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Theorem o2timesd 20112
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 20176 depends on the (right) distributivity and the existence of a (left) multiplicative identity only. (Contributed by GΓ©rard Lang, 4-Dec-2014.) (Revised by AV, 1-Feb-2025.)
Hypotheses
Ref Expression
o2timesd.e (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)))
o2timesd.u (πœ‘ β†’ 1 ∈ 𝐡)
o2timesd.i (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 ( 1 Β· π‘₯) = π‘₯)
o2timesd.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
Assertion
Ref Expression
o2timesd (πœ‘ β†’ (𝑋 + 𝑋) = (( 1 + 1 ) Β· 𝑋))
Distinct variable groups:   π‘₯,𝐡,𝑦,𝑧   π‘₯,𝑋,𝑦,𝑧   π‘₯, 1 ,𝑦,𝑧   π‘₯, Β· ,𝑦,𝑧   π‘₯, + ,𝑦,𝑧
Allowed substitution hints:   πœ‘(π‘₯,𝑦,𝑧)

Proof of Theorem o2timesd
StepHypRef Expression
1 o2timesd.x . . . 4 (πœ‘ β†’ 𝑋 ∈ 𝐡)
2 o2timesd.i . . . 4 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 ( 1 Β· π‘₯) = π‘₯)
3 oveq2 7412 . . . . . . 7 (π‘₯ = 𝑋 β†’ ( 1 Β· π‘₯) = ( 1 Β· 𝑋))
4 id 22 . . . . . . 7 (π‘₯ = 𝑋 β†’ π‘₯ = 𝑋)
53, 4eqeq12d 2742 . . . . . 6 (π‘₯ = 𝑋 β†’ (( 1 Β· π‘₯) = π‘₯ ↔ ( 1 Β· 𝑋) = 𝑋))
65rspcva 3604 . . . . 5 ((𝑋 ∈ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 ( 1 Β· π‘₯) = π‘₯) β†’ ( 1 Β· 𝑋) = 𝑋)
76eqcomd 2732 . . . 4 ((𝑋 ∈ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 ( 1 Β· π‘₯) = π‘₯) β†’ 𝑋 = ( 1 Β· 𝑋))
81, 2, 7syl2anc 583 . . 3 (πœ‘ β†’ 𝑋 = ( 1 Β· 𝑋))
98, 8oveq12d 7422 . 2 (πœ‘ β†’ (𝑋 + 𝑋) = (( 1 Β· 𝑋) + ( 1 Β· 𝑋)))
10 o2timesd.u . . . 4 (πœ‘ β†’ 1 ∈ 𝐡)
1110, 10, 13jca 1125 . . 3 (πœ‘ β†’ ( 1 ∈ 𝐡 ∧ 1 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡))
12 o2timesd.e . . 3 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)))
13 oveq1 7411 . . . . . 6 (π‘₯ = 1 β†’ (π‘₯ + 𝑦) = ( 1 + 𝑦))
1413oveq1d 7419 . . . . 5 (π‘₯ = 1 β†’ ((π‘₯ + 𝑦) Β· 𝑧) = (( 1 + 𝑦) Β· 𝑧))
15 oveq1 7411 . . . . . 6 (π‘₯ = 1 β†’ (π‘₯ Β· 𝑧) = ( 1 Β· 𝑧))
1615oveq1d 7419 . . . . 5 (π‘₯ = 1 β†’ ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)) = (( 1 Β· 𝑧) + (𝑦 Β· 𝑧)))
1714, 16eqeq12d 2742 . . . 4 (π‘₯ = 1 β†’ (((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)) ↔ (( 1 + 𝑦) Β· 𝑧) = (( 1 Β· 𝑧) + (𝑦 Β· 𝑧))))
18 oveq2 7412 . . . . . 6 (𝑦 = 1 β†’ ( 1 + 𝑦) = ( 1 + 1 ))
1918oveq1d 7419 . . . . 5 (𝑦 = 1 β†’ (( 1 + 𝑦) Β· 𝑧) = (( 1 + 1 ) Β· 𝑧))
20 oveq1 7411 . . . . . 6 (𝑦 = 1 β†’ (𝑦 Β· 𝑧) = ( 1 Β· 𝑧))
2120oveq2d 7420 . . . . 5 (𝑦 = 1 β†’ (( 1 Β· 𝑧) + (𝑦 Β· 𝑧)) = (( 1 Β· 𝑧) + ( 1 Β· 𝑧)))
2219, 21eqeq12d 2742 . . . 4 (𝑦 = 1 β†’ ((( 1 + 𝑦) Β· 𝑧) = (( 1 Β· 𝑧) + (𝑦 Β· 𝑧)) ↔ (( 1 + 1 ) Β· 𝑧) = (( 1 Β· 𝑧) + ( 1 Β· 𝑧))))
23 oveq2 7412 . . . . 5 (𝑧 = 𝑋 β†’ (( 1 + 1 ) Β· 𝑧) = (( 1 + 1 ) Β· 𝑋))
24 oveq2 7412 . . . . . 6 (𝑧 = 𝑋 β†’ ( 1 Β· 𝑧) = ( 1 Β· 𝑋))
2524, 24oveq12d 7422 . . . . 5 (𝑧 = 𝑋 β†’ (( 1 Β· 𝑧) + ( 1 Β· 𝑧)) = (( 1 Β· 𝑋) + ( 1 Β· 𝑋)))
2623, 25eqeq12d 2742 . . . 4 (𝑧 = 𝑋 β†’ ((( 1 + 1 ) Β· 𝑧) = (( 1 Β· 𝑧) + ( 1 Β· 𝑧)) ↔ (( 1 + 1 ) Β· 𝑋) = (( 1 Β· 𝑋) + ( 1 Β· 𝑋))))
2717, 22, 26rspc3v 3622 . . 3 (( 1 ∈ 𝐡 ∧ 1 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)) β†’ (( 1 + 1 ) Β· 𝑋) = (( 1 Β· 𝑋) + ( 1 Β· 𝑋))))
2811, 12, 27sylc 65 . 2 (πœ‘ β†’ (( 1 + 1 ) Β· 𝑋) = (( 1 Β· 𝑋) + ( 1 Β· 𝑋)))
299, 28eqtr4d 2769 1 (πœ‘ β†’ (𝑋 + 𝑋) = (( 1 + 1 ) Β· 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  (class class class)co 7404
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 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407
This theorem is referenced by:  rglcom4d  20113  srgo2times  20114  ringo2times  20171
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