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Theorem mgm2nsgrplem3 17529
Description: Lemma 3 for mgm2nsgrp 17531. (Contributed by AV, 28-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
mgm2nsgrplem3 ((𝐴𝑉𝐵𝑊) → (𝐴 (𝐴 𝐵)) = 𝐵)
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 mgm2nsgrplem3
StepHypRef Expression
1 prid1g 4402 . . 3 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
2 mgm2nsgrp.s . . 3 𝑆 = {𝐴, 𝐵}
31, 2syl6eleqr 2814 . 2 (𝐴𝑉𝐴𝑆)
4 prid2g 4403 . . 3 (𝐵𝑊𝐵 ∈ {𝐴, 𝐵})
54, 2syl6eleqr 2814 . 2 (𝐵𝑊𝐵𝑆)
6 mgm2nsgrp.p . . . . 5 = (+g𝑀)
7 mgm2nsgrp.o . . . . 5 (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))
86, 7eqtri 2746 . . . 4 = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴))
98a1i 11 . . 3 ((𝐴𝑆𝐵𝑆) → = (𝑥𝑆, 𝑦𝑆 ↦ if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴)))
10 simprl 811 . . . . 5 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = (𝐴 𝐵))) → 𝑥 = 𝐴)
11 simpr 479 . . . . . 6 ((𝑥 = 𝐴𝑦 = (𝐴 𝐵)) → 𝑦 = (𝐴 𝐵))
12 ifeq1 4198 . . . . . . . . . . 11 (𝐵 = 𝐴 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = if((𝑥 = 𝐴𝑦 = 𝐴), 𝐴, 𝐴))
13 ifid 4233 . . . . . . . . . . 11 if((𝑥 = 𝐴𝑦 = 𝐴), 𝐴, 𝐴) = 𝐴
1412, 13syl6eq 2774 . . . . . . . . . 10 (𝐵 = 𝐴 → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
1514a1d 25 . . . . . . . . 9 (𝐵 = 𝐴 → ((𝑥 = 𝐴𝑦 = 𝐵) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴))
16 eqeq1 2728 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝐵 → (𝑦 = 𝐴𝐵 = 𝐴))
1716biimpcd 239 . . . . . . . . . . . . . . . 16 (𝑦 = 𝐴 → (𝑦 = 𝐵𝐵 = 𝐴))
1817adantl 473 . . . . . . . . . . . . . . 15 ((𝑥 = 𝐴𝑦 = 𝐴) → (𝑦 = 𝐵𝐵 = 𝐴))
1918com12 32 . . . . . . . . . . . . . 14 (𝑦 = 𝐵 → ((𝑥 = 𝐴𝑦 = 𝐴) → 𝐵 = 𝐴))
2019adantl 473 . . . . . . . . . . . . 13 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥 = 𝐴𝑦 = 𝐴) → 𝐵 = 𝐴))
2120con3d 148 . . . . . . . . . . . 12 ((𝑥 = 𝐴𝑦 = 𝐵) → (¬ 𝐵 = 𝐴 → ¬ (𝑥 = 𝐴𝑦 = 𝐴)))
2221impcom 445 . . . . . . . . . . 11 ((¬ 𝐵 = 𝐴 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ¬ (𝑥 = 𝐴𝑦 = 𝐴))
2322iffalsed 4205 . . . . . . . . . 10 ((¬ 𝐵 = 𝐴 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
2423ex 449 . . . . . . . . 9 𝐵 = 𝐴 → ((𝑥 = 𝐴𝑦 = 𝐵) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴))
2515, 24pm2.61i 176 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
2625adantl 473 . . . . . . 7 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐴)
27 simpl 474 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → 𝐴𝑆)
28 simpr 479 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → 𝐵𝑆)
299, 26, 27, 28, 27ovmpt2d 6905 . . . . . 6 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐵) = 𝐴)
3011, 29sylan9eqr 2780 . . . . 5 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = (𝐴 𝐵))) → 𝑦 = 𝐴)
3110, 30jca 555 . . . 4 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = (𝐴 𝐵))) → (𝑥 = 𝐴𝑦 = 𝐴))
3231iftrued 4202 . . 3 (((𝐴𝑆𝐵𝑆) ∧ (𝑥 = 𝐴𝑦 = (𝐴 𝐵))) → if((𝑥 = 𝐴𝑦 = 𝐴), 𝐵, 𝐴) = 𝐵)
3329, 27eqeltrd 2803 . . 3 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐵) ∈ 𝑆)
349, 32, 27, 33, 28ovmpt2d 6905 . 2 ((𝐴𝑆𝐵𝑆) → (𝐴 (𝐴 𝐵)) = 𝐵)
353, 5, 34syl2an 495 1 ((𝐴𝑉𝐵𝑊) → (𝐴 (𝐴 𝐵)) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1596  wcel 2103  ifcif 4194  {cpr 4287  cfv 6001  (class class class)co 6765  cmpt2 6767  Basecbs 15980  +gcplusg 16064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-iota 5964  df-fun 6003  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770
This theorem is referenced by:  mgm2nsgrplem4  17530
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