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Theorem ghmcmn 18158
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 18129 . . . 4 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
97, 8syl 17 . . 3 (𝜑𝐺 ∈ Mnd)
101, 2, 3, 4, 5, 6, 9mhmmnd 17458 . 2 (𝜑𝐻 ∈ Mnd)
11 simp-6l 809 . . . . . . . . . . 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 806 . . . . . . . . . 10 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → 𝑎𝑋)
14 simplr 791 . . . . . . . . . 10 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → 𝑏𝑋)
152, 4cmncom 18130 . . . . . . . . . 10 ((𝐺 ∈ CMnd ∧ 𝑎𝑋𝑏𝑋) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
1612, 13, 14, 15syl3anc 1323 . . . . . . . . 9 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
1716fveq2d 6152 . . . . . . . 8 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑏 + 𝑎)))
1811, 1syl3an1 1356 . . . . . . . . 9 ((((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) ∧ 𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
1918, 13, 14mhmlem 17456 . . . . . . . 8 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝐹‘(𝑎 + 𝑏)) = ((𝐹𝑎) (𝐹𝑏)))
2018, 14, 13mhmlem 17456 . . . . . . . 8 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝐹‘(𝑏 + 𝑎)) = ((𝐹𝑏) (𝐹𝑎)))
2117, 19, 203eqtr3d 2663 . . . . . . 7 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → ((𝐹𝑎) (𝐹𝑏)) = ((𝐹𝑏) (𝐹𝑎)))
22 simpllr 798 . . . . . . . 8 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝐹𝑎) = 𝑖)
23 simpr 477 . . . . . . . 8 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝐹𝑏) = 𝑗)
2422, 23oveq12d 6622 . . . . . . 7 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → ((𝐹𝑎) (𝐹𝑏)) = (𝑖 𝑗))
2523, 22oveq12d 6622 . . . . . . 7 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → ((𝐹𝑏) (𝐹𝑎)) = (𝑗 𝑖))
2621, 24, 253eqtr3d 2663 . . . . . 6 (((((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) ∧ 𝑏𝑋) ∧ (𝐹𝑏) = 𝑗) → (𝑖 𝑗) = (𝑗 𝑖))
27 foelrni 6201 . . . . . . . . 9 ((𝐹:𝑋onto𝑌𝑗𝑌) → ∃𝑏𝑋 (𝐹𝑏) = 𝑗)
286, 27sylan 488 . . . . . . . 8 ((𝜑𝑗𝑌) → ∃𝑏𝑋 (𝐹𝑏) = 𝑗)
2928adantlr 750 . . . . . . 7 (((𝜑𝑖𝑌) ∧ 𝑗𝑌) → ∃𝑏𝑋 (𝐹𝑏) = 𝑗)
3029ad2antrr 761 . . . . . 6 (((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) → ∃𝑏𝑋 (𝐹𝑏) = 𝑗)
3126, 30r19.29a 3071 . . . . 5 (((((𝜑𝑖𝑌) ∧ 𝑗𝑌) ∧ 𝑎𝑋) ∧ (𝐹𝑎) = 𝑖) → (𝑖 𝑗) = (𝑗 𝑖))
32 foelrni 6201 . . . . . . 7 ((𝐹:𝑋onto𝑌𝑖𝑌) → ∃𝑎𝑋 (𝐹𝑎) = 𝑖)
336, 32sylan 488 . . . . . 6 ((𝜑𝑖𝑌) → ∃𝑎𝑋 (𝐹𝑎) = 𝑖)
3433adantr 481 . . . . 5 (((𝜑𝑖𝑌) ∧ 𝑗𝑌) → ∃𝑎𝑋 (𝐹𝑎) = 𝑖)
3531, 34r19.29a 3071 . . . 4 (((𝜑𝑖𝑌) ∧ 𝑗𝑌) → (𝑖 𝑗) = (𝑗 𝑖))
3635anasss 678 . . 3 ((𝜑 ∧ (𝑖𝑌𝑗𝑌)) → (𝑖 𝑗) = (𝑗 𝑖))
3736ralrimivva 2965 . 2 (𝜑 → ∀𝑖𝑌𝑗𝑌 (𝑖 𝑗) = (𝑗 𝑖))
383, 5iscmn 18121 . 2 (𝐻 ∈ CMnd ↔ (𝐻 ∈ Mnd ∧ ∀𝑖𝑌𝑗𝑌 (𝑖 𝑗) = (𝑗 𝑖)))
3910, 37, 38sylanbrc 697 1 (𝜑𝐻 ∈ CMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  ontowfo 5845  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  Mndcmnd 17215  CMndccmn 18114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855  df-riota 6565  df-ov 6607  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-cmn 18116
This theorem is referenced by:  ghmabl  18159
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