Theorem mgmidpfod 18864

Description: The operation of a magma with identity as a function is an onto function. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 16-Aug-2026.)

Hypotheses

Ref	Expression
mgmidpfod.b	⊢ 𝐵 = (Base‘𝐺)
mgmidpfod.p	⊢ + = (+g‘𝐺)
mgmidpfod.g	⊢ (𝜑 → 𝐺 ∈ Mgm)
mgmidpfod.e	⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
mgmidpfod.f	⊢ ⨣ = (+𝑓‘𝐺)

Assertion

Ref	Expression
mgmidpfod	⊢ (𝜑 → ⨣ :(𝐵 × 𝐵)–onto→𝐵)

Distinct variable groups:   𝐵,𝑒,𝑥   𝜑,𝑒,𝑥   ⨣ ,𝑒,𝑥

Allowed substitution hints:   + (𝑥, 𝑒)   𝐺(𝑥, 𝑒)

Proof of Theorem mgmidpfod

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mgmidpfod.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Mgm)
2		mgmidpfod.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3		mgmidpfod.f	. . . 4 ⊢ ⨣ = (+𝑓‘𝐺)
4	2, 3	mgmplusf 18833	. . 3 ⊢ (𝐺 ∈ Mgm → ⨣ :(𝐵 × 𝐵)⟶𝐵)
5	1, 4	syl 18	. 2 ⊢ (𝜑 → ⨣ :(𝐵 × 𝐵)⟶𝐵)
6		mgmidpfod.e	. . . 4 ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
7		mgmidpfod.p	. . . . . . . . . 10 ⊢ + = (+g‘𝐺)
8	2, 7, 3	plusfval 18830	. . . . . . . . 9 ⊢ ((𝑒 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑒 ⨣ 𝑥) = (𝑒 + 𝑥))
9	8	adantll 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑒 ⨣ 𝑥) = (𝑒 + 𝑥))
10	9	eqeq1d 2762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑒 ⨣ 𝑥) = 𝑥 ↔ (𝑒 + 𝑥) = 𝑥))
11	10	anbi1d 643	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (((𝑒 ⨣ 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
12	11	ralbidva 3183	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ((𝑒 ⨣ 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
13	12	rexbidva 3184	. . . 4 ⊢ (𝜑 → (∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 ⨣ 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
14	6, 13	mpbird 260	. . 3 ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 ⨣ 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
15		simpl 488	. . . . . . . 8 ⊢ (((𝑒 ⨣ 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) → (𝑒 ⨣ 𝑥) = 𝑥)
16	15	ralimi 3099	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 ((𝑒 ⨣ 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) → ∀𝑥 ∈ 𝐵 (𝑒 ⨣ 𝑥) = 𝑥)
17		oveq2 7430	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑒 ⨣ 𝑥) = (𝑒 ⨣ 𝑦))
18		id 23	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
19	17, 18	eqeq12d 2776	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑒 ⨣ 𝑥) = 𝑥 ↔ (𝑒 ⨣ 𝑦) = 𝑦))
20	19	rspcv 3572	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑒 ⨣ 𝑥) = 𝑥 → (𝑒 ⨣ 𝑦) = 𝑦))
21		eqcom 2767	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑒 ⨣ 𝑥) ↔ (𝑒 ⨣ 𝑥) = 𝑦)
22	17	eqeq1d 2762	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑒 ⨣ 𝑥) = 𝑦 ↔ (𝑒 ⨣ 𝑦) = 𝑦))
23	21, 22	bitrid 286	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑦 = (𝑒 ⨣ 𝑥) ↔ (𝑒 ⨣ 𝑦) = 𝑦))
24	23	rspcev 3576	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝑒 ⨣ 𝑦) = 𝑦) → ∃𝑥 ∈ 𝐵 𝑦 = (𝑒 ⨣ 𝑥))
25	24	ex 418	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → ((𝑒 ⨣ 𝑦) = 𝑦 → ∃𝑥 ∈ 𝐵 𝑦 = (𝑒 ⨣ 𝑥)))
26	20, 25	syld 48	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑒 ⨣ 𝑥) = 𝑥 → ∃𝑥 ∈ 𝐵 𝑦 = (𝑒 ⨣ 𝑥)))
27	16, 26	syl5 35	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ((𝑒 ⨣ 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) → ∃𝑥 ∈ 𝐵 𝑦 = (𝑒 ⨣ 𝑥)))
28	27	reximdv 3177	. . . . 5 ⊢ (𝑦 ∈ 𝐵 → (∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 ⨣ 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) → ∃𝑒 ∈ 𝐵 ∃𝑥 ∈ 𝐵 𝑦 = (𝑒 ⨣ 𝑥)))
29	28	impcom 413	. . . 4 ⊢ ((∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 ⨣ 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ∧ 𝑦 ∈ 𝐵) → ∃𝑒 ∈ 𝐵 ∃𝑥 ∈ 𝐵 𝑦 = (𝑒 ⨣ 𝑥))
30	29	ralrimiva 3154	. . 3 ⊢ (∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 ⨣ 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) → ∀𝑦 ∈ 𝐵 ∃𝑒 ∈ 𝐵 ∃𝑥 ∈ 𝐵 𝑦 = (𝑒 ⨣ 𝑥))
31	14, 30	syl 18	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑒 ∈ 𝐵 ∃𝑥 ∈ 𝐵 𝑦 = (𝑒 ⨣ 𝑥))
32		foov 7597	. 2 ⊢ ( ⨣ :(𝐵 × 𝐵)–onto→𝐵 ↔ ( ⨣ :(𝐵 × 𝐵)⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑒 ∈ 𝐵 ∃𝑥 ∈ 𝐵 𝑦 = (𝑒 ⨣ 𝑥)))
33	5, 31, 32	sylanbrc 595	1 ⊢ (𝜑 → ⨣ :(𝐵 × 𝐵)–onto→𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   × cxp 5648  ⟶wf 6537  –onto→wfo 6539  ‘cfv 6541  (class class class)co 7422  Basecbs 17394  +_gcplusg 17435  +_𝑓cplusf 18820  Mgmcmgm 18821

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-fo 6547  df-fv 6549  df-ov 7425  df-oprab 7426  df-mpo 7427  df-1st 8003  df-2nd 8004  df-plusf 18822  df-mgm 18823

This theorem is used by:  mgmidprnd  18865  mgmfod  18866  mndpfo  18954