Theorem lidrididd 13025

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 13024) is equal to the two-sided identity element. (Contributed by AV, 26-Dec-2023.)

Hypotheses

Ref	Expression
lidrideqd.l	⊢ (𝜑 → 𝐿 ∈ 𝐵)
lidrideqd.r	⊢ (𝜑 → 𝑅 ∈ 𝐵)
lidrideqd.li	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐿 + 𝑥) = 𝑥)
lidrideqd.ri	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥)
lidrideqd.b	⊢ 𝐵 = (Base‘𝐺)
lidrideqd.p	⊢ + = (+g‘𝐺)
lidrididd.o	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
lidrididd	⊢ (𝜑 → 𝐿 = 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.

Step	Hyp	Ref	Expression
1		lidrideqd.b	. 2 ⊢ 𝐵 = (Base‘𝐺)
2		lidrididd.o	. 2 ⊢ 0 = (0g‘𝐺)
3		lidrideqd.p	. 2 ⊢ + = (+g‘𝐺)
4		lidrideqd.l	. 2 ⊢ (𝜑 → 𝐿 ∈ 𝐵)
5		lidrideqd.li	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐿 + 𝑥) = 𝑥)
6		oveq2 5930	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐿 + 𝑥) = (𝐿 + 𝑦))
7		id 19	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
8	6, 7	eqeq12d 2211	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐿 + 𝑥) = 𝑥 ↔ (𝐿 + 𝑦) = 𝑦))
9	8	rspcv 2864	. . 3 ⊢ (𝑦 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝐿 + 𝑥) = 𝑥 → (𝐿 + 𝑦) = 𝑦))
10	5, 9	mpan9 281	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐿 + 𝑦) = 𝑦)
11		lidrideqd.ri	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥)
12		lidrideqd.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝐵)
13	4, 12, 5, 11	lidrideqd 13024	. . . 4 ⊢ (𝜑 → 𝐿 = 𝑅)
14		oveq1 5929	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 + 𝑅) = (𝑦 + 𝑅))
15	14, 7	eqeq12d 2211	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 + 𝑅) = 𝑥 ↔ (𝑦 + 𝑅) = 𝑦))
16	15	rspcv 2864	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥 → (𝑦 + 𝑅) = 𝑦))
17		oveq2 5930	. . . . . . . . 9 ⊢ (𝐿 = 𝑅 → (𝑦 + 𝐿) = (𝑦 + 𝑅))
18	17	adantl 277	. . . . . . . 8 ⊢ (((𝑦 + 𝑅) = 𝑦 ∧ 𝐿 = 𝑅) → (𝑦 + 𝐿) = (𝑦 + 𝑅))
19		simpl 109	. . . . . . . 8 ⊢ (((𝑦 + 𝑅) = 𝑦 ∧ 𝐿 = 𝑅) → (𝑦 + 𝑅) = 𝑦)
20	18, 19	eqtrd 2229	. . . . . . 7 ⊢ (((𝑦 + 𝑅) = 𝑦 ∧ 𝐿 = 𝑅) → (𝑦 + 𝐿) = 𝑦)
21	20	ex 115	. . . . . 6 ⊢ ((𝑦 + 𝑅) = 𝑦 → (𝐿 = 𝑅 → (𝑦 + 𝐿) = 𝑦))
22	16, 21	syl6com 35	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥 → (𝑦 ∈ 𝐵 → (𝐿 = 𝑅 → (𝑦 + 𝐿) = 𝑦)))
23	22	com23 78	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥 → (𝐿 = 𝑅 → (𝑦 ∈ 𝐵 → (𝑦 + 𝐿) = 𝑦)))
24	11, 13, 23	sylc 62	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → (𝑦 + 𝐿) = 𝑦))
25	24	imp 124	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 + 𝐿) = 𝑦)
26	1, 2, 3, 4, 10, 25	ismgmid2 13023	1 ⊢ (𝜑 → 𝐿 = 0 )

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ‘cfv 5258  (class class class)co 5922  Basecbs 12678  +_gcplusg 12755  0_gc0g 12927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-inn 8991  df-ndx 12681  df-slot 12682  df-base 12684  df-0g 12929

This theorem is referenced by: (None)