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Theorem mgm2nsgrplem2 18876
Description: Lemma 2 for mgm2nsgrp 18879. (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 4767 . . 3 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
2 mgm2nsgrp.s . . 3 𝑆 = {𝐴, 𝐵}
31, 2eleqtrrdi 2839 . 2 (𝐴𝑉𝐴𝑆)
4 prid2g 4768 . . 3 (𝐵𝑊𝐵 ∈ {𝐴, 𝐵})
54, 2eleqtrrdi 2839 . 2 (𝐵𝑊𝐵𝑆)
6 mgm2nsgrp.p . . . . 5 = (+g𝑀)
7 mgm2nsgrp.o . . . . 5 (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))
86, 7eqtri 2755 . . . 4 = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))
98a1i 11 . . 3 ((𝐴𝑆𝐵𝑆) → = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴)))
10 ifeq1 4534 . . . . . . 7 (𝐵 = 𝐴 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = if((𝑥 = 𝐴𝑦 = 𝐴), 𝐴, 𝐴))
11 ifid 4570 . . . . . . 7 if((𝑥 = 𝐴𝑦 = 𝐴), 𝐴, 𝐴) = 𝐴
1210, 11eqtrdi 2783 . . . . . 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 2731 . . . . . . . . . . 11 (𝑦 = 𝐵 → (𝑦 = 𝐴𝐵 = 𝐴))
1514bicomd 222 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝐵 = 𝐴𝑦 = 𝐴))
1615notbid 317 . . . . . . . . 9 (𝑦 = 𝐵 → (¬ 𝐵 = 𝐴 ↔ ¬ 𝑦 = 𝐴))
1716biimpac 477 . . . . . . . 8 ((¬ 𝐵 = 𝐴𝑦 = 𝐵) → ¬ 𝑦 = 𝐴)
1817intnand 487 . . . . . . 7 ((¬ 𝐵 = 𝐴𝑦 = 𝐵) → ¬ (𝑥 = 𝐴𝑦 = 𝐴))
1918iffalsed 4541 . . . . . 6 ((¬ 𝐵 = 𝐴𝑦 = 𝐵) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
2019ex 411 . . . . 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 727 . . 3 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = (𝐴 𝐴) ∧ 𝑦 = 𝐵)) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
23 iftrue 4536 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐴) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐵)
2423adantl 480 . . . . 5 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = 𝐴)) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐵)
25 simpl 481 . . . . 5 ((𝐴𝑆𝐵𝑆) → 𝐴𝑆)
26 simpr 483 . . . . 5 ((𝐴𝑆𝐵𝑆) → 𝐵𝑆)
279, 24, 25, 25, 26ovmpod 7577 . . . 4 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐴) = 𝐵)
2827, 26eqeltrd 2828 . . 3 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐴) ∈ 𝑆)
299, 22, 28, 26, 25ovmpod 7577 . 2 ((𝐴𝑆𝐵𝑆) → ((𝐴 𝐴) 𝐵) = 𝐴)
303, 5, 29syl2an 594 1 ((𝐴𝑉𝐵𝑊) → ((𝐴 𝐴) 𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  ifcif 4530  {cpr 4632  cfv 6551  (class class class)co 7424  cmpo 7426  Basecbs 17185  +gcplusg 17238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-iota 6503  df-fun 6553  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429
This theorem is referenced by:  mgm2nsgrplem4  18878
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