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Theorem sgrp2rid2 18737
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 4722 . . . 4 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
2 mgm2nsgrp.s . . . 4 𝑆 = {𝐴, 𝐵}
31, 2eleqtrrdi 2849 . . 3 (𝐴𝑉𝐴𝑆)
4 prid2g 4723 . . . 4 (𝐵𝑊𝐵 ∈ {𝐴, 𝐵})
54, 2eleqtrrdi 2849 . . 3 (𝐵𝑊𝐵𝑆)
6 simpl 484 . . . . 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 18735 . . . . 5 ((𝐴𝑆𝐴𝑆) → (𝐴 𝐴) = 𝐴)
116, 10syldan 592 . . . 4 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐴) = 𝐴)
12 oveq1 7365 . . . . . . 7 (𝐴 = 𝐵 → (𝐴 𝐴) = (𝐵 𝐴))
13 id 22 . . . . . . 7 (𝐴 = 𝐵𝐴 = 𝐵)
1412, 13eqeq12d 2753 . . . . . 6 (𝐴 = 𝐵 → ((𝐴 𝐴) = 𝐴 ↔ (𝐵 𝐴) = 𝐵))
1511, 14imbitrid 243 . . . . 5 (𝐴 = 𝐵 → ((𝐴𝑆𝐵𝑆) → (𝐵 𝐴) = 𝐵))
16 simprl 770 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → 𝐴𝑆)
17 simprr 772 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → 𝐵𝑆)
18 neqne 2952 . . . . . . . 8 𝐴 = 𝐵𝐴𝐵)
1918adantr 482 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → 𝐴𝐵)
202, 7, 8, 9sgrp2nmndlem3 18736 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵 𝐴) = 𝐵)
2116, 17, 19, 20syl3anc 1372 . . . . . 6 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → (𝐵 𝐴) = 𝐵)
2221ex 414 . . . . 5 𝐴 = 𝐵 → ((𝐴𝑆𝐵𝑆) → (𝐵 𝐴) = 𝐵))
2315, 22pm2.61i 182 . . . 4 ((𝐴𝑆𝐵𝑆) → (𝐵 𝐴) = 𝐵)
242, 7, 8, 9sgrp2nmndlem2 18735 . . . . 5 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐵) = 𝐴)
2513, 13oveq12d 7376 . . . . . . . 8 (𝐴 = 𝐵 → (𝐴 𝐴) = (𝐵 𝐵))
2625, 13eqeq12d 2753 . . . . . . 7 (𝐴 = 𝐵 → ((𝐴 𝐴) = 𝐴 ↔ (𝐵 𝐵) = 𝐵))
2711, 26imbitrid 243 . . . . . 6 (𝐴 = 𝐵 → ((𝐴𝑆𝐵𝑆) → (𝐵 𝐵) = 𝐵))
282, 7, 8, 9sgrp2nmndlem3 18736 . . . . . . . 8 ((𝐵𝑆𝐵𝑆𝐴𝐵) → (𝐵 𝐵) = 𝐵)
2917, 17, 19, 28syl3anc 1372 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → (𝐵 𝐵) = 𝐵)
3029ex 414 . . . . . 6 𝐴 = 𝐵 → ((𝐴𝑆𝐵𝑆) → (𝐵 𝐵) = 𝐵))
3127, 30pm2.61i 182 . . . . 5 ((𝐴𝑆𝐵𝑆) → (𝐵 𝐵) = 𝐵)
3224, 31jca 513 . . . 4 ((𝐴𝑆𝐵𝑆) → ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵))
3311, 23, 32jca31 516 . . 3 ((𝐴𝑆𝐵𝑆) → (((𝐴 𝐴) = 𝐴 ∧ (𝐵 𝐴) = 𝐵) ∧ ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵)))
343, 5, 33syl2an 597 . 2 ((𝐴𝑉𝐵𝑊) → (((𝐴 𝐴) = 𝐴 ∧ (𝐵 𝐴) = 𝐵) ∧ ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵)))
352raleqi 3312 . . . . 5 (∀𝑦𝑆 (𝑦 𝑥) = 𝑦 ↔ ∀𝑦 ∈ {𝐴, 𝐵} (𝑦 𝑥) = 𝑦)
36 oveq1 7365 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 𝑥) = (𝐴 𝑥))
37 id 22 . . . . . . 7 (𝑦 = 𝐴𝑦 = 𝐴)
3836, 37eqeq12d 2753 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 𝑥) = 𝑦 ↔ (𝐴 𝑥) = 𝐴))
