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Theorem mndfo 18619
Description: The addition operation of a monoid is an onto function (assuming it is a function). (Contributed by Mario Carneiro, 11-Oct-2013.) (Proof shortened by AV, 23-Jan-2020.)
Hypotheses
Ref Expression
mndfo.b 𝐵 = (Base‘𝐺)
mndfo.p + = (+g𝐺)
Assertion
Ref Expression
mndfo ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)–onto𝐵)

Proof of Theorem mndfo
StepHypRef Expression
1 mndfo.b . . . 4 𝐵 = (Base‘𝐺)
2 eqid 2729 . . . 4 (+𝑓𝐺) = (+𝑓𝐺)
31, 2mndpfo 18618 . . 3 (𝐺 ∈ Mnd → (+𝑓𝐺):(𝐵 × 𝐵)–onto𝐵)
43adantr 480 . 2 ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → (+𝑓𝐺):(𝐵 × 𝐵)–onto𝐵)
5 mndfo.p . . . . . 6 + = (+g𝐺)
61, 5, 2plusfeq 18509 . . . . 5 ( + Fn (𝐵 × 𝐵) → (+𝑓𝐺) = + )
76eqcomd 2735 . . . 4 ( + Fn (𝐵 × 𝐵) → + = (+𝑓𝐺))
87adantl 481 . . 3 ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + = (+𝑓𝐺))
9 foeq1 6726 . . 3 ( + = (+𝑓𝐺) → ( + :(𝐵 × 𝐵)–onto𝐵 ↔ (+𝑓𝐺):(𝐵 × 𝐵)–onto𝐵))
108, 9syl 17 . 2 ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → ( + :(𝐵 × 𝐵)–onto𝐵 ↔ (+𝑓𝐺):(𝐵 × 𝐵)–onto𝐵))
114, 10mpbird 257 1 ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)–onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2109   × cxp 5611   Fn wfn 6471  ontowfo 6474  cfv 6476  Basecbs 17107  +gcplusg 17148  +𝑓cplusf 18498  Mndcmnd 18595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5367  ax-un 7662
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4940  df-br 5089  df-opab 5151  df-mpt 5170  df-id 5508  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fo 6482  df-fv 6484  df-riota 7297  df-ov 7343  df-oprab 7344  df-mpo 7345  df-1st 7915  df-2nd 7916  df-0g 17332  df-plusf 18500  df-mgm 18501  df-sgrp 18580  df-mnd 18596
This theorem is referenced by: (None)
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