Theorem mgmidpfod 18907

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)⟶onto𝐵)

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 18876	. . 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 18873	. . . . . . . . 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 2765	. . . . . . 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 3187	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ((𝑒 ⨣ 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
13	12	rexbidva 3188	. . . 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 3103	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 ((𝑒 ⨣ 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) → ∀𝑥 ∈ 𝐵 (𝑒 ⨣ 𝑥) = 𝑥)
17		oveq2 7459	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑒 ⨣ 𝑥) = (𝑒 ⨣ 𝑦))
18		id 23	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
19	17, 18	eqeq12d 2779	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑒 ⨣ 𝑥) = 𝑥 ↔ (𝑒 ⨣ 𝑦) = 𝑦))
20	19	rspcv 3576	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑒 ⨣ 𝑥) = 𝑥 → (𝑒 ⨣ 𝑦) = 𝑦))
21		eqcom 2770	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑒 ⨣ 𝑥) ↔ (𝑒 ⨣ 𝑥) = 𝑦)
22	17	eqeq1d 2765	. . . . . . . . . . 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 3580	. . . . . . . . 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 3181	. . . . 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 3158	. . 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 7627	. 2 ⊢ ( ⨣ :(𝐵 × 𝐵)⟶onto𝐵 ↔ ( ⨣ :(𝐵 × 𝐵)⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑒 ∈ 𝐵 ∃𝑥 ∈ 𝐵 𝑦 = (𝑒 ⨣ 𝑥)))
33	5, 31, 32	sylanbrc 595	1 ⊢ (𝜑 → ⨣ :(𝐵 × 𝐵)⟶onto𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   × cxp 5656  ⟶wf 6562  ⟶ontowfo 6564  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  +_gcplusg 17472  +_𝑓cplusf 18863  Mgmcmgm 18864

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fo 6572  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-plusf 18865  df-mgm 18866

This theorem is used by:  mgmidprnd  18908  mgmfod  18909  mndpfo  18997