39 oveq1 7365 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 𝑥) = (𝐵 𝑥))
40 id 22 . . . . . . 7 (𝑦 = 𝐵𝑦 = 𝐵)
4139, 40eqeq12d 2753 . . . . . 6 (𝑦 = 𝐵 → ((𝑦 𝑥) = 𝑦 ↔ (𝐵 𝑥) = 𝐵))
4238, 41ralprg 4656 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∀𝑦 ∈ {𝐴, 𝐵} (𝑦 𝑥) = 𝑦 ↔ ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵)))
4335, 42bitrid 283 . . . 4 ((𝐴𝑉𝐵𝑊) → (∀𝑦𝑆 (𝑦 𝑥) = 𝑦 ↔ ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵)))
4443ralbidv 3175 . . 3 ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑆𝑦𝑆 (𝑦 𝑥) = 𝑦 ↔ ∀𝑥𝑆 ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵)))
452raleqi 3312 . . . 4 (∀𝑥𝑆 ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵) ↔ ∀𝑥 ∈ {𝐴, 𝐵} ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵))
46 oveq2 7366 . . . . . . 7 (𝑥 = 𝐴 → (𝐴 𝑥) = (𝐴 𝐴))
4746eqeq1d 2739 . . . . . 6 (𝑥 = 𝐴 → ((𝐴 𝑥) = 𝐴 ↔ (𝐴 𝐴) = 𝐴))
48 oveq2 7366 . . . . . . 7 (𝑥 = 𝐴 → (𝐵 𝑥) = (𝐵 𝐴))
4948eqeq1d 2739 . . . . . 6 (𝑥 = 𝐴 → ((𝐵 𝑥) = 𝐵 ↔ (𝐵 𝐴) = 𝐵))
5047, 49anbi12d 632 . . . . 5 (𝑥 = 𝐴 → (((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵) ↔ ((𝐴 𝐴) = 𝐴 ∧ (𝐵 𝐴) = 𝐵)))
51 oveq2 7366 . . . . . . 7 (𝑥 = 𝐵 → (𝐴 𝑥) = (𝐴 𝐵))
5251eqeq1d 2739 . . . . . 6 (𝑥 = 𝐵 → ((𝐴 𝑥) = 𝐴 ↔ (𝐴 𝐵) = 𝐴))
53 oveq2 7366 . . . . . . 7 (𝑥 = 𝐵 → (𝐵 𝑥) = (𝐵 𝐵))
5453eqeq1d 2739 . . . . . 6 (𝑥 = 𝐵 → ((𝐵 𝑥) = 𝐵 ↔ (𝐵 𝐵) = 𝐵))
5552, 54anbi12d 632 . . . . 5 (𝑥 = 𝐵 → (((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵) ↔ ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵)))
5650, 55ralprg 4656 . . . 4 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵} ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵) ↔ (((𝐴 𝐴) = 𝐴 ∧ (𝐵 𝐴) = 𝐵) ∧ ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵))))
5745, 56bitrid 283 . . 3 ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑆 ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵) ↔ (((𝐴 𝐴) = 𝐴 ∧ (𝐵 𝐴) = 𝐵) ∧ ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵))))
5844, 57bitrd 279 . 2 ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑆𝑦𝑆 (𝑦 𝑥) = 𝑦 ↔ (((𝐴 𝐴) = 𝐴 ∧ (𝐵 𝐴) = 𝐵) ∧ ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵))))
5934, 58mpbird 257 1 ((𝐴𝑉𝐵𝑊) → ∀𝑥𝑆𝑦𝑆 (𝑦 𝑥) = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2944  wral 3065  ifcif 4487  {cpr 4589  cfv 6497  (class class class)co 7358  cmpo 7360  Basecbs 17084  +gcplusg 17134
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 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363
This theorem is referenced by:  sgrp2rid2ex  18738
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