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Theorem mhmid 17744
 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 2771 . 2 (0g𝐻) = (0g𝐻)
3 ghmgrp.q . 2 = (+g𝐻)
4 ghmgrp.1 . . . 4 (𝜑𝐹:𝑋onto𝑌)
5 fof 6257 . . . 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 17516 . . . 4 (𝐺 ∈ Mnd → 0𝑋)
117, 10syl 17 . . 3 (𝜑0𝑋)
126, 11ffvelrnd 6505 . 2 (𝜑 → (𝐹0 ) ∈ 𝑌)
13 simplll 758 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝜑)
14 ghmgrp.f . . . . . . 7 ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
1513, 14syl3an1 1166 . . . . . 6 (((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
167ad3antrrr 709 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝐺 ∈ Mnd)
1716, 10syl 17 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 0𝑋)
18 simplr 752 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝑖𝑋)
1915, 17, 18mhmlem 17743 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘( 0 + 𝑖)) = ((𝐹0 ) (𝐹𝑖)))
20 ghmgrp.p . . . . . . . 8 + = (+g𝐺)
218, 20, 9mndlid 17519 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝑖𝑋) → ( 0 + 𝑖) = 𝑖)
2216, 18, 21syl2anc 573 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ( 0 + 𝑖) = 𝑖)
2322fveq2d 6337 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘( 0 + 𝑖)) = (𝐹𝑖))
2419, 23eqtr3d 2807 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹0 ) (𝐹𝑖)) = (𝐹𝑖))
25 simpr 471 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹𝑖) = 𝑎)
2625oveq2d 6812 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹0 ) (𝐹𝑖)) = ((𝐹0 ) 𝑎))
2724, 26, 253eqtr3d 2813 . . 3 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹0 ) 𝑎) = 𝑎)
28 foelrni 6388 . . . 4 ((𝐹:𝑋onto𝑌𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
294, 28sylan 569 . . 3 ((𝜑𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
3027, 29r19.29a 3226 . 2 ((𝜑𝑎𝑌) → ((𝐹0 ) 𝑎) = 𝑎)
3115, 18, 17mhmlem 17743 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(𝑖 + 0 )) = ((𝐹𝑖) (𝐹0 )))
328, 20, 9mndrid 17520 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝑖𝑋) → (𝑖 + 0 ) = 𝑖)
3316, 18, 32syl2anc 573 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝑖 + 0 ) = 𝑖)
3433fveq2d 6337 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(𝑖 + 0 )) = (𝐹𝑖))
3531, 34eqtr3d 2807 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹𝑖) (𝐹0 )) = (𝐹𝑖))
3625oveq1d 6811 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹𝑖) (𝐹0 )) = (𝑎 (𝐹0 )))
3735, 36, 253eqtr3d 2813 . . 3 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝑎 (𝐹0 )) = 𝑎)
3837, 29r19.29a 3226 . 2 ((𝜑𝑎𝑌) → (𝑎 (𝐹0 )) = 𝑎)
391, 2, 3, 12, 30, 38ismgmid2 17475 1 (𝜑 → (𝐹0 ) = (0g𝐻))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∃wrex 3062  ⟶wf 6026  –onto→wfo 6028  ‘cfv 6030  (class class class)co 6796  Basecbs 16064  +gcplusg 16149  0gc0g 16308  Mndcmnd 17502 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 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fo 6036  df-fv 6038  df-riota 6757  df-ov 6799  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503 This theorem is referenced by:  mhmfmhm  17746  ghmgrp  17747
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