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Theorem mhmid 17583
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 2651 . 2 (0g𝐻) = (0g𝐻)
3 ghmgrp.q . 2 = (+g𝐻)
4 ghmgrp.1 . . . 4 (𝜑𝐹:𝑋onto𝑌)
5 fof 6153 . . . 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 17355 . . . 4 (𝐺 ∈ Mnd → 0𝑋)
117, 10syl 17 . . 3 (𝜑0𝑋)
126, 11ffvelrnd 6400 . 2 (𝜑 → (𝐹0 ) ∈ 𝑌)
13 simplll 813 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝜑)
14 ghmgrp.f . . . . . . 7 ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
1513, 14syl3an1 1399 . . . . . 6 (((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
167ad3antrrr 766 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝐺 ∈ Mnd)
1716, 10syl 17 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 0𝑋)
18 simplr 807 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝑖𝑋)
1915, 17, 18mhmlem 17582 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘( 0 + 𝑖)) = ((𝐹0 ) (𝐹𝑖)))
20 ghmgrp.p . . . . . . . 8 + = (+g𝐺)
218, 20, 9mndlid 17358 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝑖𝑋) → ( 0 + 𝑖) = 𝑖)
2216, 18, 21syl2anc 694 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ( 0 + 𝑖) = 𝑖)
2322fveq2d 6233 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘( 0 + 𝑖)) = (𝐹𝑖))
2419, 23eqtr3d 2687 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹0 ) (𝐹𝑖)) = (𝐹𝑖))
25 simpr 476 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹𝑖) = 𝑎)
2625oveq2d 6706 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹0 ) (𝐹𝑖)) = ((𝐹0 ) 𝑎))
2724, 26, 253eqtr3d 2693 . . 3 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹0 ) 𝑎) = 𝑎)
28 foelrni 6283 . . . 4 ((𝐹:𝑋onto𝑌𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
294, 28sylan 487 . . 3 ((𝜑𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
3027, 29r19.29a 3107 . 2 ((𝜑𝑎𝑌) → ((𝐹0 ) 𝑎) = 𝑎)
3115, 18, 17mhmlem 17582 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(𝑖 + 0 )) = ((𝐹𝑖) (𝐹0 )))
328, 20, 9mndrid 17359 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝑖𝑋) → (𝑖 + 0 ) = 𝑖)
3316, 18, 32syl2anc 694 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝑖 + 0 ) = 𝑖)
3433fveq2d 6233 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(𝑖 + 0 )) = (𝐹𝑖))
3531, 34eqtr3d 2687 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹𝑖) (𝐹0 )) = (𝐹𝑖))
3625oveq1d 6705 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹𝑖) (𝐹0 )) = (𝑎 (𝐹0 )))
3735, 36, 253eqtr3d 2693 . . 3 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝑎 (𝐹0 )) = 𝑎)
3837, 29r19.29a 3107 . 2 ((𝜑𝑎𝑌) → (𝑎 (𝐹0 )) = 𝑎)
391, 2, 3, 12, 30, 38ismgmid2 17314 1 (𝜑 → (𝐹0 ) = (0g𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wrex 2942  wf 5922  ontowfo 5924  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  0gc0g 16147  Mndcmnd 17341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-riota 6651  df-ov 6693  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342
This theorem is referenced by:  mhmfmhm  17585  ghmgrp  17586
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