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Theorem sgrp2nmndlem3 17352
Description: Lemma 3 for sgrp2nmnd 17357. (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 2643 . . 3 = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
43a1i 11 . 2 ((𝐶𝑆𝐵𝑆𝐴𝐵) → = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵)))
5 df-ne 2791 . . . . . 6 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
6 eqcom 2628 . . . . . . . . 9 (𝐴 = 𝑥𝑥 = 𝐴)
7 eqeq2 2632 . . . . . . . . . 10 (𝑥 = 𝐵 → (𝐴 = 𝑥𝐴 = 𝐵))
87adantr 481 . . . . . . . . 9 ((𝑥 = 𝐵𝑦 = 𝐶) → (𝐴 = 𝑥𝐴 = 𝐵))
96, 8syl5rbbr 275 . . . . . . . 8 ((𝑥 = 𝐵𝑦 = 𝐶) → (𝐴 = 𝐵𝑥 = 𝐴))
109notbid 308 . . . . . . 7 ((𝑥 = 𝐵𝑦 = 𝐶) → (¬ 𝐴 = 𝐵 ↔ ¬ 𝑥 = 𝐴))
1110biimpcd 239 . . . . . 6 𝐴 = 𝐵 → ((𝑥 = 𝐵𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
125, 11sylbi 207 . . . . 5 (𝐴𝐵 → ((𝑥 = 𝐵𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
13123ad2ant3 1082 . . . 4 ((𝐶𝑆𝐵𝑆𝐴𝐵) → ((𝑥 = 𝐵𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
1413imp 445 . . 3 (((𝐶𝑆𝐵𝑆𝐴𝐵) ∧ (𝑥 = 𝐵𝑦 = 𝐶)) → ¬ 𝑥 = 𝐴)
1514iffalsed 4075 . 2 (((𝐶𝑆𝐵𝑆𝐴𝐵) ∧ (𝑥 = 𝐵𝑦 = 𝐶)) → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐵)
16 simp2 1060 . 2 ((𝐶𝑆𝐵𝑆𝐴𝐵) → 𝐵𝑆)
17 simp1 1059 . 2 ((𝐶𝑆𝐵𝑆𝐴𝐵) → 𝐶𝑆)
184, 15, 16, 17, 16ovmpt2d 6753 1 ((𝐶𝑆𝐵𝑆𝐴𝐵) → (𝐵 𝐶) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  ifcif 4064  {cpr 4157  cfv 5857  (class class class)co 6615  cmpt2 6617  Basecbs 15800  +gcplusg 15881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620
This theorem is referenced by:  sgrp2rid2  17353  sgrp2nmndlem4  17355  sgrp2nmndlem5  17356
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