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

Theorem mgm2nsgrplem2 18165
 Description: Lemma 2 for mgm2nsgrp 18168. (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 4657 . . 3 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
2 mgm2nsgrp.s . . 3 𝑆 = {𝐴, 𝐵}
31, 2eleqtrrdi 2864 . 2 (𝐴𝑉𝐴𝑆)
4 prid2g 4658 . . 3 (𝐵𝑊𝐵 ∈ {𝐴, 𝐵})
54, 2eleqtrrdi 2864 . 2 (𝐵𝑊𝐵𝑆)
6 mgm2nsgrp.p . . . . 5 = (+g𝑀)
7 mgm2nsgrp.o . . . . 5 (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))
86, 7eqtri 2782 . . . 4 = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))
98a1i 11 . . 3 ((𝐴𝑆𝐵𝑆) → = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴)))
10 ifeq1 4428 . . . . . . 7 (𝐵 = 𝐴 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = if((𝑥 = 𝐴𝑦 = 𝐴), 𝐴, 𝐴))
11 ifid 4464 . . . . . . 7 if((𝑥 = 𝐴𝑦 = 𝐴), 𝐴, 𝐴) = 𝐴
1210, 11eqtrdi 2810 . . . . . 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 2763 . . . . . . . . . . 11 (𝑦 = 𝐵 → (𝑦 = 𝐴𝐵 = 𝐴))
1514bicomd 226 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝐵 = 𝐴𝑦 = 𝐴))
1615notbid 321 . . . . . . . . 9 (𝑦 = 𝐵 → (¬ 𝐵 = 𝐴 ↔ ¬ 𝑦 = 𝐴))
1716biimpac 482 . . . . . . . 8 ((¬ 𝐵 = 𝐴𝑦 = 𝐵) → ¬ 𝑦 = 𝐴)
1817intnand 492 . . . . . . 7 ((¬ 𝐵 = 𝐴𝑦 = 𝐵) → ¬ (𝑥 = 𝐴𝑦 = 𝐴))
1918iffalsed 4435 . . . . . 6 ((¬ 𝐵 = 𝐴𝑦 = 𝐵) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
2019ex 416 . . . . 5 𝐵 = 𝐴 → (𝑦 = 𝐵 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴))
2113, 20pm2.61i 185 . . . 4 (𝑦 = 𝐵 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
2221ad2antll 728 . . 3 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = (𝐴 𝐴) ∧ 𝑦 = 𝐵)) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
23 iftrue 4430 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐴) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐵)
2423adantl 485 . . . . 5 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐵)
25 simpl 486 . . . . 5 ((𝐴𝑆𝐵𝑆) → 𝐴𝑆)
26 simpr 488 . . . . 5 ((𝐴𝑆𝐵𝑆) → 𝐵𝑆)
279, 24, 25, 25, 26ovmpod 7304 . . . 4 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐴) = 𝐵)
2827, 26eqeltrd 2853 . . 3 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐴) ∈ 𝑆)
299, 22, 28, 26, 25ovmpod 7304 . 2 ((𝐴𝑆𝐵𝑆) → ((𝐴 𝐴) 𝐵) = 𝐴)
303, 5, 29syl2an 598 1 ((𝐴𝑉𝐵𝑊) → ((𝐴 𝐴) 𝐵) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1539   ∈ wcel 2112  ifcif 4424  {cpr 4528  ‘cfv 6341  (class class class)co 7157   ∈ cmpo 7159  Basecbs 16556  +gcplusg 16638 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3700  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-br 5038  df-opab 5100  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-iota 6300  df-fun 6343  df-fv 6349  df-ov 7160  df-oprab 7161  df-mpo 7162 This theorem is referenced by:  mgm2nsgrplem4  18167
 Copyright terms: Public domain W3C validator