴𝑦 = 𝐵) → ¬ 𝑦 = 𝐴)
1817intnand 488 . . . . . . 7 ((¬ 𝐵 = 𝐴𝑦 = 𝐵) → ¬ (𝑥 = 𝐴𝑦 = 𝐴))
1918iffalsed 4542 . . . . . 6 ((¬ 𝐵 = 𝐴𝑦 = 𝐵) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
2019ex 412 . . . . 5 𝐵 = 𝐴 → (𝑦 = 𝐵 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴))
2113, 20pm2.61i 182 . . . 4 (𝑦 = 𝐵 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
2221ad2antll 729 . . 3 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = (𝐴 𝐴) ∧ 𝑦 = 𝐵)) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
23 iftrue 4537 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐴) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐵)
2423adantl 481 . . . . 5 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐵)
25 simpl 482 . . . . 5 ((𝐴𝑆𝐵𝑆) → 𝐴𝑆)
26 simpr 484 . . . . 5 ((𝐴𝑆𝐵𝑆) → 𝐵𝑆)
279, 24, 25, 25, 26ovmpod 7585 . . . 4 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐴) = 𝐵)
2827, 26eqeltrd 2839 . . 3 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐴) ∈ 𝑆)
299, 22, 28, 26, 25ovmpod 7585 . 2 ((𝐴𝑆𝐵𝑆) → ((𝐴 𝐴) 𝐵) = 𝐴)
303, 5, 29syl2an 596 1 ((𝐴𝑉𝐵𝑊) → ((𝐴 𝐴) 𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  ifcif 4531  {cpr 4633  cfv 6563  (class class class)co 7431  cmpo 7433  Basecbs 17245  +gcplusg 17298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  mgm2nsgrplem4  18947
  Copyright terms: Public domain W3C validator