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Theorem mhmmnd 18697

Description: The image of a monoid 𝐺 under a monoid homomorphism 𝐹 is a monoid. (Contributed by Thierry Arnoux, 25-Jan-2020.)

**Hypotheses**
Ref	Expression
ghmgrp.f	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
ghmgrp.x	⊢ 𝑋 = (Base‘𝐺)
ghmgrp.y	⊢ 𝑌 = (Base‘𝐻)
ghmgrp.p	⊢ + = (+_g‘𝐺)
ghmgrp.q	⊢ ⨣ = (+_g‘𝐻)
ghmgrp.1	⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
mhmmnd.3	⊢ (𝜑 → 𝐺 ∈ Mnd)

**Assertion**
Ref	Expression
mhmmnd	⊢ (𝜑 → 𝐻 ∈ Mnd)

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

Proof of Theorem mhmmnd Dummy variables 𝑎 𝑑 𝑖 𝑗 𝑘 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpllr 773	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → (𝐹‘𝑖) = 𝑎)
2		simpr 485	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → (𝐹‘𝑗) = 𝑏)
3	1, 2	oveq12d 7293	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → ((𝐹‘𝑖) ⨣ (𝐹‘𝑗)) = (𝑎 ⨣ 𝑏))
4		simp-5l 782	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → 𝜑)
5		ghmgrp.f	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
6	4, 5	syl3an1 1162	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
7		simp-4r 781	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → 𝑖 ∈ 𝑋)
8		simplr 766	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → 𝑗 ∈ 𝑋)
9	6, 7, 8	mhmlem 18695	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → (𝐹‘(𝑖 + 𝑗)) = ((𝐹‘𝑖) ⨣ (𝐹‘𝑗)))
10		ghmgrp.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
11		fof 6688	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
12	10, 11	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
13	12	ad5antr 731	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → 𝐹:𝑋⟶𝑌)
14		mhmmnd.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ Mnd)
15	14	ad5antr 731	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → 𝐺 ∈ Mnd)
16		ghmgrp.x	. . . . . . . . . . 11 ⊢ 𝑋 = (Base‘𝐺)
17		ghmgrp.p	. . . . . . . . . . 11 ⊢ + = (+_g‘𝐺)
18	16, 17	mndcl 18393	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋) → (𝑖 + 𝑗) ∈ 𝑋)
19	15, 7, 8, 18	syl3anc 1370	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → (𝑖 + 𝑗) ∈ 𝑋)
20	13, 19	ffvelrnd 6962	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → (𝐹‘(𝑖 + 𝑗)) ∈ 𝑌)
21	9, 20	eqeltrrd 2840	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → ((𝐹‘𝑖) ⨣ (𝐹‘𝑗)) ∈ 𝑌)
22	3, 21	eqeltrrd 2840	. . . . . 6 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → (𝑎 ⨣ 𝑏) ∈ 𝑌)
23		simpr 485	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌) → 𝑏 ∈ 𝑌)
24		foelrni 6831	. . . . . . . 8 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑏 ∈ 𝑌) → ∃𝑗 ∈ 𝑋 (𝐹‘𝑗) = 𝑏)
25	10, 23, 24	syl2an 596	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) → ∃𝑗 ∈ 𝑋 (𝐹‘𝑗) = 𝑏)
26	25	ad2antrr 723	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ∃𝑗 ∈ 𝑋 (𝐹‘𝑗) = 𝑏)
27	22, 26	r19.29a 3218	. . . . 5 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝑎 ⨣ 𝑏) ∈ 𝑌)
28		simpl 483	. . . . . 6 ⊢ ((𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌) → 𝑎 ∈ 𝑌)
29		foelrni 6831	. . . . . 6 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
30	10, 28, 29	syl2an 596	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
31	27, 30	r19.29a 3218	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) → (𝑎 ⨣ 𝑏) ∈ 𝑌)
