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Theorem sgrp2rid2 18736
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 4721 . . . 4 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
2 mgm2nsgrp.s . . . 4 𝑆 = {𝐴, 𝐵}
31, 2eleqtrrdi 2849 . . 3 (𝐴𝑉𝐴𝑆)
4 prid2g 4722 . . . 4 (𝐵𝑊𝐵 ∈ {𝐴, 𝐵})
54, 2eleqtrrdi 2849 . . 3 (𝐵𝑊𝐵𝑆)
6 simpl 483 . . . . 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 18734 . . . . 5 ((𝐴𝑆𝐴𝑆) → (𝐴 𝐴) = 𝐴)
116, 10syldan 591 . . . 4 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐴) = 𝐴)
12 oveq1 7364 . . . . . . 7 (𝐴 = 𝐵 → (𝐴 𝐴) = (𝐵 𝐴))
13 id 22 . . . . . . 7 (𝐴 = 𝐵𝐴 = 𝐵)
1412, 13eqeq12d 2752 . . . . . 6 (𝐴 = 𝐵 → ((𝐴 𝐴) = 𝐴 ↔ (𝐵 𝐴) = 𝐵))
1511, 14imbitrid 243 . . . . 5 (𝐴 = 𝐵 → ((𝐴𝑆𝐵𝑆) → (𝐵 𝐴) = 𝐵))
16 simprl 769 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → 𝐴𝑆)
17 simprr 771 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → 𝐵𝑆)
18 neqne 2951 . . . . . . . 8 𝐴 = 𝐵𝐴𝐵)
1918adantr 481 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → 𝐴𝐵)
202, 7, 8, 9sgrp2nmndlem3 18735 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵 𝐴) = 𝐵)
2116, 17, 19, 20syl3anc 1371 . . . . . 6 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → (𝐵 𝐴) = 𝐵)
2221ex 413 . . . . 5 𝐴 = 𝐵 → ((𝐴𝑆𝐵𝑆) → (𝐵 𝐴) = 𝐵))
2315, 22pm2.61i 182 . . . 4 ((𝐴𝑆𝐵𝑆) → (𝐵 𝐴) = 𝐵)
242, 7, 8, 9sgrp2nmndlem2 18734 . . . . 5 ((𝐴𝑆𝐵𝑆) → (𝐴 𝐵) = 𝐴)
2513, 13oveq12d 7375 . . . . . . . 8 (𝐴 = 𝐵 → (𝐴 𝐴) = (𝐵 𝐵))
2625, 13eqeq12d 2752 . . . . . . 7 (𝐴 = 𝐵 → ((𝐴 𝐴) = 𝐴 ↔ (𝐵 𝐵) = 𝐵))
2711, 26imbitrid 243 . . . . . 6 (𝐴 = 𝐵 → ((𝐴𝑆𝐵𝑆) → (𝐵 𝐵) = 𝐵))
282, 7, 8, 9sgrp2nmndlem3 18735 . . . . . . . 8 ((𝐵𝑆𝐵𝑆𝐴𝐵) → (𝐵 𝐵) = 𝐵)
2917, 17, 19, 28syl3anc 1371 . . . . . . 7 ((¬ 𝐴 = 𝐵 ∧ (𝐴𝑆𝐵𝑆)) → (𝐵 𝐵) = 𝐵)
3029ex 413 . . . . . 6 𝐴 = 𝐵 → ((𝐴𝑆𝐵𝑆) → (𝐵 𝐵) = 𝐵))
3127, 30pm2.61i 182 . . . . 5 ((𝐴𝑆𝐵𝑆) → (𝐵 𝐵) = 𝐵)
3224, 31jca 512 . . . 4 ((𝐴𝑆𝐵𝑆) → ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵))
3311, 23, 32jca31 515 . . 3 ((𝐴𝑆𝐵𝑆) → (((𝐴 𝐴) = 𝐴 ∧ (𝐵 𝐴) = 𝐵) ∧ ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵)))
343, 5, 33syl2an 596 . 2 ((𝐴𝑉𝐵𝑊) → (((𝐴 𝐴) = 𝐴 ∧ (𝐵 𝐴) = 𝐵) ∧ ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵)))
352raleqi 3311 . . . . 5 (∀𝑦𝑆 (𝑦 𝑥) = 𝑦 ↔ ∀𝑦 ∈ {𝐴, 𝐵} (𝑦 𝑥) = 𝑦)
36 oveq1 7364 . . . . . . 7 (𝑦 = 𝐴 → (𝑦 𝑥) = (𝐴 𝑥))
37 id 22 . . . . . . 7 (𝑦 = 𝐴𝑦 = 𝐴)
