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Theorem mndpfo 18196
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 = (+𝑓𝐺)
Assertion
Ref Expression
mndpfo (𝐺 ∈ Mnd → :(𝐵 × 𝐵)–onto𝐵)

Proof of Theorem mndpfo
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mndpf.b . . 3 𝐵 = (Base‘𝐺)
2 mndpf.p . . 3 = (+𝑓𝐺)
31, 2mndplusf 18191 . 2 (𝐺 ∈ Mnd → :(𝐵 × 𝐵)⟶𝐵)
4 simpr 488 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → 𝑥𝐵)
5 eqid 2737 . . . . . . 7 (0g𝐺) = (0g𝐺)
61, 5mndidcl 18188 . . . . . 6 (𝐺 ∈ Mnd → (0g𝐺) ∈ 𝐵)
76adantr 484 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → (0g𝐺) ∈ 𝐵)
8 eqid 2737 . . . . . . 7 (+g𝐺) = (+g𝐺)
91, 8, 5mndrid 18194 . . . . . 6 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → (𝑥(+g𝐺)(0g𝐺)) = 𝑥)
109eqcomd 2743 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → 𝑥 = (𝑥(+g𝐺)(0g𝐺)))
11 rspceov 7260 . . . . 5 ((𝑥𝐵 ∧ (0g𝐺) ∈ 𝐵𝑥 = (𝑥(+g𝐺)(0g𝐺))) → ∃𝑦𝐵𝑧𝐵 𝑥 = (𝑦(+g𝐺)𝑧))
124, 7, 10, 11syl3anc 1373 . . . 4 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → ∃𝑦𝐵𝑧𝐵 𝑥 = (𝑦(+g𝐺)𝑧))
131, 8, 2plusfval 18121 . . . . . 6 ((𝑦𝐵𝑧𝐵) → (𝑦 𝑧) = (𝑦(+g𝐺)𝑧))
1413eqeq2d 2748 . . . . 5 ((𝑦𝐵𝑧𝐵) → (𝑥 = (𝑦 𝑧) ↔ 𝑥 = (𝑦(+g𝐺)𝑧)))
15142rexbiia 3217 . . . 4 (∃𝑦𝐵𝑧𝐵 𝑥 = (𝑦 𝑧) ↔ ∃𝑦𝐵𝑧𝐵 𝑥 = (𝑦(+g𝐺)𝑧))
1612, 15sylibr 237 . . 3 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → ∃𝑦𝐵𝑧𝐵 𝑥 = (𝑦 𝑧))
1716ralrimiva 3105 . 2 (𝐺 ∈ Mnd → ∀𝑥𝐵𝑦𝐵𝑧𝐵 𝑥 = (𝑦 𝑧))
18 foov 7382 . 2 ( :(𝐵 × 𝐵)–onto𝐵 ↔ ( :(𝐵 × 𝐵)⟶𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 𝑥 = (𝑦 𝑧)))
193, 17, 18sylanbrc 586 1 (𝐺 ∈ Mnd → :(𝐵 × 𝐵)–onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  wral 3061  wrex 3062   × cxp 5549  wf 6376  ontowfo 6378  cfv 6380  (class class class)co 7213  Basecbs 16760  +gcplusg 16802  0gc0g 16944  +𝑓cplusf 18111  Mndcmnd 18173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fo 6386  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-0g 16946  df-plusf 18113  df-mgm 18114  df-sgrp 18163  df-mnd 18174
This theorem is referenced by:  mndfo  18197  grpplusfo  18380
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