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Theorem sgrp2rid2 17329
Description: A small semigroup (with two elements) with two right identities which are different if 𝐴𝐵. (Contributed by AV, 10-Feb-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
sgrp2rid2 ((𝐴𝑉𝐵𝑊) → ∀𝑥𝑆𝑦𝑆 (𝑦 𝑥) = 𝑦)
Distinct variable groups:   𝑥,𝑆,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑀   𝑥,𝑉   𝑥,𝑊   𝑥, ,𝑦
Allowed substitution hints:   𝑀(𝑦)   𝑉(𝑦)   𝑊(𝑦)

Proof of Theorem sgrp2rid2
StepHypRef Expression
1 prid1g 4270 . . . 4 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
2 mgm2nsgrp.s . . . 4 𝑆 = {𝐴, 𝐵}
31, 2syl6eleqr 2715 . . 3 (𝐴𝑉𝐴𝑆)
4 prid2g 4271 . . . 4 (𝐵𝑊𝐵 ∈ {𝐴, 𝐵})
54, 2syl6eleqr 2715 . . 3 (𝐵𝑊𝐵𝑆)
6 simpl 473 . . . . 5 ((𝐴𝑆𝐵𝑆) → 𝐴𝑆)
7 mgm2nsgrp.b . . . . . 6 (Base‘𝑀) = 𝑆
8 sgrp2nmnd.o . . . . . 6 (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
9 sgrp2nmnd.p . . . . . 6 = (+g𝑀)
102, 7, 8, 9sgrp2nmndlem2 17327 . . . . 5 ((𝐴𝑆𝐴𝑆) → (𝐴 𝐴) = 𝐴)
116, 10syldan 487 . . . 4 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐴) = 𝐴)
12 oveq1 6612 . . . . . . 7 (𝐴 = 𝐵 → (𝐴 𝐴) = (𝐵 𝐴))
13 id 22 . . . . . . 7 (𝐴 = 𝐵𝐴 = 𝐵)
1412, 13eqeq12d 2641 . . . . . 6 (𝐴 = 𝐵 → ((𝐴 𝐴) = 𝐴 ↔ (𝐵 𝐴) = 𝐵))
1511, 14syl5ib 234 . . . . 5 (𝐴 = 𝐵 → ((𝐴𝑆𝐵𝑆) → (𝐵 𝐴) = 𝐵))
16 simprl 793 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → 𝐴𝑆)
17 simprr 795 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → 𝐵𝑆)
18 df-ne 2797 . . . . . . . . 9 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
1918biimpri 218 . . . . . . . 8 𝐴 = 𝐵𝐴𝐵)
2019adantr 481 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → 𝐴𝐵)
212, 7, 8, 9sgrp2nmndlem3 17328 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵 𝐴) = 𝐵)
2216, 17, 20, 21syl3anc 1323 . . . . . 6 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → (𝐵 𝐴) = 𝐵)
2322ex 450 . . . . 5 𝐴 = 𝐵 → ((𝐴𝑆𝐵𝑆) → (𝐵 𝐴) = 𝐵))
2415, 23pm2.61i 176 . . . 4 ((𝐴𝑆𝐵𝑆) → (𝐵 𝐴) = 𝐵)
252, 7, 8, 9sgrp2nmndlem2 17327 . . . . 5 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐵) = 𝐴)
2613, 13oveq12d 6623 . . . . . . . 8 (𝐴 = 𝐵 → (𝐴 𝐴) = (𝐵 𝐵))
2726, 13eqeq12d 2641 . . . . . . 7 (𝐴 = 𝐵 → ((𝐴 𝐴) = 𝐴 ↔ (𝐵 𝐵) = 𝐵))
2811, 27syl5ib 234 . . . . . 6 (𝐴 = 𝐵 → ((𝐴𝑆𝐵𝑆) → (𝐵 𝐵) = 𝐵))
292, 7, 8, 9sgrp2nmndlem3 17328 . . . . . . . 8 ((𝐵𝑆𝐵𝑆𝐴𝐵) → (𝐵 𝐵) = 𝐵)
3017, 17, 20, 29syl3anc 1323 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → (𝐵 𝐵) = 𝐵)
3130ex 450 . . . . . 6 𝐴 = 𝐵 → ((𝐴𝑆𝐵𝑆) → (𝐵 𝐵) = 𝐵))
3228, 31pm2.61i 176 . . . . 5 ((𝐴𝑆𝐵𝑆) → (𝐵 𝐵) = 𝐵)
3325, 32jca 554 . . . 4 ((𝐴𝑆𝐵𝑆) → ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵))
3411, 24, 33jca31 556 . . 3 ((𝐴𝑆𝐵𝑆) → (((𝐴 𝐴) = 𝐴 ∧ (𝐵 𝐴) = 𝐵) ∧ ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵)))
353, 5, 34syl2an 494 . 2 ((𝐴𝑉𝐵𝑊) → (((𝐴 𝐴) = 𝐴 ∧ (𝐵 𝐴) = 𝐵) ∧ ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵)))
