Theorem mndfo 18685

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 2729	. . . 4 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
3	1, 2	mndpfo 18684	. . 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 18575	. . . . 5 ⊢ ( + Fn (𝐵 × 𝐵) → (+𝑓‘𝐺) = + )
7	6	eqcomd 2735	. . . 4 ⊢ ( + Fn (𝐵 × 𝐵) → + = (+𝑓‘𝐺))
8	7	adantl 481	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + = (+𝑓‘𝐺))
9		foeq1 6768	. . 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 1540   ∈ wcel 2109   × cxp 5636   Fn wfn 6506  –onto→wfo 6509  ‘cfv 6511  Basecbs 17179  +_gcplusg 17220  +_𝑓cplusf 18564  Mndcmnd 18661

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 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fo 6517  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-0g 17404  df-plusf 18566  df-mgm 18567  df-sgrp 18646  df-mnd 18662

This theorem is referenced by: (None)