Theorem mhmmnd 12836

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 532	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → (𝐹‘𝑖) = 𝑎)
2		simpr 110	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → (𝐹‘𝑗) = 𝑏)
3	1, 2	oveq12d 5880	. . . . . . 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 541	. . . . . . . . . 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 1269	. . . . . . . . 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 540	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → 𝑖 ∈ 𝑋)
8		simplr 528	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → 𝑗 ∈ 𝑋)
9	6, 7, 8	mhmlem 12834	. . . . . . . 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 5427	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
12	10, 11	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
13	12	ad5antr 496	. . . . . . . . 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 496	. . . . . . . . . 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 12686	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋) → (𝑖 + 𝑗) ∈ 𝑋)
19	15, 7, 8, 18	syl3anc 1236	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → (𝑖 + 𝑗) ∈ 𝑋)
20	13, 19	ffvelrnd 5641	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → (𝐹‘(𝑖 + 𝑗)) ∈ 𝑌)
21	9, 20	eqeltrrd 2251	. . . . . . 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 2251	. . . . . 6 ⊢ ((((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑗 ∈ 𝑋) ∧ (𝐹‘𝑗) = 𝑏) → (𝑎 ⨣ 𝑏) ∈ 𝑌)
23		simpr 110	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌) → 𝑏 ∈ 𝑌)
24		foelrni 5557	. . . . . . . 8 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑏 ∈ 𝑌) → ∃𝑗 ∈ 𝑋 (𝐹‘𝑗) = 𝑏)
25	10, 23, 24	syl2an 289	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) → ∃𝑗 ∈ 𝑋 (𝐹‘𝑗) = 𝑏)
26	25	ad2antrr 488	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ∃𝑗 ∈ 𝑋 (𝐹‘𝑗) = 𝑏)
27	22, 26	r19.29a 2616	. . . . 5 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝑎 ⨣ 𝑏) ∈ 𝑌)
28		simpl 109	. . . . . 6 ⊢ ((𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌) → 𝑎 ∈ 𝑌)
29		foelrni 5557	. . . . . 6 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
30	10, 28, 29	syl2an 289	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
31	27, 30	r19.29a 2616	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) → (𝑎 ⨣ 𝑏) ∈ 𝑌)
32		simpll 527	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑐 ∈ 𝑌) → 𝜑)
33		simplrl 533	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑐 ∈ 𝑌) → 𝑎 ∈ 𝑌)
34		simplrr 534	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑐 ∈ 𝑌) → 𝑏 ∈ 𝑌)
35		simpr 110	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑐 ∈ 𝑌) → 𝑐 ∈ 𝑌)
36	14	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) → 𝐺 ∈ Mnd)
37	36	ad5antr 496	. . . . . . . . . . . . . 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 544	. . . . . . . . . . . . . 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 540	. . . . . . . . . . . . . 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 528	. . . . . . . . . . . . . 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 12687	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ (𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑋)) → ((𝑖 + 𝑗) + 𝑘) = (𝑖 + (𝑗 + 𝑘)))
42	37, 38, 39, 40, 41	syl13anc 1238	. . . . . . . . . . . . 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 5508	. . . . . . . . . . . 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 545	. . . . . . . . . . . . . 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 1269	. . . . . . . . . . . . 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 1236	. . . . . . . . . . . . 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 12834	. . . . . . . . . . . 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 12686	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ 𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑋) → (𝑗 + 𝑘) ∈ 𝑋)
