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Theorem plusfreseq 43916
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 17849 . . . 4 Fn (𝐵 × 𝐵)
4 fnfun 6446 . . . 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 17847 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (𝑥 𝑦) = (𝑥 + 𝑦))
109eqcomd 2824 . . . . 5 ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑥 𝑦))
1110rgen2 3200 . . . 4 𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑥 𝑦)
1211a1i 11 . . 3 (∅ ∉ ran → ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑥 𝑦))
13 fveq2 6663 . . . . . 6 (𝑝 = ⟨𝑥, 𝑦⟩ → ( +𝑝) = ( + ‘⟨𝑥, 𝑦⟩))
14 df-ov 7148 . . . . . 6 (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
1513, 14syl6eqr 2871 . . . . 5 (𝑝 = ⟨𝑥, 𝑦⟩ → ( +𝑝) = (𝑥 + 𝑦))
16 fveq2 6663 . . . . . 6 (𝑝 = ⟨𝑥, 𝑦⟩ → ( 𝑝) = ( ‘⟨𝑥, 𝑦⟩))
17 df-ov 7148 . . . . . 6 (𝑥 𝑦) = ( ‘⟨𝑥, 𝑦⟩)
1816, 17syl6eqr 2871 . . . . 5 (𝑝 = ⟨𝑥, 𝑦⟩ → ( 𝑝) = (𝑥 𝑦))
1915, 18eqeq12d 2834 . . . 4 (𝑝 = ⟨𝑥, 𝑦⟩ → (( +𝑝) = ( 𝑝) ↔ (𝑥 + 𝑦) = (𝑥 𝑦)))
2019ralxp 5705 . . 3 (∀𝑝 ∈ (𝐵 × 𝐵)( +𝑝) = ( 𝑝) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑥 𝑦))
2112, 20sylibr 235 . 2 (∅ ∉ ran → ∀𝑝 ∈ (𝐵 × 𝐵)( +𝑝) = ( 𝑝))
22 fndm 6448 . . . . 5 ( Fn (𝐵 × 𝐵) → dom = (𝐵 × 𝐵))
2322eqcomd 2824 . . . 4 ( Fn (𝐵 × 𝐵) → (𝐵 × 𝐵) = dom )
243, 23ax-mp 5 . . 3 (𝐵 × 𝐵) = dom
2524fveqressseq 6839 . 2 ((Fun ∧ ∅ ∉ ran ∧ ∀𝑝 ∈ (𝐵 × 𝐵)( +𝑝) = ( 𝑝)) → ( + ↾ (𝐵 × 𝐵)) = )
266, 7, 21, 25syl3anc 1363 1 (∅ ∉ ran → ( + ↾ (𝐵 × 𝐵)) = )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wnel 3120  wral 3135  c0 4288  cop 4563   × cxp 5546  dom cdm 5548  ran crn 5549  cres 5550  Fun wfun 6342   Fn wfn 6343  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  +𝑓cplusf 17837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-plusf 17839
This theorem is referenced by:  mgmplusfreseq  43917
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