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Theorem mhmid 17301
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 2605 . 2 (0g𝐻) = (0g𝐻)
3 ghmgrp.q . 2 = (+g𝐻)
4 ghmgrp.1 . . . 4 (𝜑𝐹:𝑋onto𝑌)
5 fof 6009 . . . 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 17073 . . . 4 (𝐺 ∈ Mnd → 0𝑋)
117, 10syl 17 . . 3 (𝜑0𝑋)
126, 11ffvelrnd 6249 . 2 (𝜑 → (𝐹0 ) ∈ 𝑌)
13 simplll 793 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝜑)
14 ghmgrp.f . . . . . . 7 ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
1513, 14syl3an1 1350 . . . . . 6 (((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
167ad3antrrr 761 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝐺 ∈ Mnd)
1716, 10syl 17 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 0𝑋)
18 simplr 787 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝑖𝑋)
1915, 17, 18mhmlem 17300 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘( 0 + 𝑖)) = ((𝐹0 ) (𝐹𝑖)))
20 ghmgrp.p . . . . . . . 8 + = (+g𝐺)
218, 20, 9mndlid 17076 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝑖𝑋) → ( 0 + 𝑖) = 𝑖)
2216, 18, 21syl2anc 690 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ( 0 + 𝑖) = 𝑖)
2322fveq2d 6088 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘( 0 + 𝑖)) = (𝐹𝑖))
2419, 23eqtr3d 2641 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹0 ) (𝐹𝑖)) = (𝐹𝑖))
25 simpr 475 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹𝑖) = 𝑎)
2625oveq2d 6539 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹0 ) (𝐹𝑖)) = ((𝐹0 ) 𝑎))
2724, 26, 253eqtr3d 2647 . . 3 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹0 ) 𝑎) = 𝑎)
28 foelrni 6135 . . . 4 ((𝐹:𝑋onto𝑌𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
294, 28sylan 486 . . 3 ((𝜑𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
3027, 29r19.29a 3055 . 2 ((𝜑𝑎𝑌) → ((𝐹0 ) 𝑎) = 𝑎)
3115, 18, 17mhmlem 17300 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(𝑖 + 0 )) = ((𝐹𝑖) (𝐹0 )))
328, 20, 9mndrid 17077 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝑖𝑋) → (𝑖 + 0 ) = 𝑖)
3316, 18, 32syl2anc 690 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝑖 + 0 ) = 𝑖)
3433fveq2d 6088 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(𝑖 + 0 )) = (𝐹𝑖))
3531, 34eqtr3d 2641 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹𝑖) (𝐹0 )) = (𝐹𝑖))
3625oveq1d 6538 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹𝑖) (𝐹0 )) = (𝑎 (𝐹0 )))
3735, 36, 253eqtr3d 2647 . . 3 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝑎 (𝐹0 )) = 𝑎)
3837, 29r19.29a 3055 . 2 ((𝜑𝑎𝑌) → (𝑎 (𝐹0 )) = 𝑎)
391, 2, 3, 12, 30, 38ismgmid2 17032 1 (𝜑 → (𝐹0 ) = (0g𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1975  wrex 2892  wf 5782  ontowfo 5784  cfv 5786  (class class class)co 6523  Basecbs 15637  +gcplusg 15710  0gc0g 15865  Mndcmnd 17059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-fo 5792  df-fv 5794  df-riota 6485  df-ov 6526  df-0g 15867  df-mgm 17007  df-sgrp 17049  df-mnd 17060
This theorem is referenced by:  mhmfmhm  17303  ghmgrp  17304
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