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Theorem mndpfo 18770
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 18765 . 2 (𝐺 ∈ Mnd → :(𝐵 × 𝐵)⟶𝐵)
4 simpr 484 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → 𝑥𝐵)
5 eqid 2737 . . . . . . 7 (0g𝐺) = (0g𝐺)
61, 5mndidcl 18762 . . . . . 6 (𝐺 ∈ Mnd → (0g𝐺) ∈ 𝐵)
76adantr 480 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → (0g𝐺) ∈ 𝐵)
8 eqid 2737 . . . . . . 7 (+g𝐺) = (+g𝐺)
91, 8, 5mndrid 18768 . . . . . 6 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → (𝑥(+g𝐺)(0g𝐺)) = 𝑥)
109eqcomd 2743 . . . . 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 1373 . . . 4 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → ∃𝑦𝐵𝑧𝐵 𝑥 = (𝑦(+g𝐺)𝑧))
131, 8, 2plusfval 18660 . . . . . 6 ((𝑦𝐵𝑧𝐵) → (𝑦 𝑧) = (𝑦(+g𝐺)𝑧))
1413eqeq2d 2748 . . . . 5 ((𝑦𝐵𝑧𝐵) → (𝑥 = (𝑦 𝑧) ↔ 𝑥 = (𝑦(+g𝐺)𝑧)))
15142rexbiia 3218 . . . 4 (∃𝑦𝐵𝑧𝐵 𝑥 = (𝑦 𝑧) ↔ ∃𝑦𝐵𝑧𝐵 𝑥 = (𝑦(+g𝐺)𝑧))
1612, 15sylibr 234 . . 3 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → ∃𝑦𝐵𝑧𝐵 𝑥 = (𝑦 𝑧))
1716ralrimiva 3146 . 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 1540  wcel 2108  wral 3061  wrex 3070   × cxp 5683  wf 6557  ontowfo 6559  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  0gc0g 17484  +𝑓cplusf 18650  Mndcmnd 18747
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-0g 17486  df-plusf 18652  df-mgm 18653  df-sgrp 18732  df-mnd 18748
This theorem is referenced by:  mndfo  18771  grpplusfo  18967
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