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Theorem mndfo 18709
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 2727 . . . 4 (+𝑓𝐺) = (+𝑓𝐺)
31, 2mndpfo 18708 . . 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 18599 . . . . 5 ( + Fn (𝐵 × 𝐵) → (+𝑓𝐺) = + )
76eqcomd 2733 . . . 4 ( + Fn (𝐵 × 𝐵) → + = (+𝑓𝐺))
87adantl 481 . . 3 ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + = (+𝑓𝐺))
9 foeq1 6801 . . 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 395   = wceq 1534  wcel 2099   × cxp 5670   Fn wfn 6537  ontowfo 6540  cfv 6542  Basecbs 17171  +gcplusg 17224  +𝑓cplusf 18588  Mndcmnd 18685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-0g 17414  df-plusf 18590  df-mgm 18591  df-sgrp 18670  df-mnd 18686
This theorem is referenced by: (None)
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