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Theorem mhmid 13704
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 2231 . 2 (0g𝐻) = (0g𝐻)
3 ghmgrp.q . 2 = (+g𝐻)
4 ghmgrp.1 . . . 4 (𝜑𝐹:𝑋onto𝑌)
5 fof 5559 . . . 4 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
64, 5syl 14 . . 3 (𝜑𝐹:𝑋𝑌)
7 mhmmnd.3 . . . 4 (𝜑𝐺 ∈ Mnd)
8 ghmgrp.x . . . . 5 𝑋 = (Base‘𝐺)
9 mhmid.0 . . . . 5 0 = (0g𝐺)
108, 9mndidcl 13515 . . . 4 (𝐺 ∈ Mnd → 0𝑋)
117, 10syl 14 . . 3 (𝜑0𝑋)
126, 11ffvelcdmd 5783 . 2 (𝜑 → (𝐹0 ) ∈ 𝑌)
13 simplll 535 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝜑)
14 ghmgrp.f . . . . . . 7 ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
1513, 14syl3an1 1306 . . . . . 6 (((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
167ad3antrrr 492 . . . . . . 7 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝐺 ∈ Mnd)
1716, 10syl 14 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 0𝑋)
18 simplr 529 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → 𝑖𝑋)
1915, 17, 18mhmlem 13703 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘( 0 + 𝑖)) = ((𝐹0 ) (𝐹𝑖)))
20 ghmgrp.p . . . . . . . 8 + = (+g𝐺)
218, 20, 9mndlid 13520 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝑖𝑋) → ( 0 + 𝑖) = 𝑖)
2216, 18, 21syl2anc 411 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ( 0 + 𝑖) = 𝑖)
2322fveq2d 5643 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘( 0 + 𝑖)) = (𝐹𝑖))
2419, 23eqtr3d 2266 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹0 ) (𝐹𝑖)) = (𝐹𝑖))
25 simpr 110 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹𝑖) = 𝑎)
2625oveq2d 6034 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹0 ) (𝐹𝑖)) = ((𝐹0 ) 𝑎))
2724, 26, 253eqtr3d 2272 . . 3 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹0 ) 𝑎) = 𝑎)
28 foelcdmi 5698 . . . 4 ((𝐹:𝑋onto𝑌𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
294, 28sylan 283 . . 3 ((𝜑𝑎𝑌) → ∃𝑖𝑋 (𝐹𝑖) = 𝑎)
3027, 29r19.29a 2676 . 2 ((𝜑𝑎𝑌) → ((𝐹0 ) 𝑎) = 𝑎)
3115, 18, 17mhmlem 13703 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(𝑖 + 0 )) = ((𝐹𝑖) (𝐹0 )))
328, 20, 9mndrid 13521 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝑖𝑋) → (𝑖 + 0 ) = 𝑖)
3316, 18, 32syl2anc 411 . . . . . 6 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝑖 + 0 ) = 𝑖)
3433fveq2d 5643 . . . . 5 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝐹‘(𝑖 + 0 )) = (𝐹𝑖))
3531, 34eqtr3d 2266 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹𝑖) (𝐹0 )) = (𝐹𝑖))
3625oveq1d 6033 . . . 4 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → ((𝐹𝑖) (𝐹0 )) = (𝑎 (𝐹0 )))
3735, 36, 253eqtr3d 2272 . . 3 ((((𝜑𝑎𝑌) ∧ 𝑖𝑋) ∧ (𝐹𝑖) = 𝑎) → (𝑎 (𝐹0 )) = 𝑎)
3837, 29r19.29a 2676 . 2 ((𝜑𝑎𝑌) → (𝑎 (𝐹0 )) = 𝑎)
391, 2, 3, 12, 30, 38ismgmid2 13465 1 (𝜑 → (𝐹0 ) = (0g𝐻))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 1004   = wceq 1397  wcel 2202  wrex 2511  wf 5322  ontowfo 5324  cfv 5326  (class class class)co 6018  Basecbs 13084  +gcplusg 13162  0gc0g 13341  Mndcmnd 13501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8123  ax-resscn 8124  ax-1re 8126  ax-addrcl 8129
This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-riota 5971  df-ov 6021  df-inn 9144  df-2 9202  df-ndx 13087  df-slot 13088  df-base 13090  df-plusg 13175  df-0g 13343  df-mgm 13441  df-sgrp 13487  df-mnd 13502
This theorem is referenced by:  mhmfmhm  13706  ghmgrp  13707
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