Theorem o2timesd 20157

Description: An element of a ring-like structure plus itself is two times the element. "Two" in such a structure is the sum of the unity element with itself. This (formerly) part of the proof for ringcom 20223 depends on the (right) distributivity and the existence of a (left) multiplicative identity only. (Contributed by Gérard Lang, 4-Dec-2014.) (Revised by AV, 1-Feb-2025.)

Hypotheses

Ref	Expression
o2timesd.e	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
o2timesd.u	⊢ (𝜑 → 1 ∈ 𝐵)
o2timesd.i	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 1 · 𝑥) = 𝑥)
o2timesd.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
o2timesd	⊢ (𝜑 → (𝑋 + 𝑋) = (( 1 + 1 ) · 𝑋))

Distinct variable groups:   𝑥,𝐵,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑥, 1 ,𝑦,𝑧   𝑥, · ,𝑦,𝑧   𝑥, + ,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem o2timesd

Step	Hyp	Ref	Expression
1		o2timesd.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
2		o2timesd.i	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ( 1 · 𝑥) = 𝑥)
3		oveq2 7434	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ( 1 · 𝑥) = ( 1 · 𝑋))
4		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
5	3, 4	eqeq12d 2744	. . . . . 6 ⊢ (𝑥 = 𝑋 → (( 1 · 𝑥) = 𝑥 ↔ ( 1 · 𝑋) = 𝑋))
6	5	rspcva 3609	. . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ( 1 · 𝑥) = 𝑥) → ( 1 · 𝑋) = 𝑋)
7	6	eqcomd 2734	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ( 1 · 𝑥) = 𝑥) → 𝑋 = ( 1 · 𝑋))
8	1, 2, 7	syl2anc 582	. . 3 ⊢ (𝜑 → 𝑋 = ( 1 · 𝑋))
9	8, 8	oveq12d 7444	. 2 ⊢ (𝜑 → (𝑋 + 𝑋) = (( 1 · 𝑋) + ( 1 · 𝑋)))
10		o2timesd.u	. . . 4 ⊢ (𝜑 → 1 ∈ 𝐵)
11	10, 10, 1	3jca 1125	. . 3 ⊢ (𝜑 → ( 1 ∈ 𝐵 ∧ 1 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵))
12		o2timesd.e	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
13		oveq1 7433	. . . . . 6 ⊢ (𝑥 = 1 → (𝑥 + 𝑦) = ( 1 + 𝑦))
14	13	oveq1d 7441	. . . . 5 ⊢ (𝑥 = 1 → ((𝑥 + 𝑦) · 𝑧) = (( 1 + 𝑦) · 𝑧))
15		oveq1 7433	. . . . . 6 ⊢ (𝑥 = 1 → (𝑥 · 𝑧) = ( 1 · 𝑧))
16	15	oveq1d 7441	. . . . 5 ⊢ (𝑥 = 1 → ((𝑥 · 𝑧) + (𝑦 · 𝑧)) = (( 1 · 𝑧) + (𝑦 · 𝑧)))
17	14, 16	eqeq12d 2744	. . . 4 ⊢ (𝑥 = 1 → (((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)) ↔ (( 1 + 𝑦) · 𝑧) = (( 1 · 𝑧) + (𝑦 · 𝑧))))
18		oveq2 7434	. . . . . 6 ⊢ (𝑦 = 1 → ( 1 + 𝑦) = ( 1 + 1 ))
19	18	oveq1d 7441	. . . . 5 ⊢ (𝑦 = 1 → (( 1 + 𝑦) · 𝑧) = (( 1 + 1 ) · 𝑧))
20		oveq1 7433	. . . . . 6 ⊢ (𝑦 = 1 → (𝑦 · 𝑧) = ( 1 · 𝑧))
21	20	oveq2d 7442	. . . . 5 ⊢ (𝑦 = 1 → (( 1 · 𝑧) + (𝑦 · 𝑧)) = (( 1 · 𝑧) + ( 1 · 𝑧)))
22	19, 21	eqeq12d 2744	. . . 4 ⊢ (𝑦 = 1 → ((( 1 + 𝑦) · 𝑧) = (( 1 · 𝑧) + (𝑦 · 𝑧)) ↔ (( 1 + 1 ) · 𝑧) = (( 1 · 𝑧) + ( 1 · 𝑧))))
23		oveq2 7434	. . . . 5 ⊢ (𝑧 = 𝑋 → (( 1 + 1 ) · 𝑧) = (( 1 + 1 ) · 𝑋))
24		oveq2 7434	. . . . . 6 ⊢ (𝑧 = 𝑋 → ( 1 · 𝑧) = ( 1 · 𝑋))
25	24, 24	oveq12d 7444	. . . . 5 ⊢ (𝑧 = 𝑋 → (( 1 · 𝑧) + ( 1 · 𝑧)) = (( 1 · 𝑋) + ( 1 · 𝑋)))
26	23, 25	eqeq12d 2744	. . . 4 ⊢ (𝑧 = 𝑋 → ((( 1 + 1 ) · 𝑧) = (( 1 · 𝑧) + ( 1 · 𝑧)) ↔ (( 1 + 1 ) · 𝑋) = (( 1 · 𝑋) + ( 1 · 𝑋))))
27	17, 22, 26	rspc3v 3627	. . 3 ⊢ (( 1 ∈ 𝐵 ∧ 1 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)) → (( 1 + 1 ) · 𝑋) = (( 1 · 𝑋) + ( 1 · 𝑋))))
28	11, 12, 27	sylc 65	. 2 ⊢ (𝜑 → (( 1 + 1 ) · 𝑋) = (( 1 · 𝑋) + ( 1 · 𝑋)))
29	9, 28	eqtr4d 2771	1 ⊢ (𝜑 → (𝑋 + 𝑋) = (( 1 + 1 ) · 𝑋))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3058  (class class class)co 7426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429

This theorem is referenced by:  rglcom4d  20158  srgo2times  20159  ringo2times  20218