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Theorem lidrididd 18382
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 18381) 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 7303 . . . . 5 (𝑥 = 𝑦 → (𝐿 + 𝑥) = (𝐿 + 𝑦))
7 id 22 . . . . 5 (𝑥 = 𝑦𝑥 = 𝑦)
86, 7eqeq12d 2749 . . . 4 (𝑥 = 𝑦 → ((𝐿 + 𝑥) = 𝑥 ↔ (𝐿 + 𝑦) = 𝑦))
98rspcv 3559 . . 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 18381 . . . 4 (𝜑𝐿 = 𝑅)
14 oveq1 7302 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 + 𝑅) = (𝑦 + 𝑅))
1514, 7eqeq12d 2749 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥 + 𝑅) = 𝑥 ↔ (𝑦 + 𝑅) = 𝑦))
1615rspcv 3559 . . . . . 6 (𝑦𝐵 → (∀𝑥𝐵 (𝑥 + 𝑅) = 𝑥 → (𝑦 + 𝑅) = 𝑦))
17 oveq2 7303 . . . . . . . . 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 2773 . . . . . . 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 18380 1 (𝜑𝐿 = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2101  wral 3059  cfv 6447  (class class class)co 7295  Basecbs 16940  +gcplusg 16990  0gc0g 17178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pr 5355
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3222  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-iota 6399  df-fun 6449  df-fv 6455  df-riota 7252  df-ov 7298  df-0g 17180
This theorem is referenced by: (None)
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