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Theorem ghmcmn 19864
Description: The image of a commutative monoid 𝐺 under a group homomorphism 𝐹 is a commutative monoid. (Contributed by Thierry Arnoux, 26-Jan-2020.)
Hypotheses
Ref Expression
ghmabl.x 𝑋 = (Base‘𝐺)
ghmabl.y 𝑌 = (Base‘𝐻)
ghmabl.p + = (+g𝐺)
ghmabl.q = (+g𝐻)
ghmabl.f ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
ghmabl.1 (𝜑𝐹:𝑋onto𝑌)
ghmcmn.3 (𝜑𝐺 ∈ CMnd)
Assertion
Ref Expression
ghmcmn (𝜑𝐻 ∈ CMnd)
Distinct variable groups:   𝑥, + ,𝑦   𝑥, ,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦

Proof of Theorem ghmcmn
Dummy variables 𝑎 𝑏 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghmabl.f . . 3 ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
2 ghmabl.x . . 3 𝑋 = (Base‘𝐺)
3 ghmabl.y . . 3 𝑌 = (Base‘𝐻)
4 ghmabl.p . . 3 + = (+g𝐺)
5 ghmabl.q . . 3 = (+g𝐻)
6 ghmabl.1 . . 3 (𝜑𝐹:𝑋onto𝑌)
7 ghmcmn.3 . . . 4 (𝜑𝐺 ∈ CMnd)
8 cmnmnd 19830 . . . 4 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
97, 8syl 17 . . 3 (𝜑𝐺 ∈ Mnd)
101, 2, 3, 4, 5, 6, 9mhmmnd 19095 . 2 (𝜑𝐻 ∈ Mnd)
11 simp-6l 787 . . . . . . . . . . 11 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → 𝜑)
1211, 7syl 17 . . . . . . . . . 10 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → 𝐺 ∈ CMnd)
13 simp-4r 784 . . . . . . . . . 10 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → 𝑎𝑋)
14 simplr 769 . . . . . . . . . 10 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → 𝑏𝑋)
152, 4cmncom 19831 . . . . . . . . . 10 ((𝐺 ∈ CMnd ∧ 𝑎𝑋𝑏𝑋) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
1612, 13, 14, 15syl3anc 1370 . . . . . . . . 9 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
1716fveq2d 6911 . . . . . . . 8 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑏 + 𝑎)))
1811, 1syl3an1 1162 . . . . . . . . 9 ((((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) ∧ 𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
1918, 13, 14mhmlem 19093 . . . . . . . 8 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝐹‘(𝑎 + 𝑏)) = ((𝐹𝑎) (𝐹𝑏)))
2018, 14, 13mhmlem 19093 . . . . . . . 8 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝐹‘(𝑏 + 𝑎)) = ((𝐹𝑏) (𝐹𝑎)))
2117, 19, 203eqtr3d 2783 . . . . . . 7 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → ((𝐹𝑎) (𝐹𝑏)) = ((𝐹𝑏) (𝐹𝑎)))
22 simpllr 776 . . . . . . . 8 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝐹𝑎) = 𝑖)
23 simpr 484 . . . . . . . 8 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝐹𝑏) = 𝑗)
2422, 23oveq12d 7449 . . . . . . 7 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → ((𝐹𝑎) (𝐹𝑏)) = (𝑖 𝑗))
2523, 22oveq12d 7449 . . . . . . 7 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → ((𝐹𝑏) (𝐹𝑎)) = (𝑗 𝑖))
2621, 24, 253eqtr3d 2783 . . . . . 6 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝑖 𝑗) = (𝑗 𝑖))
27 foelcdmi 6970 . . . . . . . 8 ((𝐹:𝑋onto𝑌𝑗𝑌) → ∃𝑏𝑋 (𝐹𝑏) = 𝑗)
286, 27sylan 580 . . . . . . 7 ((𝜑𝑗𝑌) → ∃𝑏𝑋 (𝐹𝑏) = 𝑗)
2928ad5ant13 757 . . . . . 6 (((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) → ∃𝑏𝑋 (𝐹𝑏) = 𝑗)
3026, 29r19.29a 3160 . . . . 5 (((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) → (𝑖 𝑗) = (𝑗 𝑖))
31 foelcdmi 6970 . . . . . . 7 ((𝐹:𝑋onto𝑌𝑖𝑌) → ∃𝑎𝑋 (𝐹𝑎) = 𝑖)
326, 31sylan 580 . . . . . 6 ((𝜑𝑖𝑌) → ∃𝑎𝑋 (𝐹𝑎) = 𝑖)
3332adantr 480 . . . . 5 (((𝜑𝑖𝑌) ∧ 𝑗𝑌) → ∃𝑎𝑋 (𝐹𝑎) = 𝑖)
3430, 33r19.29a 3160 . . . 4 (((𝜑𝑖𝑌) ∧ 𝑗𝑌) → (𝑖 𝑗) = (𝑗 𝑖))
3534anasss 466 . . 3 ((𝜑 ∧ (𝑖𝑌𝑗𝑌)) → (𝑖 𝑗) = (𝑗 𝑖))
3635ralrimivva 3200 . 2 (𝜑 → ∀𝑖𝑌𝑗𝑌 (𝑖 𝑗) = (𝑗 𝑖))
373, 5iscmn 19822 . 2 (𝐻 ∈ CMnd ↔ (𝐻 ∈ Mnd ∧ ∀𝑖𝑌𝑗𝑌 (𝑖 𝑗) = (𝑗 𝑖)))
3810, 36, 37sylanbrc 583 1 (𝜑𝐻 ∈ CMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  ontowfo 6561  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Mndcmnd 18760  CMndccmn 19813
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-pr 5438
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-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  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-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-cmn 19815
This theorem is referenced by:  ghmabl  19865
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