49	37, 39, 40, 48	syl3anc 1236	. . . . . . . . . . . . 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 12834	. . . . . . . . . . . 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 2214	. . . . . . . . . . 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 995	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋) → 𝜑)
53	52, 5	syl3an1 1269	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
54		simp2 996	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋) → 𝑖 ∈ 𝑋)
55		simp3 997	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋) → 𝑗 ∈ 𝑋)
56	53, 54, 55	mhmlem 12834	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ 𝑗 ∈ 𝑋) → (𝐹‘(𝑖 + 𝑗)) = ((𝐹‘𝑖) ⨣ (𝐹‘𝑗)))
57	44, 38, 39, 56	syl3anc 1236	. . . . . . . . . . . 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 5877	. . . . . . . . . . 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 12834	. . . . . . . . . . . 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 5878	. . . . . . . . . . 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 2214	. . . . . . . . . 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 542	. . . . . . . . . . . 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 532	. . . . . . . . . . . 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 5880	. . . . . . . . . . 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 110	. . . . . . . . . . 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 5880	. . . . . . . . . 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 5880	. . . . . . . . . . 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 5880	. . . . . . . . . 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 2214	. . . . . . . . 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 5557	. . . . . . . . . . . 12 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑐 ∈ 𝑌) → ∃𝑘 ∈ 𝑋 (𝐹‘𝑘) = 𝑐)
71	10, 70	sylan 283	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑌) → ∃𝑘 ∈ 𝑋 (𝐹‘𝑘) = 𝑐)
72	71	3ad2antr3 1162	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) → ∃𝑘 ∈ 𝑋 (𝐹‘𝑘) = 𝑐)
73	72	ad4antr 494	. . . . . . . . 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 2616	. . . . . . . 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 1156	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) → ∃𝑗 ∈ 𝑋 (𝐹‘𝑗) = 𝑏)
76	75	ad2antrr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ∃𝑗 ∈ 𝑋 (𝐹‘𝑗) = 𝑏)
77	74, 76	r19.29a 2616	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐)))
78	30	3adantr3 1156	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
79	77, 78	r19.29a 2616	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌 ∧ 𝑐 ∈ 𝑌)) → ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐)))
80	32, 33, 34, 35, 79	syl13anc 1238	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) ∧ 𝑐 ∈ 𝑌) → ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐)))
81	80	ralrimiva 2546	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) → ∀𝑐 ∈ 𝑌 ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐)))
82	31, 81	jca 306	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑌 ∧ 𝑏 ∈ 𝑌)) → ((𝑎 ⨣ 𝑏) ∈ 𝑌 ∧ ∀𝑐 ∈ 𝑌 ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐))))
83	82	ralrimivva 2555	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑌 ∀𝑏 ∈ 𝑌 ((𝑎 ⨣ 𝑏) ∈ 𝑌 ∧ ∀𝑐 ∈ 𝑌 ((𝑎 ⨣ 𝑏) ⨣ 𝑐) = (𝑎 ⨣ (𝑏 ⨣ 𝑐))))
84		eqid 2173	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
85	16, 84	mndidcl 12693	. . . . 5 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝑋)
86	14, 85	syl 14	. . . 4 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝑋)
87	12, 86	ffvelrnd 5641	. . 3 ⊢ (𝜑 → (𝐹‘(0g‘𝐺)) ∈ 𝑌)
88		simplll 531	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝜑)
89	88, 5	syl3an1 1269	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
90	14	ad3antrrr 492	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝐺 ∈ Mnd)
91	90, 85	syl 14	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (0g‘𝐺) ∈ 𝑋)
92		simplr 528	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝑖 ∈ 𝑋)
