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Theorem o2timesd 20228
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 20294 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 → ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
o2timesd.u (𝜑1𝐵)
o2timesd.i (𝜑 → ∀𝑥𝐵 ( 1 · 𝑥) = 𝑥)
o2timesd.x (𝜑𝑋𝐵)
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
StepHypRef Expression
1 o2timesd.x . . . 4 (𝜑𝑋𝐵)
2 o2timesd.i . . . 4 (𝜑 → ∀𝑥𝐵 ( 1 · 𝑥) = 𝑥)
3 oveq2 7439 . . . . . . 7 (𝑥 = 𝑋 → ( 1 · 𝑥) = ( 1 · 𝑋))
4 id 22 . . . . . . 7 (𝑥 = 𝑋𝑥 = 𝑋)
53, 4eqeq12d 2751 . . . . . 6 (𝑥 = 𝑋 → (( 1 · 𝑥) = 𝑥 ↔ ( 1 · 𝑋) = 𝑋))
65rspcva 3620 . . . . 5 ((𝑋𝐵 ∧ ∀𝑥𝐵 ( 1 · 𝑥) = 𝑥) → ( 1 · 𝑋) = 𝑋)
76eqcomd 2741 . . . 4 ((𝑋𝐵 ∧ ∀𝑥𝐵 ( 1 · 𝑥) = 𝑥) → 𝑋 = ( 1 · 𝑋))
81, 2, 7syl2anc 584 . . 3 (𝜑𝑋 = ( 1 · 𝑋))
98, 8oveq12d 7449 . 2 (𝜑 → (𝑋 + 𝑋) = (( 1 · 𝑋) + ( 1 · 𝑋)))
10 o2timesd.u . . . 4 (𝜑1𝐵)
1110, 10, 13jca 1127 . . 3 (𝜑 → ( 1𝐵1𝐵𝑋𝐵))
12 o2timesd.e . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))
13 oveq1 7438 . . . . . 6 (𝑥 = 1 → (𝑥 + 𝑦) = ( 1 + 𝑦))
1413oveq1d 7446 . . . . 5 (𝑥 = 1 → ((𝑥 + 𝑦) · 𝑧) = (( 1 + 𝑦) · 𝑧))
15 oveq1 7438 . . . . . 6 (𝑥 = 1 → (𝑥 · 𝑧) = ( 1 · 𝑧))
1615oveq1d 7446 . . . . 5 (𝑥 = 1 → ((𝑥 · 𝑧) + (𝑦 · 𝑧)) = (( 1 · 𝑧) + (𝑦 · 𝑧)))
1714, 16eqeq12d 2751 . . . 4 (𝑥 = 1 → (((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)) ↔ (( 1 + 𝑦) · 𝑧) = (( 1 · 𝑧) + (𝑦 · 𝑧))))
18 oveq2 7439 . . . . . 6 (𝑦 = 1 → ( 1 + 𝑦) = ( 1 + 1 ))
1918oveq1d 7446 . . . . 5 (𝑦 = 1 → (( 1 + 𝑦) · 𝑧) = (( 1 + 1 ) · 𝑧))
20 oveq1 7438 . . . . . 6 (𝑦 = 1 → (𝑦 · 𝑧) = ( 1 · 𝑧))
2120oveq2d 7447 . . . . 5 (𝑦 = 1 → (( 1 · 𝑧) + (𝑦 · 𝑧)) = (( 1 · 𝑧) + ( 1 · 𝑧)))
2219, 21eqeq12d 2751 . . . 4 (𝑦 = 1 → ((( 1 + 𝑦) · 𝑧) = (( 1 · 𝑧) + (𝑦 · 𝑧)) ↔ (( 1 + 1 ) · 𝑧) = (( 1 · 𝑧) + ( 1 · 𝑧))))
23 oveq2 7439 . . . . 5 (𝑧 = 𝑋 → (( 1 + 1 ) · 𝑧) = (( 1 + 1 ) · 𝑋))
24 oveq2 7439 . . . . . 6 (𝑧 = 𝑋 → ( 1 · 𝑧) = ( 1 · 𝑋))
2524, 24oveq12d 7449 . . . . 5 (𝑧 = 𝑋 → (( 1 · 𝑧) + ( 1 · 𝑧)) = (( 1 · 𝑋) + ( 1 · 𝑋)))
2623, 25eqeq12d 2751 . . . 4 (𝑧 = 𝑋 → ((( 1 + 1 ) · 𝑧) = (( 1 · 𝑧) + ( 1 · 𝑧)) ↔ (( 1 + 1 ) · 𝑋) = (( 1 · 𝑋) + ( 1 · 𝑋))))
2717, 22, 26rspc3v 3638 . . 3 (( 1𝐵1𝐵𝑋𝐵) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)) → (( 1 + 1 ) · 𝑋) = (( 1 · 𝑋) + ( 1 · 𝑋))))
2811, 12, 27sylc 65 . 2 (𝜑 → (( 1 + 1 ) · 𝑋) = (( 1 · 𝑋) + ( 1 · 𝑋)))
299, 28eqtr4d 2778 1 (𝜑 → (𝑋 + 𝑋) = (( 1 + 1 ) · 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  (class class class)co 7431
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-ext 2706
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-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  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-iota 6516  df-fv 6571  df-ov 7434
This theorem is referenced by:  rglcom4d  20229  srgo2times  20230  ringo2times  20289
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