32		simpll 764	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑐 ∈ 𝑌) → 𝜑)
33		simplrl 774	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑐 ∈ 𝑌) → 𝑎 ∈ 𝑌)
34		simplrr 775	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑐 ∈ 𝑌) → 𝑏 ∈ 𝑌)
35		simpr 485	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑐 ∈ 𝑌) → 𝑐 ∈ 𝑌)
36	14	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) → 𝐺 ∈ Mnd)
37	36	ad5antr 731	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → 𝐺 ∈ Mnd)
38		simp-6r 785	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → 𝑖 ∈ 𝑋)
39		simp-4r 781	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → 𝑗 ∈ 𝑋)
40		simplr 766	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → 𝑘 ∈ 𝑋)
41	16, 17	mndass 18394	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑋)) → ((𝑖 + 𝑗) + 𝑘) = (𝑖 + (𝑗 + 𝑘)))
42	37, 38, 39, 40, 41	syl13anc 1371	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → ((𝑖 + 𝑗) + 𝑘) = (𝑖 + (𝑗 + 𝑘)))
43	42	fveq2d 6778	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝐹‘((𝑖 + 𝑗) + 𝑘)) = (𝐹‘(𝑖 + (𝑗 + 𝑘))))
44		simp-7l 786	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → 𝜑)
45	44, 5	syl3an1 1162	. . . . . . . . . . . . 13 ⊢ (((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
46	37, 38, 39, 18	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝑖 + 𝑗) ∈ 𝑋)
47	45, 46, 40	mhmlem 18695	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝐹‘((𝑖 + 𝑗) + 𝑘)) = ((𝐹‘(𝑖 + 𝑗)) ⨣ (𝐹‘𝑘)))
48	16, 17	mndcl 18393	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ 𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) → (𝑗 + 𝑘) ∈ 𝑋)
49	37, 39, 40, 48	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝑗 + 𝑘) ∈ 𝑋)
50	45, 38, 49	mhmlem 18695	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝐹‘(𝑖 + (𝑗 + 𝑘))) = ((𝐹‘𝑖) ⨣ (𝐹‘(𝑗 + 𝑘))))
51	43, 47, 50	3eqtr3d 2786	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → ((𝐹‘(𝑖 + 𝑗)) ⨣ (𝐹‘𝑘)) = ((𝐹‘𝑖) ⨣ (𝐹‘(𝑗 + 𝑘))))
52		simp1 1135	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋) → 𝜑)
53	52, 5	syl3an1 1162	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
54		simp2 1136	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋) → 𝑖 ∈ 𝑋)
55		simp3 1137	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋) → 𝑗 ∈ 𝑋)
56	53, 54, 55	mhmlem 18695	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋) → (𝐹‘(𝑖 + 𝑗)) = ((𝐹‘𝑖) ⨣ (𝐹‘𝑗)))
57	44, 38, 39, 56	syl3anc 1370	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝐹‘(𝑖 + 𝑗)) = ((𝐹‘𝑖) ⨣ (𝐹‘𝑗)))
58	57	oveq1d 7290	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → ((𝐹‘(𝑖 + 𝑗)) ⨣ (𝐹‘𝑘)) = (((𝐹‘𝑖) ⨣ (𝐹‘𝑗)) ⨣ (𝐹‘𝑘)))
59	45, 39, 40	mhmlem 18695	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝐹‘(𝑗 + 𝑘)) = ((𝐹‘𝑗) ⨣ (𝐹‘𝑘)))
60	59	oveq2d 7291	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → ((𝐹‘𝑖) ⨣ (𝐹‘(𝑗 + 𝑘))) = ((𝐹‘𝑖) ⨣ ((𝐹‘𝑗) ⨣ (𝐹‘𝑘))))
61	51, 58, 60	3eqtr3d 2786	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (((𝐹‘𝑖) ⨣ (𝐹‘𝑗)) ⨣ (𝐹‘𝑘)) = ((𝐹‘𝑖) ⨣ ((𝐹‘𝑗) ⨣ (𝐹‘𝑘))))
62		simp-5r 783	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝐹‘𝑖) = 𝑎)
63		simpllr 773	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝐹‘𝑗) = 𝑏)
64	62, 63	oveq12d 7293	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → ((𝐹‘𝑖) ⨣ (𝐹‘𝑗)) = (𝑎 ⨣ 𝑏))
