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Theorem sgrp2rid2 17460
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 4327 . . . 4 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
2 mgm2nsgrp.s . . . 4 𝑆 = {𝐴, 𝐵}
31, 2syl6eleqr 2741 . . 3 (𝐴𝑉𝐴𝑆)
4 prid2g 4328 . . . 4 (𝐵𝑊𝐵 ∈ {𝐴, 𝐵})
54, 2syl6eleqr 2741 . . 3 (𝐵𝑊𝐵𝑆)
6 simpl 472 . . . . 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 17458 . . . . 5 ((𝐴𝑆𝐴𝑆) → (𝐴 𝐴) = 𝐴)
116, 10syldan 486 . . . 4 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐴) = 𝐴)
12 oveq1 6697 . . . . . . 7 (𝐴 = 𝐵 → (𝐴 𝐴) = (𝐵 𝐴))
13 id 22 . . . . . . 7 (𝐴 = 𝐵𝐴 = 𝐵)
1412, 13eqeq12d 2666 . . . . . 6 (𝐴 = 𝐵 → ((𝐴 𝐴) = 𝐴 ↔ (𝐵 𝐴) = 𝐵))
1511, 14syl5ib 234 . . . . 5 (𝐴 = 𝐵 → ((𝐴𝑆𝐵𝑆) → (𝐵 𝐴) = 𝐵))
16 simprl 809 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → 𝐴𝑆)
17 simprr 811 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → 𝐵𝑆)
18 df-ne 2824 . . . . . . . . 9 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
1918biimpri 218 . . . . . . . 8 𝐴 = 𝐵𝐴𝐵)
2019adantr 480 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → 𝐴𝐵)
212, 7, 8, 9sgrp2nmndlem3 17459 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵 𝐴) = 𝐵)
2216, 17, 20, 21syl3anc 1366 . . . . . 6 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → (𝐵 𝐴) = 𝐵)
2322ex 449 . . . . 5 𝐴 = 𝐵 → ((𝐴𝑆𝐵𝑆) → (𝐵 𝐴) = 𝐵))
2415, 23pm2.61i 176 . . . 4 ((𝐴𝑆𝐵𝑆) → (𝐵 𝐴) = 𝐵)
252, 7, 8, 9sgrp2nmndlem2 17458 . . . . 5 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐵) = 𝐴)
2613, 13oveq12d 6708 . . . . . . . 8 (𝐴 = 𝐵 → (𝐴 𝐴) = (𝐵 𝐵))
2726, 13eqeq12d 2666 . . . . . . 7 (𝐴 = 𝐵 → ((𝐴 𝐴) = 𝐴 ↔ (𝐵 𝐵) = 𝐵))
2811, 27syl5ib 234 . . . . . 6 (𝐴 = 𝐵 → ((𝐴𝑆𝐵𝑆) → (𝐵 𝐵) = 𝐵))
292, 7, 8, 9sgrp2nmndlem3 17459 . . . . . . . 8 ((𝐵𝑆𝐵𝑆𝐴𝐵) → (𝐵 𝐵) = 𝐵)
3017, 17, 20, 29syl3anc 1366 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → (𝐵 𝐵) = 𝐵)
3130ex 449 . . . . . 6 𝐴 = 𝐵 → ((𝐴𝑆𝐵𝑆) → (𝐵 𝐵) = 𝐵))
3228, 31pm2.61i 176 . . . . 5 ((𝐴𝑆𝐵𝑆) → (𝐵 𝐵) = 𝐵)
3325, 32jca 553 . . . 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 493 . 2 ((𝐴𝑉𝐵𝑊) → (((𝐴 𝐴) = 𝐴 ∧ (𝐵 𝐴) = 𝐵) ∧ ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵)))
362raleqi 3172 . . . . 5 (∀𝑦𝑆 (𝑦 𝑥) = 𝑦 ↔ ∀𝑦 ∈ {𝐴, 𝐵} (𝑦 𝑥) = 𝑦)
37 oveq1 6697 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 𝑥) = (𝐴 𝑥))
38 id 22 . . . . . . 7 (𝑦 = 𝐴𝑦 = 𝐴)
3937, 38eqeq12d 2666 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 𝑥) = 𝑦 ↔ (𝐴 𝑥) = 𝐴))
40 oveq1 6697 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 𝑥) = (𝐵 𝑥))
41 id 22 . . . . . . 7 (𝑦 = 𝐵𝑦 = 𝐵)
4240, 41eqeq12d 2666 . . . . . 6 (𝑦 = 𝐵 → ((𝑦 𝑥) = 𝑦 ↔ (𝐵 𝑥) = 𝐵))
4339, 42ralprg 4266 . . . . 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 3015 . . 3 ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑆𝑦𝑆 (𝑦 𝑥) = 𝑦 ↔ ∀𝑥𝑆 ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵)))
462raleqi 3172 . . . 4 (∀𝑥𝑆 ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵) ↔ ∀𝑥 ∈ {𝐴, 𝐵} ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵))
47 oveq2 6698 . . . . . . 7 (𝑥 = 𝐴 → (𝐴 𝑥) = (𝐴 𝐴))
4847eqeq1d 2653 . . . . . 6 (𝑥 = 𝐴 → ((𝐴 𝑥) = 𝐴 ↔ (𝐴 𝐴) = 𝐴))
49 oveq2 6698 . . . . . . 7 (𝑥 = 𝐴 → (𝐵 𝑥) = (𝐵 𝐴))
5049eqeq1d 2653 . . . . . 6 (𝑥 = 𝐴 → ((𝐵 𝑥) = 𝐵 ↔ (𝐵 𝐴) = 𝐵))
5148, 50anbi12d 747 . . . . 5 (𝑥 = 𝐴 → (((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵) ↔ ((𝐴 𝐴) = 𝐴 ∧ (𝐵 𝐴) = 𝐵)))
52 oveq2 6698 . . . . . . 7 (𝑥 = 𝐵 → (𝐴 𝑥) = (𝐴 𝐵))
5352eqeq1d 2653 . . . . . 6 (𝑥 = 𝐵 → ((𝐴 𝑥) = 𝐴 ↔ (𝐴 𝐵) = 𝐴))
54 oveq2 6698 . . . . . . 7 (𝑥 = 𝐵 → (𝐵 𝑥) = (𝐵 𝐵))
5554eqeq1d 2653 . . . . . 6 (𝑥 = 𝐵 → ((𝐵 𝑥) = 𝐵 ↔ (𝐵 𝐵) = 𝐵))
5653, 55anbi12d 747 . . . . 5 (𝑥 = 𝐵 → (((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵) ↔ ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵)))
5751, 56ralprg 4266 . . . 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 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  ifcif 4119  {cpr 4212  cfv 5926  (class class class)co 6690  cmpt2 6692  Basecbs 15904  +gcplusg 15988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695
This theorem is referenced by:  sgrp2rid2ex  17461
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