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Theorem mhmid 18946
Description: A surjective monoid morphism preserves identity element. (Contributed by Thierry Arnoux, 25-Jan-2020.)
Hypotheses
Ref Expression
ghmgrp.f ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
ghmgrp.x 𝑋 = (Base‘𝐺)
ghmgrp.y 𝑌 = (Base‘𝐻)
ghmgrp.p + = (+g𝐺)
ghmgrp.q = (+g𝐻)
ghmgrp.1 (𝜑𝐹:𝑋onto𝑌)
mhmmnd.3 (𝜑𝐺 ∈ Mnd)
mhmid.0 0 = (0g𝐺)
Assertion
Ref Expression
mhmid (𝜑 → (𝐹0 ) = (0g𝐻))
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥, + ,𝑦   𝑥,𝐻,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥, ,𝑦   𝜑,𝑥,𝑦   𝑥, 0 ,𝑦

Proof of Theorem mhmid
Dummy variables 𝑎 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghmgrp.y . 2 𝑌 = (Base‘𝐻)
2 eqid 2733 . 2 (0g𝐻) = (0g𝐻)
3 ghmgrp.q . 2 = (+g𝐻)
4 ghmgrp.1 . . . 4 (𝜑𝐹:𝑋onto𝑌)
5 fof 6806 . . . 4 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
64, 5syl 17 . . 3 (𝜑𝐹:𝑋𝑌)
7 mhmmnd.3 . . . 4 (𝜑𝐺 ∈ Mnd)
8 ghmgrp.x . . . . 5 𝑋 = (Base‘𝐺)
9 mhmid.0 . . . . 5 0 = (0g𝐺)
108, 9mndidcl 18640 . . . 4 (𝐺 ∈ Mnd → 0𝑋)
117, 10syl 17 . . 3 (𝜑0𝑋)
126, 11ffvelcdmd 7088 . 2 (𝜑 → (𝐹0 ) ∈ 𝑌)
13 simplll 774 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝜑)
14 ghmgrp.f . . . . . . 7 ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
1513, 14syl3an1 1164 . . . . . 6 (((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
167ad3antrrr 729 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝐺 ∈ Mnd)
1716, 10syl 17 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 0𝑋)
18 simplr 768 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝑖𝑋)
1915, 17, 18mhmlem 18945 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘( 0 + 𝑖)) = ((𝐹0 ) (𝐹𝑖)))
20 ghmgrp.p . . . . . . . 8 + = (+g𝐺)
218, 20, 9mndlid 18645 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝑖𝑋) → ( 0 + 𝑖) = 𝑖)
2216, 18, 21syl2anc 585 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ( 0 + 𝑖) = 𝑖)
2322fveq2d 6896 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘( 0 + 𝑖)) = (𝐹𝑖))
2419, 23eqtr3d 2775 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹0 ) (𝐹𝑖)) = (𝐹𝑖))
25 simpr 486 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹𝑖) = 𝑎)
2625oveq2d 7425 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹0 ) (𝐹𝑖)) = ((𝐹0 ) 𝑎))
2724, 26, 253eqtr3d 2781 . . 3 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹0 ) 𝑎) = 𝑎)
28 foelcdmi 6954 . . . 4 ((𝐹:𝑋onto𝑌𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
294, 28sylan 581 . . 3 ((𝜑𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
3027, 29r19.29a 3163 . 2 ((𝜑𝑎𝑌) → ((𝐹0 ) 𝑎) = 𝑎)
3115, 18, 17mhmlem 18945 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(𝑖 + 0 )) = ((𝐹𝑖) (𝐹0 )))
328, 20, 9mndrid 18646 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝑖𝑋) → (𝑖 + 0 ) = 𝑖)
3316, 18, 32syl2anc 585 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝑖 + 0 ) = 𝑖)
3433fveq2d 6896 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(𝑖 + 0 )) = (𝐹𝑖))
3531, 34eqtr3d 2775 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹𝑖) (𝐹0 )) = (𝐹𝑖))
3625oveq1d 7424 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹𝑖) (𝐹0 )) = (𝑎 (𝐹0 )))
3735, 36, 253eqtr3d 2781 . . 3 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝑎 (𝐹0 )) = 𝑎)
3837, 29r19.29a 3163 . 2 ((𝜑𝑎𝑌) → (𝑎 (𝐹0 )) = 𝑎)
391, 2, 3, 12, 30, 38ismgmid2 18587 1 (𝜑 → (𝐹0 ) = (0g𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wrex 3071  wf 6540  ontowfo 6542  cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  0gc0g 17385  Mndcmnd 18625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-riota 7365  df-ov 7412  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626
This theorem is referenced by:  mhmfmhm  18948  ghmgrp  18949
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