3836, 37eqeq12d 2752 . . . . . 6 (𝑦 = 𝐴 → ((𝑦 𝑥) = 𝑦 ↔ (𝐴 𝑥) = 𝐴))
39 oveq1 7364 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 𝑥) = (𝐵 𝑥))
40 id 22 . . . . . . 7 (𝑦 = 𝐵𝑦 = 𝐵)
4139, 40eqeq12d 2752 . . . . . 6 (𝑦 = 𝐵 → ((𝑦 𝑥) = 𝑦 ↔ (𝐵 𝑥) = 𝐵))
4238, 41ralprg 4655 . . . . 5 ((𝐴𝑉𝐵𝑊) → (∀𝑦 ∈ {𝐴, 𝐵} (𝑦 𝑥) = 𝑦 ↔ ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵)))
4335, 42bitrid 282 . . . 4 ((𝐴𝑉𝐵𝑊) → (∀𝑦𝑆 (𝑦 𝑥) = 𝑦 ↔ ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵)))
4443ralbidv 3174 . . 3 ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑆𝑦𝑆 (𝑦 𝑥) = 𝑦 ↔ ∀𝑥𝑆 ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵)))
452raleqi 3311 . . . 4 (∀𝑥𝑆 ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵) ↔ ∀𝑥 ∈ {𝐴, 𝐵} ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵))
46 oveq2 7365 . . . . . . 7 (𝑥 = 𝐴 → (𝐴 𝑥) = (𝐴 𝐴))
4746eqeq1d 2738 . . . . . 6 (𝑥 = 𝐴 → ((𝐴 𝑥) = 𝐴 ↔ (𝐴 𝐴) = 𝐴))
48 oveq2 7365 . . . . . . 7 (𝑥 = 𝐴 → (𝐵 𝑥) = (𝐵 𝐴))
4948eqeq1d 2738 . . . . . 6 (𝑥 = 𝐴 → ((𝐵 𝑥) = 𝐵 ↔ (𝐵 𝐴) = 𝐵))
5047, 49anbi12d 631 . . . . 5 (𝑥 = 𝐴 → (((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵) ↔ ((𝐴 𝐴) = 𝐴 ∧ (𝐵 𝐴) = 𝐵)))
51 oveq2 7365 . . . . . . 7 (𝑥 = 𝐵 → (𝐴 𝑥) = (𝐴 𝐵))
5251eqeq1d 2738 . . . . . 6 (𝑥 = 𝐵 → ((𝐴 𝑥) = 𝐴 ↔ (𝐴 𝐵) = 𝐴))
53 oveq2 7365 . . . . . . 7 (𝑥 = 𝐵 → (𝐵 𝑥) = (𝐵 𝐵))
5453eqeq1d 2738 . . . . . 6 (𝑥 = 𝐵 → ((𝐵 𝑥) = 𝐵 ↔ (𝐵 𝐵) = 𝐵))
5552, 54anbi12d 631 . . . . 5 (𝑥 = 𝐵 → (((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵) ↔ ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵)))
5650, 55ralprg 4655 . . . 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 282 . . 3 ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑆 ((𝐴 𝑥) = 𝐴 ∧ (𝐵 𝑥) = 𝐵) ↔ (((𝐴 𝐴) = 𝐴 ∧ (𝐵 𝐴) = 𝐵) ∧ ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵))))
5844, 57bitrd 278 . 2 ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑆𝑦𝑆 (𝑦 𝑥) = 𝑦 ↔ (((𝐴 𝐴) = 𝐴 ∧ (𝐵 𝐴) = 𝐵) ∧ ((𝐴 𝐵) = 𝐴 ∧ (𝐵 𝐵) = 𝐵))))
5934, 58mpbird 256 1 ((𝐴𝑉𝐵𝑊) → ∀𝑥𝑆𝑦𝑆 (𝑦 𝑥) = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  ifcif 4486  {cpr 4588  cfv 6496  (class class class)co 7357  cmpo 7359  Basecbs 17083  +gcplusg 17133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362
This theorem is referenced by:  sgrp2rid2ex  18737
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