Theorem mndfoOLD 19000

Description: Obsolete version of mndfo 18998 as of 17-Aug-2026. 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.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
mndfoOLD.b	⊢ 𝐵 = (Base‘𝐺)
mndfoOLD.p	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
mndfoOLD	⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶onto𝐵)

Proof of Theorem mndfoOLD

Step	Hyp	Ref	Expression
1		mndfoOLD.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
2		eqid 2763	. . . 4 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
3	1, 2	mndpfo 18997	. . 3 ⊢ (𝐺 ∈ Mnd → (+𝑓‘𝐺):(𝐵 × 𝐵)⟶onto𝐵)
4	3	adantr 486	. 2 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → (+𝑓‘𝐺):(𝐵 × 𝐵)⟶onto𝐵)
5		mndfoOLD.p	. . . . . 6 ⊢ + = (+g‘𝐺)
6	1, 5, 2	plusfeq 18874	. . . . 5 ⊢ ( + Fn (𝐵 × 𝐵) → (+𝑓‘𝐺) = + )
7	6	eqcomd 2769	. . . 4 ⊢ ( + Fn (𝐵 × 𝐵) → + = (+𝑓‘𝐺))
8	7	adantl 487	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + = (+𝑓‘𝐺))
9		foeq1 6822	. . 3 ⊢ ( + = (+𝑓‘𝐺) → ( + :(𝐵 × 𝐵)⟶onto𝐵 ↔ (+𝑓‘𝐺):(𝐵 × 𝐵)⟶onto𝐵))
10	8, 9	syl 18	. 2 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → ( + :(𝐵 × 𝐵)⟶onto𝐵 ↔ (+𝑓‘𝐺):(𝐵 × 𝐵)⟶onto𝐵))
11	4, 10	mpbird 260	1 ⊢ ((𝐺 ∈ Mnd ∧ + Fn (𝐵 × 𝐵)) → + :(𝐵 × 𝐵)⟶onto𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   × cxp 5656   Fn wfn 6561  ⟶ontowfo 6564  ‘cfv 6566  Basecbs 17431  +_gcplusg 17472  +_𝑓cplusf 18863  Mndcmnd 18973

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fo 6572  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-plusf 18865  df-mgm 18866  df-sgrp 18958  df-mnd 18974

This theorem is used by: (None)