Theorem mndfo 18861

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, 17-Aug-2026.)

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

Dummy variables 𝑢 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mndfo.b	. 2 ⊢ 𝐵 = (Base‘𝐺)
2		mndfo.p	. 2 ⊢ + = (+g‘𝐺)
3		mndmgm 18843	. . 3 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mgm)
4	3	adantr 486	. 2 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → 𝐺 ∈ Mgm)
5	1, 2	mndid 18846	. . 3 ⊢ (𝐺 ∈ Mnd → ∃𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))
6	5	adantr 486	. 2 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → ∃𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))
7		simpr 490	. 2 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + Fn (𝐵 × 𝐵))
8	1, 2, 4, 6, 7	mgmfod 18772	1 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)–onto→𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   × cxp 5653   Fn wfn 6528  –onto→wfo 6531  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  +_gcplusg 17342  Mgmcmgm 18728  Mndcmnd 18836

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-plusf 18729  df-mgm 18730  df-sgrp 18821  df-mnd 18837

This theorem is used by: (None)