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Theorem plusfreseq 44331
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 17852 . . . 4 Fn (𝐵 × 𝐵)
4 fnfun 6432 . . . 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 17850 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (𝑥 𝑦) = (𝑥 + 𝑦))
109eqcomd 2828 . . . . 5 ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑥 𝑦))
1110rgen2 3193 . . . 4 𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑥 𝑦)
1211a1i 11 . . 3 (∅ ∉ ran → ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑥 𝑦))
13 fveq2 6652 . . . . . 6 (𝑝 = ⟨𝑥, 𝑦⟩ → ( +𝑝) = ( + ‘⟨𝑥, 𝑦⟩))
14 df-ov 7143 . . . . . 6 (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
1513, 14eqtr4di 2875 . . . . 5 (𝑝 = ⟨𝑥, 𝑦⟩ → ( +𝑝) = (𝑥 + 𝑦))
16 fveq2 6652 . . . . . 6 (𝑝 = ⟨𝑥, 𝑦⟩ → ( 𝑝) = ( ‘⟨𝑥, 𝑦⟩))
17 df-ov 7143 . . . . . 6 (𝑥 𝑦) = ( ‘⟨𝑥, 𝑦⟩)
1816, 17eqtr4di 2875 . . . . 5 (𝑝 = ⟨𝑥, 𝑦⟩ → ( 𝑝) = (𝑥 𝑦))
1915, 18eqeq12d 2838 . . . 4 (𝑝 = ⟨𝑥, 𝑦⟩ → (( +𝑝) = ( 𝑝) ↔ (𝑥 + 𝑦) = (𝑥 𝑦)))
2019ralxp 5689 . . 3 (∀𝑝 ∈ (𝐵 × 𝐵)( +𝑝) = ( 𝑝) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑥 𝑦))
2112, 20sylibr 237 . 2 (∅ ∉ ran → ∀𝑝 ∈ (𝐵 × 𝐵)( +𝑝) = ( 𝑝))
22 fndm 6434 . . . . 5 ( Fn (𝐵 × 𝐵) → dom = (𝐵 × 𝐵))
2322eqcomd 2828 . . . 4 ( Fn (𝐵 × 𝐵) → (𝐵 × 𝐵) = dom )
243, 23ax-mp 5 . . 3 (𝐵 × 𝐵) = dom
2524fveqressseq 6829 . 2 ((Fun ∧ ∅ ∉ ran ∧ ∀𝑝 ∈ (𝐵 × 𝐵)( +𝑝) = ( 𝑝)) → ( + ↾ (𝐵 × 𝐵)) = )
266, 7, 21, 25syl3anc 1368 1 (∅ ∉ ran → ( + ↾ (𝐵 × 𝐵)) = )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wnel 3115  wral 3130  c0 4265  cop 4545   × cxp 5530  dom cdm 5532  ran crn 5533  cres 5534  Fun wfun 6328   Fn wfn 6329  cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  +𝑓cplusf 17840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-plusf 17842
This theorem is referenced by:  mgmplusfreseq  44332
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