Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  plusfreseq Structured version   Visualization version   GIF version

Theorem plusfreseq 42278
Description: If the empty set is not contained in the range of the group addition function of an extensible structure (not necessarily a magma), the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.)
Hypotheses
Ref Expression
plusfreseq.1 𝐵 = (Base‘𝑀)
plusfreseq.2 + = (+g𝑀)
plusfreseq.3 = (+𝑓𝑀)
Assertion
Ref Expression
plusfreseq (∅ ∉ ran → ( + ↾ (𝐵 × 𝐵)) = )

Proof of Theorem plusfreseq
Dummy variables 𝑝 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 plusfreseq.1 . . . . 5 𝐵 = (Base‘𝑀)
2 plusfreseq.3 . . . . 5 = (+𝑓𝑀)
31, 2plusffn 17447 . . . 4 Fn (𝐵 × 𝐵)
4 fnfun 6145 . . . 4 ( Fn (𝐵 × 𝐵) → Fun )
53, 4ax-mp 5 . . 3 Fun
65a1i 11 . 2 (∅ ∉ ran → Fun )
7 id 22 . 2 (∅ ∉ ran → ∅ ∉ ran )
8 plusfreseq.2 . . . . . . 7 + = (+g𝑀)
91, 8, 2plusfval 17445 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (𝑥 𝑦) = (𝑥 + 𝑦))
109eqcomd 2762 . . . . 5 ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑥 𝑦))
1110rgen2a 3111 . . . 4 𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑥 𝑦)
1211a1i 11 . . 3 (∅ ∉ ran → ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑥 𝑦))
13 fveq2 6348 . . . . . 6 (𝑝 = ⟨𝑥, 𝑦⟩ → ( +𝑝) = ( + ‘⟨𝑥, 𝑦⟩))
14 df-ov 6812 . . . . . 6 (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
1513, 14syl6eqr 2808 . . . . 5 (𝑝 = ⟨𝑥, 𝑦⟩ → ( +𝑝) = (𝑥 + 𝑦))
16 fveq2 6348 . . . . . 6 (𝑝 = ⟨𝑥, 𝑦⟩ → ( 𝑝) = ( ‘⟨𝑥, 𝑦⟩))
17 df-ov 6812 . . . . . 6 (𝑥 𝑦) = ( ‘⟨𝑥, 𝑦⟩)
1816, 17syl6eqr 2808 . . . . 5 (𝑝 = ⟨𝑥, 𝑦⟩ → ( 𝑝) = (𝑥 𝑦))
1915, 18eqeq12d 2771 . . . 4 (𝑝 = ⟨𝑥, 𝑦⟩ → (( +𝑝) = ( 𝑝) ↔ (𝑥 + 𝑦) = (𝑥 𝑦)))
2019ralxp 5415 . . 3 (∀𝑝 ∈ (𝐵 × 𝐵)( +𝑝) = ( 𝑝) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑥 𝑦))
2112, 20sylibr 224 . 2 (∅ ∉ ran → ∀𝑝 ∈ (𝐵 × 𝐵)( +𝑝) = ( 𝑝))
22 fndm 6147 . . . . 5 ( Fn (𝐵 × 𝐵) → dom = (𝐵 × 𝐵))
2322eqcomd 2762 . . . 4 ( Fn (𝐵 × 𝐵) → (𝐵 × 𝐵) = dom )
243, 23ax-mp 5 . . 3 (𝐵 × 𝐵) = dom
2524fveqressseq 6514 . 2 ((Fun ∧ ∅ ∉ ran ∧ ∀𝑝 ∈ (𝐵 × 𝐵)( +𝑝) = ( 𝑝)) → ( + ↾ (𝐵 × 𝐵)) = )
266, 7, 21, 25syl3anc 1477 1 (∅ ∉ ran → ( + ↾ (𝐵 × 𝐵)) = )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1628  wcel 2135  wnel 3031  wral 3046  c0 4054  cop 4323   × cxp 5260  dom cdm 5262  ran crn 5263  cres 5264  Fun wfun 6039   Fn wfn 6040  cfv 6045  (class class class)co 6809  Basecbs 16055  +gcplusg 16139  +𝑓cplusf 17436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-nel 3032  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-1st 7329  df-2nd 7330  df-plusf 17438
This theorem is referenced by:  mgmplusfreseq  42279
  Copyright terms: Public domain W3C validator