65		simpr 485	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (𝐹‘𝑘) = 𝑐)
66	64, 65	oveq12d 7293	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → (((𝐹‘𝑖) ⨣ (𝐹‘𝑗)) ⨣ (𝐹‘𝑘)) = ((𝑎 ⨣ 𝑏) ⨣ 𝑐))
67	63, 65	oveq12d 7293	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → ((𝐹‘𝑗) ⨣ (𝐹‘𝑘)) = (𝑏 ⨣ 𝑐))
68	62, 67	oveq12d 7293	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → ((𝐹‘𝑖) ⨣ ((𝐹‘𝑗) ⨣ (𝐹‘𝑘))) = (𝑎 ⨣ (𝑏 ⨣ 𝑐)))
69	61, 66, 68	3eqtr3d 2786	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) ∧ 𝑘 ∈ 𝑋) ∧ (𝐹‘𝑘) = 𝑐) → ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐)))
70		foelrni 6831	. . . . . . . . . . . 12 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑐 ∈ 𝑌) → ∃𝑘 ∈ 𝑋 (𝐹‘𝑘) = 𝑐)
71	10, 70	sylan 580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑌) → ∃𝑘 ∈ 𝑋 (𝐹‘𝑘) = 𝑐)
72	71	3ad2antr3 1189	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) → ∃𝑘 ∈ 𝑋 (𝐹‘𝑘) = 𝑐)
73	72	ad4antr 729	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → ∃𝑘 ∈ 𝑋 (𝐹‘𝑘) = 𝑐)
74	69, 73	r19.29a 3218	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐)))
75	25	3adantr3 1170	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) → ∃𝑗 ∈ 𝑋 (𝐹‘𝑗) = 𝑏)
76	75	ad2antrr 723	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ∃𝑗 ∈ 𝑋 (𝐹‘𝑗) = 𝑏)
77	74, 76	r19.29a 3218	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐)))
78	30	3adantr3 1170	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
79	77, 78	r19.29a 3218	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) → ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐)))
80	32, 33, 34, 35, 79	syl13anc 1371	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑐 ∈ 𝑌) → ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐)))
81	80	ralrimiva 3103	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) → ∀𝑐 ∈ 𝑌 ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐)))
82	31, 81	jca 512	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) → ((𝑎 ⨣ 𝑏) ∈ 𝑌 ∧ ∀𝑐 ∈ 𝑌 ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐))))
83	82	ralrimivva 3123	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑌 ∀𝑏 ∈ 𝑌 ((𝑎 ⨣ 𝑏) ∈ 𝑌 ∧ ∀𝑐 ∈ 𝑌 ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐))))
84		eqid 2738	. . . . . 6 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
85	16, 84	mndidcl 18400	. . . . 5 ⊢ (𝐺 ∈ Mnd → (0_g‘𝐺) ∈ 𝑋)
86	14, 85	syl 17	. . . 4 ⊢ (𝜑 → (0_g‘𝐺) ∈ 𝑋)
87	12, 86	ffvelrnd 6962	. . 3 ⊢ (𝜑 → (𝐹‘(0_g‘𝐺)) ∈ 𝑌)
88		simplll 772	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝜑)
89	88, 5	syl3an1 1162	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
90	14	ad3antrrr 727	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝐺 ∈ Mnd)
91	90, 85	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (0_g‘𝐺) ∈ 𝑋)
92		simplr 766	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝑖 ∈ 𝑋)
93	89, 91, 92	mhmlem 18695	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘((0_g‘𝐺) + 𝑖)) = ((𝐹‘(0_g‘𝐺)) ⨣ (𝐹‘𝑖)))
94	16, 17, 84	mndlid 18405	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ 𝑖 ∈ 𝑋) → ((0_g‘𝐺) + 𝑖) = 𝑖)
95	90, 92, 94	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((0_g‘𝐺) + 𝑖) = 𝑖)
96	95	fveq2d 6778	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘((0_g‘𝐺) + 𝑖)) = (𝐹‘𝑖))
