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Theorem plusfreseq 46073
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 18507 . . . 4 Fn (𝐵 × 𝐵)
4 fnfun 6603 . . . 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 18505 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (𝑥 𝑦) = (𝑥 + 𝑦))
109eqcomd 2743 . . . . 5 ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑥 𝑦))
1110rgen2 3195 . . . 4 𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑥 𝑦)
1211a1i 11 . . 3 (∅ ∉ ran → ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑥 𝑦))
13 fveq2 6843 . . . . . 6 (𝑝 = ⟨𝑥, 𝑦⟩ → ( +𝑝) = ( + ‘⟨𝑥, 𝑦⟩))
14 df-ov 7361 . . . . . 6 (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
1513, 14eqtr4di 2795 . . . . 5 (𝑝 = ⟨𝑥, 𝑦⟩ → ( +𝑝) = (𝑥 + 𝑦))
16 fveq2 6843 . . . . . 6 (𝑝 = ⟨𝑥, 𝑦⟩ → ( 𝑝) = ( ‘⟨𝑥, 𝑦⟩))
17 df-ov 7361 . . . . . 6 (𝑥 𝑦) = ( ‘⟨𝑥, 𝑦⟩)
1816, 17eqtr4di 2795 . . . . 5 (𝑝 = ⟨𝑥, 𝑦⟩ → ( 𝑝) = (𝑥 𝑦))
1915, 18eqeq12d 2753 . . . 4 (𝑝 = ⟨𝑥, 𝑦⟩ → (( +𝑝) = ( 𝑝) ↔ (𝑥 + 𝑦) = (𝑥 𝑦)))
2019ralxp 5798 . . 3 (∀𝑝 ∈ (𝐵 × 𝐵)( +𝑝) = ( 𝑝) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑥 𝑦))
2112, 20sylibr 233 . 2 (∅ ∉ ran → ∀𝑝 ∈ (𝐵 × 𝐵)( +𝑝) = ( 𝑝))
22 fndm 6606 . . . . 5 ( Fn (𝐵 × 𝐵) → dom = (𝐵 × 𝐵))
2322eqcomd 2743 . . . 4 ( Fn (𝐵 × 𝐵) → (𝐵 × 𝐵) = dom )
243, 23ax-mp 5 . . 3 (𝐵 × 𝐵) = dom
2524fveqressseq 7031 . 2 ((Fun ∧ ∅ ∉ ran ∧ ∀𝑝 ∈ (𝐵 × 𝐵)( +𝑝) = ( 𝑝)) → ( + ↾ (𝐵 × 𝐵)) = )
266, 7, 21, 25syl3anc 1372 1 (∅ ∉ ran → ( + ↾ (𝐵 × 𝐵)) = )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wnel 3050  wral 3065  c0 4283  cop 4593   × cxp 5632  dom cdm 5634  ran crn 5635  cres 5636  Fun wfun 6491   Fn wfn 6492  cfv 6497  (class class class)co 7358  Basecbs 17084  +gcplusg 17134  +𝑓cplusf 18495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-plusf 18497
This theorem is referenced by:  mgmplusfreseq  46074
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