Theorem sgrp2nmndlem3 18806

Description: Lemma 3 for sgrp2nmnd 18811. (Contributed by AV, 29-Jan-2020.)

Hypotheses

Ref	Expression
mgm2nsgrp.s	⊢ 𝑆 = {𝐴, 𝐵}
mgm2nsgrp.b	⊢ (Base‘𝑀) = 𝑆
sgrp2nmnd.o	⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
sgrp2nmnd.p	⊢ ⚬ = (+g‘𝑀)

Assertion

Ref	Expression
sgrp2nmndlem3	⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (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

Step	Hyp	Ref	Expression
1		sgrp2nmnd.p	. . . 4 ⊢ ⚬ = (+g‘𝑀)
2		sgrp2nmnd.o	. . . 4 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
3	1, 2	eqtri 2761	. . 3 ⊢ ⚬ = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
4	3	a1i 11	. 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ⚬ = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵)))
5		df-ne 2942	. . . . . 6 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
6		eqeq2 2745	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝐴 = 𝑥 ↔ 𝐴 = 𝐵))
7	6	adantr 482	. . . . . . . . 9 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → (𝐴 = 𝑥 ↔ 𝐴 = 𝐵))
8		eqcom 2740	. . . . . . . . 9 ⊢ (𝐴 = 𝑥 ↔ 𝑥 = 𝐴)
9	7, 8	bitr3di 286	. . . . . . . 8 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → (𝐴 = 𝐵 ↔ 𝑥 = 𝐴))
10	9	notbid 318	. . . . . . 7 ⊢ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → (¬ 𝐴 = 𝐵 ↔ ¬ 𝑥 = 𝐴))
11	10	biimpcd 248	. . . . . 6 ⊢ (¬ 𝐴 = 𝐵 → ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
12	5, 11	sylbi 216	. . . . 5 ⊢ (𝐴 ≠ 𝐵 → ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
13	12	3ad2ant3 1136	. . . 4 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) → ¬ 𝑥 = 𝐴))
14	13	imp 408	. . 3 ⊢ (((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ (𝑥 = 𝐵 ∧ 𝑦 = 𝐶)) → ¬ 𝑥 = 𝐴)
15	14	iffalsed 4540	. 2 ⊢ (((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) ∧ (𝑥 = 𝐵 ∧ 𝑦 = 𝐶)) → if(𝑥 = 𝐴, 𝐴, 𝐵) = 𝐵)
16		simp2 1138	. 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑆)
17		simp1 1137	. 2 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 𝐶 ∈ 𝑆)
18	4, 15, 16, 17, 16	ovmpod 7560	1 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵 ⚬ 𝐶) = 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ifcif 4529  {cpr 4631  ‘cfv 6544  (class class class)co 7409   ∈ cmpo 7411  Basecbs 17144  +_gcplusg 17197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414

This theorem is referenced by:  sgrp2rid2  18807  sgrp2nmndlem4  18809  sgrp2nmndlem5  18810