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Theorem sgrp2nmndlem3 18735
Description: Lemma 3 for sgrp2nmnd 18740. (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
sgrp2nmndlem3 ((𝐶𝑆𝐵𝑆𝐴𝐵) → (𝐵 𝐶) = 𝐵)
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 sgrp2nmndlem3
StepHypRef Expression
1 sgrp2nmnd.p . . . 4 = (+g𝑀)
2 sgrp2nmnd.o . . . 4 (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
31, 2eqtri 2764 . . 3 = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
43a1i 11 . 2 ((𝐶𝑆𝐵𝑆𝐴𝐵) → = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵)))
5 df-ne 2944 . . . . . 6 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
6 eqeq2 2748 . . . . . . . . . 10 (𝑥 = 𝐵 → (𝐴 = 𝑥𝐴 = 𝐵))
76adantr 481 . . . . . . . . 9 ((𝑥 = 𝐵𝑦 = 𝐶) → (𝐴 = 𝑥𝐴 = 𝐵))
8 eqcom 2743 . . . . . . . . 9 (𝐴 = 𝑥𝑥 = 𝐴)
97, 8bitr3di 285 . . . . . . . 8 ((𝑥 = 𝐵𝑦 = 𝐶) → (𝐴 = 𝐵𝑥 = 𝐴))
109notbid 317 . . . . . . 7 ((𝑥 = 𝐵𝑦 = 𝐶) → (¬ 𝐴 = 𝐵 ↔ ¬ 𝑥 = 𝐴))
1110biimpcd 248 . . . . . 6 𝐴 = 𝐵 → ((𝑥 = 𝐵𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
125, 11sylbi 216 . . . . 5 (𝐴𝐵 → ((𝑥 = 𝐵𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
13123ad2ant3 1135 . . . 4 ((𝐶𝑆𝐵𝑆𝐴𝐵) → ((𝑥 = 𝐵𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
1413imp 407 . . 3 (((𝐶𝑆𝐵𝑆𝐴𝐵) ∧ (𝑥 = 𝐵𝑦 = 𝐶)) → ¬ 𝑥 = 𝐴)
1514iffalsed 4497 . 2 (((𝐶𝑆𝐵𝑆𝐴𝐵) ∧ (𝑥 = 𝐵𝑦 = 𝐶)) → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐵)
16 simp2 1137 . 2 ((𝐶𝑆𝐵𝑆𝐴𝐵) → 𝐵𝑆)
17 simp1 1136 . 2 ((𝐶𝑆𝐵𝑆𝐴𝐵) → 𝐶𝑆)
184, 15, 16, 17, 16ovmpod 7507 1 ((𝐶𝑆𝐵𝑆𝐴𝐵) → (𝐵 𝐶) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  ifcif 4486  {cpr 4588  cfv 6496  (class class class)co 7357  cmpo 7359  Basecbs 17083  +gcplusg 17133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362
This theorem is referenced by:  sgrp2rid2  18736  sgrp2nmndlem4  18738  sgrp2nmndlem5  18739
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