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Theorem lidrididd 18696
Description: If there is a left and right identity element for any binary operation (group operation) +, the left identity element (and therefore also the right identity element according to lidrideqd 18695) is equal to the two-sided identity element. (Contributed by AV, 26-Dec-2023.)
Hypotheses
Ref Expression
lidrideqd.l (𝜑𝐿𝐵)
lidrideqd.r (𝜑𝑅𝐵)
lidrideqd.li (𝜑 → ∀𝑥𝐵 (𝐿 + 𝑥) = 𝑥)
lidrideqd.ri (𝜑 → ∀𝑥𝐵 (𝑥 + 𝑅) = 𝑥)
lidrideqd.b 𝐵 = (Base‘𝐺)
lidrideqd.p + = (+g𝐺)
lidrididd.o 0 = (0g𝐺)
Assertion
Ref Expression
lidrididd (𝜑𝐿 = 0 )
Distinct variable groups:   𝑥,𝐵   𝑥,𝐿   𝑥,𝑅   𝑥, +
Allowed substitution hints:   𝜑(𝑥)   𝐺(𝑥)   0 (𝑥)

Proof of Theorem lidrididd
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 lidrideqd.b . 2 𝐵 = (Base‘𝐺)
2 lidrididd.o . 2 0 = (0g𝐺)
3 lidrideqd.p . 2 + = (+g𝐺)
4 lidrideqd.l . 2 (𝜑𝐿𝐵)
5 lidrideqd.li . . 3 (𝜑 → ∀𝑥𝐵 (𝐿 + 𝑥) = 𝑥)
6 oveq2 7439 . . . . 5 (𝑥 = 𝑦 → (𝐿 + 𝑥) = (𝐿 + 𝑦))
7 id 22 . . . . 5 (𝑥 = 𝑦𝑥 = 𝑦)
86, 7eqeq12d 2751 . . . 4 (𝑥 = 𝑦 → ((𝐿 + 𝑥) = 𝑥 ↔ (𝐿 + 𝑦) = 𝑦))
98rspcv 3618 . . 3 (𝑦𝐵 → (∀𝑥𝐵 (𝐿 + 𝑥) = 𝑥 → (𝐿 + 𝑦) = 𝑦))
105, 9mpan9 506 . 2 ((𝜑𝑦𝐵) → (𝐿 + 𝑦) = 𝑦)
11 lidrideqd.ri . . . 4 (𝜑 → ∀𝑥𝐵 (𝑥 + 𝑅) = 𝑥)
12 lidrideqd.r . . . . 5 (𝜑𝑅𝐵)
134, 12, 5, 11lidrideqd 18695 . . . 4 (𝜑𝐿 = 𝑅)
14 oveq1 7438 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 + 𝑅) = (𝑦 + 𝑅))
1514, 7eqeq12d 2751 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥 + 𝑅) = 𝑥 ↔ (𝑦 + 𝑅) = 𝑦))
1615rspcv 3618 . . . . . 6 (𝑦𝐵 → (∀𝑥𝐵 (𝑥 + 𝑅) = 𝑥 → (𝑦 + 𝑅) = 𝑦))
17 oveq2 7439 . . . . . . . . 9 (𝐿 = 𝑅 → (𝑦 + 𝐿) = (𝑦 + 𝑅))
1817adantl 481 . . . . . . . 8 (((𝑦 + 𝑅) = 𝑦𝐿 = 𝑅) → (𝑦 + 𝐿) = (𝑦 + 𝑅))
19 simpl 482 . . . . . . . 8 (((𝑦 + 𝑅) = 𝑦𝐿 = 𝑅) → (𝑦 + 𝑅) = 𝑦)
2018, 19eqtrd 2775 . . . . . . 7 (((𝑦 + 𝑅) = 𝑦𝐿 = 𝑅) → (𝑦 + 𝐿) = 𝑦)
2120ex 412 . . . . . 6 ((𝑦 + 𝑅) = 𝑦 → (𝐿 = 𝑅 → (𝑦 + 𝐿) = 𝑦))
2216, 21syl6com 37 . . . . 5 (∀𝑥𝐵 (𝑥 + 𝑅) = 𝑥 → (𝑦𝐵 → (𝐿 = 𝑅 → (𝑦 + 𝐿) = 𝑦)))
2322com23 86 . . . 4 (∀𝑥𝐵 (𝑥 + 𝑅) = 𝑥 → (𝐿 = 𝑅 → (𝑦𝐵 → (𝑦 + 𝐿) = 𝑦)))
2411, 13, 23sylc 65 . . 3 (𝜑 → (𝑦𝐵 → (𝑦 + 𝐿) = 𝑦))
2524imp 406 . 2 ((𝜑𝑦𝐵) → (𝑦 + 𝐿) = 𝑦)
261, 2, 3, 4, 10, 25ismgmid2 18694 1 (𝜑𝐿 = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488
This theorem is referenced by: (None)
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