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Theorem mndpfo 17922
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 17917 . 2 (𝐺 ∈ Mnd → :(𝐵 × 𝐵)⟶𝐵)
4 simpr 485 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → 𝑥𝐵)
5 eqid 2826 . . . . . . 7 (0g𝐺) = (0g𝐺)
61, 5mndidcl 17914 . . . . . 6 (𝐺 ∈ Mnd → (0g𝐺) ∈ 𝐵)
76adantr 481 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → (0g𝐺) ∈ 𝐵)
8 eqid 2826 . . . . . . 7 (+g𝐺) = (+g𝐺)
91, 8, 5mndrid 17920 . . . . . 6 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → (𝑥(+g𝐺)(0g𝐺)) = 𝑥)
109eqcomd 2832 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → 𝑥 = (𝑥(+g𝐺)(0g𝐺)))
11 rspceov 7195 . . . . 5 ((𝑥𝐵 ∧ (0g𝐺) ∈ 𝐵𝑥 = (𝑥(+g𝐺)(0g𝐺))) → ∃𝑦𝐵𝑧𝐵 𝑥 = (𝑦(+g𝐺)𝑧))
124, 7, 10, 11syl3anc 1365 . . . 4 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → ∃𝑦𝐵𝑧𝐵 𝑥 = (𝑦(+g𝐺)𝑧))
131, 8, 2plusfval 17848 . . . . . 6 ((𝑦𝐵𝑧𝐵) → (𝑦 𝑧) = (𝑦(+g𝐺)𝑧))
1413eqeq2d 2837 . . . . 5 ((𝑦𝐵𝑧𝐵) → (𝑥 = (𝑦 𝑧) ↔ 𝑥 = (𝑦(+g𝐺)𝑧)))
15142rexbiia 3303 . . . 4 (∃𝑦𝐵𝑧𝐵 𝑥 = (𝑦 𝑧) ↔ ∃𝑦𝐵𝑧𝐵 𝑥 = (𝑦(+g𝐺)𝑧))
1612, 15sylibr 235 . . 3 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → ∃𝑦𝐵𝑧𝐵 𝑥 = (𝑦 𝑧))
1716ralrimiva 3187 . 2 (𝐺 ∈ Mnd → ∀𝑥𝐵𝑦𝐵𝑧𝐵 𝑥 = (𝑦 𝑧))
18 foov 7312 . 2 ( :(𝐵 × 𝐵)–onto𝐵 ↔ ( :(𝐵 × 𝐵)⟶𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 𝑥 = (𝑦 𝑧)))
193, 17, 18sylanbrc 583 1 (𝐺 ∈ Mnd → :(𝐵 × 𝐵)–onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wral 3143  wrex 3144   × cxp 5552  wf 6348  ontowfo 6350  cfv 6352  (class class class)co 7148  Basecbs 16473  +gcplusg 16555  0gc0g 16703  +𝑓cplusf 17839  Mndcmnd 17900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fo 6358  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681  df-0g 16705  df-plusf 17841  df-mgm 17842  df-sgrp 17890  df-mnd 17901
This theorem is referenced by:  mndfo  17923  grpplusfo  18046
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