Theorem mndpfo 18725

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 18720	. 2 ⊢ (𝐺 ∈ Mnd → ⨣ :(𝐵 × 𝐵)⟶𝐵)
4		simpr 484	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
5		eqid 2736	. . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺)
6	1, 5	mndidcl 18717	. . . . . 6 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝐵)
7	6	adantr 480	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → (0g‘𝐺) ∈ 𝐵)
8		eqid 2736	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
9	1, 8, 5	mndrid 18723	. . . . . 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 7416	. . . . 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 1374	. . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦(+g‘𝐺)𝑧))
13	1, 8, 2	plusfval 18615	. . . . . 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 3198	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ⨣ 𝑧) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦(+g‘𝐺)𝑧))
16	12, 15	sylibr 234	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ⨣ 𝑧))
17	16	ralrimiva 3129	. 2 ⊢ (𝐺 ∈ Mnd → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 𝑥 = (𝑦 ⨣ 𝑧))
18		foov 7541	. 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 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061   × cxp 5629  ⟶wf 6494  –onto→wfo 6496  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  0_gc0g 17402  +_𝑓cplusf 18605  Mndcmnd 18702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fo 6504  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-0g 17404  df-plusf 18607  df-mgm 18608  df-sgrp 18687  df-mnd 18703

This theorem is referenced by:  mndfo  18726  grpplusfo  18925