362raleqi 3136 . . . . 5 (∀𝑦𝑆 (𝑦 𝑥) = 𝑦 ↔ ∀𝑦 ∈ {𝐴, 𝐵} (𝑦 𝑥) = 𝑦)
37 oveq1 6612 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 𝑥) = (𝐴 𝑥))
38 id 22 . . . . . . 7 (𝑦 = 𝐴𝑦 = 𝐴)
3937, 38eqeq12d 2641 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 𝑥) = 𝑦 ↔ (𝐴 𝑥) = 𝐴))
40 oveq1 6612 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 𝑥) = (𝐵 𝑥))
41 id 22 . . . . . . 7 (𝑦 = 𝐵𝑦 = 𝐵)
4240, 41eqeq12d 2641 . . . . . 6 (𝑦 = 𝐵 → ((𝑦 𝑥) = 𝑦 ↔ (𝐵 𝑥) = 𝐵))
4339, 42ralprg 4210 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∀𝑦 ∈ {𝐴, 𝐵} (𝑦 𝑥) = 𝑦 ↔ ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵)))
4436, 43syl5bb 272 . . . 4 ((𝐴𝑉𝐵𝑊) → (∀𝑦𝑆 (𝑦 𝑥) = 𝑦 ↔ ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵)))
4544ralbidv 2985 . . 3 ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑆𝑦𝑆 (𝑦 𝑥) = 𝑦 ↔ ∀𝑥𝑆 ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵)))
462raleqi 3136 . . . 4 (∀𝑥𝑆 ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵) ↔ ∀𝑥 ∈ {𝐴, 𝐵} ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵))
47 oveq2 6613 . . . . . . 7 (𝑥 = 𝐴 → (𝐴 𝑥) = (𝐴 𝐴))
4847eqeq1d 2628 . . . . . 6 (𝑥 = 𝐴 → ((𝐴 𝑥) = 𝐴 ↔ (𝐴 𝐴) = 𝐴))
49 oveq2 6613 . . . . . . 7 (𝑥 = 𝐴 → (𝐵 𝑥) = (𝐵 𝐴))
5049eqeq1d 2628 . . . . . 6 (𝑥 = 𝐴 → ((𝐵 𝑥) = 𝐵 ↔ (𝐵 𝐴) = 𝐵))
5148, 50anbi12d 746 . . . . 5 (𝑥 = 𝐴 → (((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵) ↔ ((𝐴 𝐴) = 𝐴 ∧ (𝐵 𝐴) = 𝐵)))
52 oveq2 6613 . . . . . . 7 (𝑥 = 𝐵 → (𝐴 𝑥) = (𝐴 𝐵))
5352eqeq1d 2628 . . . . . 6 (𝑥 = 𝐵 → ((𝐴 𝑥) = 𝐴 ↔ (𝐴 𝐵) = 𝐴))
54 oveq2 6613 . . . . . . 7 (𝑥 = 𝐵 → (𝐵 𝑥) = (𝐵 𝐵))
5554eqeq1d 2628 . . . . . 6 (𝑥 = 𝐵 → ((𝐵 𝑥) = 𝐵 ↔ (𝐵 𝐵) = 𝐵))
5653, 55anbi12d 746 . . . . 5 (𝑥 = 𝐵 → (((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵) ↔ ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵)))
5751, 56ralprg 4210 . . . 4 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵} ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵) ↔ (((𝐴 𝐴) = 𝐴 ∧ (𝐵 𝐴) = 𝐵) ∧ ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵))))
5846, 57syl5bb 272 . . 3 ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑆 ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵) ↔ (((𝐴 𝐴) = 𝐴 ∧ (𝐵 𝐴) = 𝐵) ∧ ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵))))
5945, 58bitrd 268 . 2 ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑆𝑦𝑆 (𝑦 𝑥) = 𝑦 ↔ (((𝐴 𝐴) = 𝐴 ∧ (𝐵 𝐴) = 𝐵) ∧ ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵))))
6035, 59mpbird 247 1 ((𝐴𝑉𝐵𝑊) → ∀𝑥𝑆𝑦𝑆 (𝑦 𝑥) = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1992  wne 2796  wral 2912  ifcif 4063  {cpr 4155  cfv 5850  (class class class)co 6605  cmpt2 6607  Basecbs 15776  +gcplusg 15857
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610
This theorem is referenced by:  sgrp2rid2ex  17330
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