97	93, 96	eqtr3d 2780	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘(0_g‘𝐺)) ⨣ (𝐹‘𝑖)) = (𝐹‘𝑖))
98		simpr 485	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘𝑖) = 𝑎)
99	98	oveq2d 7291	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘(0_g‘𝐺)) ⨣ (𝐹‘𝑖)) = ((𝐹‘(0_g‘𝐺)) ⨣ 𝑎))
100	97, 99, 98	3eqtr3d 2786	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘(0_g‘𝐺)) ⨣ 𝑎) = 𝑎)
101	89, 92, 91	mhmlem 18695	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(𝑖 + (0_g‘𝐺))) = ((𝐹‘𝑖) ⨣ (𝐹‘(0_g‘𝐺))))
102	16, 17, 84	mndrid 18406	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ 𝑖 ∈ 𝑋) → (𝑖 + (0_g‘𝐺)) = 𝑖)
103	90, 92, 102	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝑖 + (0_g‘𝐺)) = 𝑖)
104	103	fveq2d 6778	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(𝑖 + (0_g‘𝐺))) = (𝐹‘𝑖))
105	101, 104	eqtr3d 2780	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘𝑖) ⨣ (𝐹‘(0_g‘𝐺))) = (𝐹‘𝑖))
106	98	oveq1d 7290	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘𝑖) ⨣ (𝐹‘(0_g‘𝐺))) = (𝑎 ⨣ (𝐹‘(0_g‘𝐺))))
107	105, 106, 98	3eqtr3d 2786	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝑎 ⨣ (𝐹‘(0_g‘𝐺))) = 𝑎)
108	100, 107	jca 512	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (((𝐹‘(0_g‘𝐺)) ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ (𝐹‘(0_g‘𝐺))) = 𝑎))
109	10, 29	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
110	108, 109	r19.29a 3218	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → (((𝐹‘(0_g‘𝐺)) ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ (𝐹‘(0_g‘𝐺))) = 𝑎))
111	110	ralrimiva 3103	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑌 (((𝐹‘(0_g‘𝐺)) ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ (𝐹‘(0_g‘𝐺))) = 𝑎))
112		oveq1 7282	. . . . . 6 ⊢ (𝑑 = (𝐹‘(0_g‘𝐺)) → (𝑑 ⨣ 𝑎) = ((𝐹‘(0_g‘𝐺)) ⨣ 𝑎))
113	112	eqeq1d 2740	. . . . 5 ⊢ (𝑑 = (𝐹‘(0_g‘𝐺)) → ((𝑑 ⨣ 𝑎) = 𝑎 ↔ ((𝐹‘(0_g‘𝐺)) ⨣ 𝑎) = 𝑎))
114	113	ovanraleqv 7299	. . . 4 ⊢ (𝑑 = (𝐹‘(0_g‘𝐺)) → (∀𝑎 ∈ 𝑌 ((𝑑 ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ 𝑑) = 𝑎) ↔ ∀𝑎 ∈ 𝑌 (((𝐹‘(0_g‘𝐺)) ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ (𝐹‘(0_g‘𝐺))) = 𝑎)))
115	114	rspcev 3561	. . 3 ⊢ (((𝐹‘(0_g‘𝐺)) ∈ 𝑌 ∧ ∀𝑎 ∈ 𝑌 (((𝐹‘(0_g‘𝐺)) ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ (𝐹‘(0_g‘𝐺))) = 𝑎)) → ∃𝑑 ∈ 𝑌 ∀𝑎 ∈ 𝑌 ((𝑑 ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ 𝑑) = 𝑎))
116	87, 111, 115	syl2anc 584	. 2 ⊢ (𝜑 → ∃𝑑 ∈ 𝑌 ∀𝑎 ∈ 𝑌 ((𝑑 ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ 𝑑) = 𝑎))
117		ghmgrp.y	. . 3 ⊢ 𝑌 = (Base‘𝐻)
118		ghmgrp.q	. . 3 ⊢ ⨣ = (+_g‘𝐻)
119	117, 118	ismnd 18388	. 2 ⊢ (𝐻 ∈ Mnd ↔ (∀𝑎 ∈ 𝑌 ∀𝑏 ∈ 𝑌 ((𝑎 ⨣ 𝑏) ∈ 𝑌 ∧ ∀𝑐 ∈ 𝑌 ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐))) ∧ ∃𝑑 ∈ 𝑌 ∀𝑎 ∈ 𝑌 ((𝑑 ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ 𝑑) = 𝑎)))
120	83, 116, 119	sylanbrc 583	1 ⊢ (𝜑 → 𝐻 ∈ Mnd)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ⟶wf 6429 –onto→wfo 6431 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 +_gcplusg 16962 0_gc0g 17150 Mndcmnd 18385

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fo 6439 df-fv 6441 df-riota 7232 df-ov 7278 df-0g 17152 df-mgm 18326 df-sgrp 18375 df-mnd 18386

This theorem is referenced by: mhmfmhm 18698 ghmgrp 18699 ghmcmn 19433

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