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Theorem mndfo 18649
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 2733 . . . 4 (+𝑓𝐺) = (+𝑓𝐺)
31, 2mndpfo 18648 . . 3 (𝐺 ∈ Mnd → (+𝑓𝐺):(𝐵 × 𝐵)–onto𝐵)
43adantr 482 . 2 ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → (+𝑓𝐺):(𝐵 × 𝐵)–onto𝐵)
5 mndfo.p . . . . . 6 + = (+g𝐺)
61, 5, 2plusfeq 18569 . . . . 5 ( + Fn (𝐵 × 𝐵) → (+𝑓𝐺) = + )
76eqcomd 2739 . . . 4 ( + Fn (𝐵 × 𝐵) → + = (+𝑓𝐺))
87adantl 483 . . 3 ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + = (+𝑓𝐺))
9 foeq1 6802 . . 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 205  wa 397   = wceq 1542  wcel 2107   × cxp 5675   Fn wfn 6539  ontowfo 6542  cfv 6544  Basecbs 17144  +gcplusg 17197  +𝑓cplusf 18558  Mndcmnd 18625
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 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-0g 17387  df-plusf 18560  df-mgm 18561  df-sgrp 18610  df-mnd 18626
This theorem is referenced by: (None)
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