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Theorem sgrp2nmndlem2 17533
Description: Lemma 2 for sgrp2nmnd 17539. (Contributed by AV, 29-Jan-2020.)
Hypotheses
Ref Expression
mgm2nsgrp.s 𝑆 = {𝐴, 𝐵}
mgm2nsgrp.b (Base‘𝑀) = 𝑆
sgrp2nmnd.o (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
sgrp2nmnd.p = (+g𝑀)
Assertion
Ref Expression
sgrp2nmndlem2 ((𝐴𝑆𝐶𝑆) → (𝐴 𝐶) = 𝐴)
Distinct variable groups:   𝑥,𝑆,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑀   𝑥,𝐶,𝑦
Allowed substitution hints:   𝑀(𝑦)   (𝑥,𝑦)

Proof of Theorem sgrp2nmndlem2
StepHypRef Expression
1 sgrp2nmnd.p . . . 4 = (+g𝑀)
2 sgrp2nmnd.o . . . 4 (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
31, 2eqtri 2746 . . 3 = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
43a1i 11 . 2 ((𝐴𝑆𝐶𝑆) → = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵)))
5 iftrue 4200 . . 3 (𝑥 = 𝐴 → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐴)
65ad2antrl 766 . 2 (((𝐴𝑆𝐶𝑆) ∧ (𝑥 = 𝐴𝑦 = 𝐶)) → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐴)
7 simpl 474 . 2 ((𝐴𝑆𝐶𝑆) → 𝐴𝑆)
8 simpr 479 . 2 ((𝐴𝑆𝐶𝑆) → 𝐶𝑆)
94, 6, 7, 8, 7ovmpt2d 6905 1 ((𝐴𝑆𝐶𝑆) → (𝐴 𝐶) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  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:  sgrp2rid2  17535  sgrp2nmndlem4  17537  sgrp2nmndlem5  17538
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