Theorem mndfo 17629

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 2800	. . . 4 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
3	1, 2	mndpfo 17628	. . 3 ⊢ (𝐺 ∈ Mnd → (+𝑓‘𝐺):(𝐵 × 𝐵)–onto→𝐵)
4	3	adantr 473	. 2 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → (+𝑓‘𝐺):(𝐵 × 𝐵)–onto→𝐵)
5		mndfo.p	. . . . . 6 ⊢ + = (+g‘𝐺)
6	1, 5, 2	plusfeq 17563	. . . . 5 ⊢ ( + Fn (𝐵 × 𝐵) → (+𝑓‘𝐺) = + )
7	6	eqcomd 2806	. . . 4 ⊢ ( + Fn (𝐵 × 𝐵) → + = (+𝑓‘𝐺))
8	7	adantl 474	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + = (+𝑓‘𝐺))
9		foeq1 6328	. . 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 249	1 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)–onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   = wceq 1653   ∈ wcel 2157   × cxp 5311   Fn wfn 6097  –onto→wfo 6100  ‘cfv 6102  Basecbs 16183  +_gcplusg 16266  +_𝑓cplusf 17553  Mndcmnd 17608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-fo 6108  df-fv 6110  df-riota 6840  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-1st 7402  df-2nd 7403  df-0g 16416  df-plusf 17555  df-mgm 17556  df-sgrp 17598  df-mnd 17609

This theorem is referenced by: (None)