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Theorem ghmcmn 18444
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 18415 . . . 4 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
97, 8syl 17 . . 3 (𝜑𝐺 ∈ Mnd)
101, 2, 3, 4, 5, 6, 9mhmmnd 17745 . 2 (𝜑𝐻 ∈ Mnd)
11 simp-6l 776 . . . . . . . . . . 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 770 . . . . . . . . . 10 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → 𝑎𝑋)
14 simplr 752 . . . . . . . . . 10 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → 𝑏𝑋)
152, 4cmncom 18416 . . . . . . . . . 10 ((𝐺 ∈ CMnd ∧ 𝑎𝑋𝑏𝑋) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
1612, 13, 14, 15syl3anc 1476 . . . . . . . . 9 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
1716fveq2d 6336 . . . . . . . 8 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑏 + 𝑎)))
1811, 1syl3an1 1166 . . . . . . . . 9 ((((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) ∧ 𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
1918, 13, 14mhmlem 17743 . . . . . . . 8 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝐹‘(𝑎 + 𝑏)) = ((𝐹𝑎) (𝐹𝑏)))
2018, 14, 13mhmlem 17743 . . . . . . . 8 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝐹‘(𝑏 + 𝑎)) = ((𝐹𝑏) (𝐹𝑎)))
2117, 19, 203eqtr3d 2813 . . . . . . 7 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → ((𝐹𝑎) (𝐹𝑏)) = ((𝐹𝑏) (𝐹𝑎)))
22 simpllr 760 . . . . . . . 8 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝐹𝑎) = 𝑖)
23 simpr 471 . . . . . . . 8 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝐹𝑏) = 𝑗)
2422, 23oveq12d 6811 . . . . . . 7 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → ((𝐹𝑎) (𝐹𝑏)) = (𝑖 𝑗))
2523, 22oveq12d 6811 . . . . . . 7 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → ((𝐹𝑏) (𝐹𝑎)) = (𝑗 𝑖))
2621, 24, 253eqtr3d 2813 . . . . . 6 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝑖 𝑗) = (𝑗 𝑖))
27 foelrni 6386 . . . . . . . . 9 ((𝐹:𝑋onto𝑌𝑗𝑌) → ∃𝑏𝑋 (𝐹𝑏) = 𝑗)
286, 27sylan 569 . . . . . . . 8 ((𝜑𝑗𝑌) → ∃𝑏𝑋 (𝐹𝑏) = 𝑗)
2928adantlr 694 . . . . . . 7 (((𝜑𝑖𝑌) ∧ 𝑗𝑌) → ∃𝑏𝑋 (𝐹𝑏) = 𝑗)
3029ad2antrr 705 . . . . . 6 (((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) → ∃𝑏𝑋 (𝐹𝑏) = 𝑗)
3126, 30r19.29a 3226 . . . . 5 (((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) → (𝑖 𝑗) = (𝑗 𝑖))
32 foelrni 6386 . . . . . . 7 ((𝐹:𝑋onto𝑌𝑖𝑌) → ∃𝑎𝑋 (𝐹𝑎) = 𝑖)
336, 32sylan 569 . . . . . 6 ((𝜑𝑖𝑌) → ∃𝑎𝑋 (𝐹𝑎) = 𝑖)
3433adantr 466 . . . . 5 (((𝜑𝑖𝑌) ∧ 𝑗𝑌) → ∃𝑎𝑋 (𝐹𝑎) = 𝑖)
3531, 34r19.29a 3226 . . . 4 (((𝜑𝑖𝑌) ∧ 𝑗𝑌) → (𝑖 𝑗) = (𝑗 𝑖))
3635anasss 457 . . 3 ((𝜑 ∧ (𝑖𝑌𝑗𝑌)) → (𝑖 𝑗) = (𝑗 𝑖))
3736ralrimivva 3120 . 2 (𝜑 → ∀𝑖𝑌𝑗𝑌 (𝑖 𝑗) = (𝑗 𝑖))
383, 5iscmn 18407 . 2 (𝐻 ∈ CMnd ↔ (𝐻 ∈ Mnd ∧ ∀𝑖𝑌𝑗𝑌 (𝑖 𝑗) = (𝑗 𝑖)))
3910, 37, 38sylanbrc 572 1 (𝜑𝐻 ∈ CMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wrex 3062  ontowfo 6029  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  Mndcmnd 17502  CMndccmn 18400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-riota 6754  df-ov 6796  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-cmn 18402
This theorem is referenced by:  ghmabl  18445
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