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Theorem lidrididd 18008
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 18007) 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 7190 . . . . 5 (𝑥 = 𝑦 → (𝐿 + 𝑥) = (𝐿 + 𝑦))
7 id 22 . . . . 5 (𝑥 = 𝑦𝑥 = 𝑦)
86, 7eqeq12d 2755 . . . 4 (𝑥 = 𝑦 → ((𝐿 + 𝑥) = 𝑥 ↔ (𝐿 + 𝑦) = 𝑦))
98rspcv 3524 . . 3 (𝑦𝐵 → (∀𝑥𝐵 (𝐿 + 𝑥) = 𝑥 → (𝐿 + 𝑦) = 𝑦))
105, 9mpan9 510 . 2 ((𝜑𝑦𝐵) → (𝐿 + 𝑦) = 𝑦)
11 lidrideqd.ri . . . 4 (𝜑 → ∀𝑥𝐵 (𝑥 + 𝑅) = 𝑥)
12 lidrideqd.r . . . . 5 (𝜑𝑅𝐵)
134, 12, 5, 11lidrideqd 18007 . . . 4 (𝜑𝐿 = 𝑅)
14 oveq1 7189 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 + 𝑅) = (𝑦 + 𝑅))
1514, 7eqeq12d 2755 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥 + 𝑅) = 𝑥 ↔ (𝑦 + 𝑅) = 𝑦))
1615rspcv 3524 . . . . . 6 (𝑦𝐵 → (∀𝑥𝐵 (𝑥 + 𝑅) = 𝑥 → (𝑦 + 𝑅) = 𝑦))
17 oveq2 7190 . . . . . . . . 9 (𝐿 = 𝑅 → (𝑦 + 𝐿) = (𝑦 + 𝑅))
1817adantl 485 . . . . . . . 8 (((𝑦 + 𝑅) = 𝑦𝐿 = 𝑅) → (𝑦 + 𝐿) = (𝑦 + 𝑅))
19 simpl 486 . . . . . . . 8 (((𝑦 + 𝑅) = 𝑦𝐿 = 𝑅) → (𝑦 + 𝑅) = 𝑦)
2018, 19eqtrd 2774 . . . . . . 7 (((𝑦 + 𝑅) = 𝑦𝐿 = 𝑅) → (𝑦 + 𝐿) = 𝑦)
2120ex 416 . . . . . 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 410 . 2 ((𝜑𝑦𝐵) → (𝑦 + 𝐿) = 𝑦)
261, 2, 3, 4, 10, 25ismgmid2 18006 1 (𝜑𝐿 = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  wral 3054  cfv 6349  (class class class)co 7182  Basecbs 16598  +gcplusg 16680  0gc0g 16828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-iota 6307  df-fun 6351  df-fv 6357  df-riota 7139  df-ov 7185  df-0g 16830
This theorem is referenced by: (None)
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