Theorem mndpfo 18453

Description: The addition operation of a monoid as a function is an onto function. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 11-Oct-2013.) (Revised by AV, 23-Jan-2020.)

Hypotheses

Ref	Expression
mndpf.b	⊢ 𝐵 = (Base‘𝐺)
mndpf.p	⊢ ⨣ = (+𝑓‘𝐺)

Assertion

Ref	Expression
mndpfo	⊢ (𝐺 ∈ Mnd → ⨣ :(𝐵 × 𝐵)–onto→𝐵)

Proof of Theorem mndpfo

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mndpf.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		mndpf.p	. . 3 ⊢ ⨣ = (+𝑓‘𝐺)
3	1, 2	mndplusf 18448	. 2 ⊢ (𝐺 ∈ Mnd → ⨣ :(𝐵 × 𝐵)⟶𝐵)
4		simpr 486	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
5		eqid 2736	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
6	1, 5	mndidcl 18445	. . . . . 6 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝐵)
7	6	adantr 482	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (0g‘𝐺) ∈ 𝐵)
8		eqid 2736	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
9	1, 8, 5	mndrid 18451	. . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥)
10	9	eqcomd 2742	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → 𝑥 = (𝑥(+g‘𝐺)(0g‘𝐺)))
11		rspceov 7354	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ (0g‘𝐺) ∈ 𝐵 ∧ 𝑥 = (𝑥(+g‘𝐺)(0g‘𝐺))) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦(+g‘𝐺)𝑧))
12	4, 7, 10, 11	syl3anc 1371	. . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦(+g‘𝐺)𝑧))
13	1, 8, 2	plusfval 18378	. . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦 ⨣ 𝑧) = (𝑦(+g‘𝐺)𝑧))
14	13	eqeq2d 2747	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥 = (𝑦 ⨣ 𝑧) ↔ 𝑥 = (𝑦(+g‘𝐺)𝑧)))
15	14	2rexbiia 3206	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ⨣ 𝑧) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦(+g‘𝐺)𝑧))
16	12, 15	sylibr 233	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ⨣ 𝑧))
17	16	ralrimiva 3140	. 2 ⊢ (𝐺 ∈ Mnd → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ⨣ 𝑧))
18		foov 7478	. 2 ⊢ ( ⨣ :(𝐵 × 𝐵)–onto→𝐵 ↔ ( ⨣ :(𝐵 × 𝐵)⟶𝐵 ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ⨣ 𝑧)))
19	3, 17, 18	sylanbrc 584	1 ⊢ (𝐺 ∈ Mnd → ⨣ :(𝐵 × 𝐵)–onto→𝐵)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ∀wral 3062  ∃wrex 3071   × cxp 5598  ⟶wf 6454  –onto→wfo 6456  ‘cfv 6458  (class class class)co 7307  Basecbs 16957  +_gcplusg 17007  0_gc0g 17195  +_𝑓cplusf 18368  Mndcmnd 18430

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-fo 6464  df-fv 6466  df-riota 7264  df-ov 7310  df-oprab 7311  df-mpo 7312  df-1st 7863  df-2nd 7864  df-0g 17197  df-plusf 18370  df-mgm 18371  df-sgrp 18420  df-mnd 18431

This theorem is referenced by:  mndfo  18454  grpplusfo  18637