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Theorem plusfreseq 42297
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 17458 . . . 4 Fn (𝐵 × 𝐵)
4 fnfun 6127 . . . 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 17456 . . . . . 6 ((𝑥𝐵𝑦𝐵) → (𝑥 𝑦) = (𝑥 + 𝑦))
109eqcomd 2777 . . . . 5 ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑥 𝑦))
1110rgen2a 3126 . . . 4 𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑥 𝑦)
1211a1i 11 . . 3 (∅ ∉ ran → ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑥 𝑦))
13 fveq2 6333 . . . . . 6 (𝑝 = ⟨𝑥, 𝑦⟩ → ( +𝑝) = ( + ‘⟨𝑥, 𝑦⟩))
14 df-ov 6799 . . . . . 6 (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
1513, 14syl6eqr 2823 . . . . 5 (𝑝 = ⟨𝑥, 𝑦⟩ → ( +𝑝) = (𝑥 + 𝑦))
16 fveq2 6333 . . . . . 6 (𝑝 = ⟨𝑥, 𝑦⟩ → ( 𝑝) = ( ‘⟨𝑥, 𝑦⟩))
17 df-ov 6799 . . . . . 6 (𝑥 𝑦) = ( ‘⟨𝑥, 𝑦⟩)
1816, 17syl6eqr 2823 . . . . 5 (𝑝 = ⟨𝑥, 𝑦⟩ → ( 𝑝) = (𝑥 𝑦))
1915, 18eqeq12d 2786 . . . 4 (𝑝 = ⟨𝑥, 𝑦⟩ → (( +𝑝) = ( 𝑝) ↔ (𝑥 + 𝑦) = (𝑥 𝑦)))
2019ralxp 5401 . . 3 (∀𝑝 ∈ (𝐵 × 𝐵)( +𝑝) = ( 𝑝) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑥 𝑦))
2112, 20sylibr 224 . 2 (∅ ∉ ran → ∀𝑝 ∈ (𝐵 × 𝐵)( +𝑝) = ( 𝑝))
22 fndm 6129 . . . . 5 ( Fn (𝐵 × 𝐵) → dom = (𝐵 × 𝐵))
2322eqcomd 2777 . . . 4 ( Fn (𝐵 × 𝐵) → (𝐵 × 𝐵) = dom )
243, 23ax-mp 5 . . 3 (𝐵 × 𝐵) = dom
2524fveqressseq 6500 . 2 ((Fun ∧ ∅ ∉ ran ∧ ∀𝑝 ∈ (𝐵 × 𝐵)( +𝑝) = ( 𝑝)) → ( + ↾ (𝐵 × 𝐵)) = )
266, 7, 21, 25syl3anc 1476 1 (∅ ∉ ran → ( + ↾ (𝐵 × 𝐵)) = )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wnel 3046  wral 3061  c0 4063  cop 4323   × cxp 5248  dom cdm 5250  ran crn 5251  cres 5252  Fun wfun 6024   Fn wfn 6025  cfv 6030  (class class class)co 6796  Basecbs 16064  +gcplusg 16149  +𝑓cplusf 17447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-plusf 17449
This theorem is referenced by:  mgmplusfreseq  42298
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