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Theorem mndpfo 18783
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 18778 . 2 (𝐺 ∈ Mnd → :(𝐵 × 𝐵)⟶𝐵)
4 simpr 484 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → 𝑥𝐵)
5 eqid 2735 . . . . . . 7 (0g𝐺) = (0g𝐺)
61, 5mndidcl 18775 . . . . . 6 (𝐺 ∈ Mnd → (0g𝐺) ∈ 𝐵)
76adantr 480 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → (0g𝐺) ∈ 𝐵)
8 eqid 2735 . . . . . . 7 (+g𝐺) = (+g𝐺)
91, 8, 5mndrid 18781 . . . . . 6 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → (𝑥(+g𝐺)(0g𝐺)) = 𝑥)
109eqcomd 2741 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → 𝑥 = (𝑥(+g𝐺)(0g𝐺)))
11 rspceov 7480 . . . . 5 ((𝑥𝐵 ∧ (0g𝐺) ∈ 𝐵𝑥 = (𝑥(+g𝐺)(0g𝐺))) → ∃𝑦𝐵𝑧𝐵 𝑥 = (𝑦(+g𝐺)𝑧))
124, 7, 10, 11syl3anc 1370 . . . 4 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → ∃𝑦𝐵𝑧𝐵 𝑥 = (𝑦(+g𝐺)𝑧))
131, 8, 2plusfval 18673 . . . . . 6 ((𝑦𝐵𝑧𝐵) → (𝑦 𝑧) = (𝑦(+g𝐺)𝑧))
1413eqeq2d 2746 . . . . 5 ((𝑦𝐵𝑧𝐵) → (𝑥 = (𝑦 𝑧) ↔ 𝑥 = (𝑦(+g𝐺)𝑧)))
15142rexbiia 3216 . . . 4 (∃𝑦𝐵𝑧𝐵 𝑥 = (𝑦 𝑧) ↔ ∃𝑦𝐵𝑧𝐵 𝑥 = (𝑦(+g𝐺)𝑧))
1612, 15sylibr 234 . . 3 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → ∃𝑦𝐵𝑧𝐵 𝑥 = (𝑦 𝑧))
1716ralrimiva 3144 . 2 (𝐺 ∈ Mnd → ∀𝑥𝐵𝑦𝐵𝑧𝐵 𝑥 = (𝑦 𝑧))
18 foov 7607 . 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 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068   × cxp 5687  wf 6559  ontowfo 6561  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  +𝑓cplusf 18663  Mndcmnd 18760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-0g 17488  df-plusf 18665  df-mgm 18666  df-sgrp 18745  df-mnd 18761
This theorem is referenced by:  mndfo  18784  grpplusfo  18980
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