Theorem mndfo 18784

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	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
mndfo	⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)–onto→𝐵)

Proof of Theorem mndfo

Step	Hyp	Ref	Expression
1		mndfo.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
2		eqid 2735	. . . 4 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
3	1, 2	mndpfo 18783	. . 3 ⊢ (𝐺 ∈ Mnd → (+𝑓‘𝐺):(𝐵 × 𝐵)–onto→𝐵)
4	3	adantr 480	. 2 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → (+𝑓‘𝐺):(𝐵 × 𝐵)–onto→𝐵)
5		mndfo.p	. . . . . 6 ⊢ + = (+g‘𝐺)
6	1, 5, 2	plusfeq 18674	. . . . 5 ⊢ ( + Fn (𝐵 × 𝐵) → (+𝑓‘𝐺) = + )
7	6	eqcomd 2741	. . . 4 ⊢ ( + Fn (𝐵 × 𝐵) → + = (+𝑓‘𝐺))
8	7	adantl 481	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + = (+𝑓‘𝐺))
9		foeq1 6817	. . 3 ⊢ ( + = (+𝑓‘𝐺) → ( + :(𝐵 × 𝐵)–onto→𝐵 ↔ (+𝑓‘𝐺):(𝐵 × 𝐵)–onto→𝐵))
10	8, 9	syl 17	. 2 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → ( + :(𝐵 × 𝐵)–onto→𝐵 ↔ (+𝑓‘𝐺):(𝐵 × 𝐵)–onto→𝐵))
11	4, 10	mpbird 257	1 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)–onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   × cxp 5687   Fn wfn 6558  –onto→wfo 6561  ‘cfv 6563  Basecbs 17245  +_gcplusg 17298  +_𝑓cplusf 18663  Mndcmnd 18760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-0g 17488  df-plusf 18665  df-mgm 18666  df-sgrp 18745  df-mnd 18761

This theorem is referenced by: (None)