93	89, 91, 92	mhmlem 12834	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘((0g‘𝐺) + 𝑖)) = ((𝐹‘(0g‘𝐺)) ⨣ (𝐹‘𝑖)))
94	16, 17, 84	mndlid 12698	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ 𝑖 ∈ 𝑋) → ((0g‘𝐺) + 𝑖) = 𝑖)
95	90, 92, 94	syl2anc 411	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((0g‘𝐺) + 𝑖) = 𝑖)
96	95	fveq2d 5508	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘((0g‘𝐺) + 𝑖)) = (𝐹‘𝑖))
97	93, 96	eqtr3d 2208	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘(0g‘𝐺)) ⨣ (𝐹‘𝑖)) = (𝐹‘𝑖))
98		simpr 110	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘𝑖) = 𝑎)
99	98	oveq2d 5878	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘(0g‘𝐺)) ⨣ (𝐹‘𝑖)) = ((𝐹‘(0g‘𝐺)) ⨣ 𝑎))
100	97, 99, 98	3eqtr3d 2214	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘(0g‘𝐺)) ⨣ 𝑎) = 𝑎)
101	89, 92, 91	mhmlem 12834	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(𝑖 + (0g‘𝐺))) = ((𝐹‘𝑖) ⨣ (𝐹‘(0g‘𝐺))))
102	16, 17, 84	mndrid 12699	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Mnd ∧ 𝑖 ∈ 𝑋) → (𝑖 + (0g‘𝐺)) = 𝑖)
103	90, 92, 102	syl2anc 411	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝑖 + (0g‘𝐺)) = 𝑖)
104	103	fveq2d 5508	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(𝑖 + (0g‘𝐺))) = (𝐹‘𝑖))
105	101, 104	eqtr3d 2208	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘𝑖) ⨣ (𝐹‘(0g‘𝐺))) = (𝐹‘𝑖))
106	98	oveq1d 5877	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘𝑖) ⨣ (𝐹‘(0g‘𝐺))) = (𝑎 ⨣ (𝐹‘(0g‘𝐺))))
107	105, 106, 98	3eqtr3d 2214	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝑎 ⨣ (𝐹‘(0g‘𝐺))) = 𝑎)
108	100, 107	jca 306	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (((𝐹‘(0g‘𝐺)) ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ (𝐹‘(0g‘𝐺))) = 𝑎))
109	10, 29	sylan 283	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
110	108, 109	r19.29a 2616	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → (((𝐹‘(0g‘𝐺)) ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ (𝐹‘(0g‘𝐺))) = 𝑎))
111	110	ralrimiva 2546	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑌 (((𝐹‘(0g‘𝐺)) ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ (𝐹‘(0g‘𝐺))) = 𝑎))
112		oveq1 5869	. . . . . 6 ⊢ (𝑑 = (𝐹‘(0g‘𝐺)) → (𝑑 ⨣ 𝑎) = ((𝐹‘(0g‘𝐺)) ⨣ 𝑎))
113	112	eqeq1d 2182	. . . . 5 ⊢ (𝑑 = (𝐹‘(0g‘𝐺)) → ((𝑑 ⨣ 𝑎) = 𝑎 ↔ ((𝐹‘(0g‘𝐺)) ⨣ 𝑎) = 𝑎))
114	113	ovanraleqv 5886	. . . 4 ⊢ (𝑑 = (𝐹‘(0g‘𝐺)) → (∀𝑎 ∈ 𝑌 ((𝑑 ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ 𝑑) = 𝑎) ↔ ∀𝑎 ∈ 𝑌 (((𝐹‘(0g‘𝐺)) ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ (𝐹‘(0g‘𝐺))) = 𝑎)))
115	114	rspcev 2837	. . 3 ⊢ (((𝐹‘(0g‘𝐺)) ∈ 𝑌 ∧ ∀𝑎 ∈ 𝑌 (((𝐹‘(0g‘𝐺)) ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ (𝐹‘(0g‘𝐺))) = 𝑎)) → ∃𝑑 ∈ 𝑌 ∀𝑎 ∈ 𝑌 ((𝑑 ⨣ 𝑎) = 𝑎 ∧ (𝑎 ⨣ 𝑑) = 𝑎))
116	87, 111, 115	syl2anc 411	. 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 12682	. 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 417	1 ⊢ (𝜑 → 𝐻 ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 976   = wceq 1351   ∈ wcel 2144  ∀wral 2451  ∃wrex 2452  ⟶wf 5201  –onto→wfo 5203  ‘cfv 5205  (class class class)co 5862  Basecbs 12425  +_gcplusg 12489  0_gc0g 12623  Mndcmnd 12679

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 707  ax-5 1443  ax-7 1444  ax-gen 1445  ax-ie1 1489  ax-ie2 1490  ax-8 1500  ax-10 1501  ax-11 1502  ax-i12 1503  ax-bndl 1505  ax-4 1506  ax-17 1522  ax-i9 1526  ax-ial 1530  ax-i5r 1531  ax-13 2146  ax-14 2147  ax-ext 2155  ax-sep 4113  ax-pow 4166  ax-pr 4200  ax-un 4424  ax-cnex 7874  ax-resscn 7875  ax-1re 7877  ax-addrcl 7880

This theorem depends on definitions:  df-bi 117  df-3an 978  df-tru 1354  df-nf 1457  df-sb 1759  df-eu 2025  df-mo 2026  df-clab 2160  df-cleq 2166  df-clel 2169  df-nfc 2304  df-ral 2456  df-rex 2457  df-reu 2458  df-rmo 2459  df-rab 2460  df-v 2735  df-sbc 2959  df-csb 3053  df-un 3128  df-in 3130  df-ss 3137  df-pw 3571  df-sn 3592  df-pr 3593  df-op 3595  df-uni 3803  df-int 3838  df-br 3996  df-opab 4057  df-mpt 4058  df-id 4284  df-xp 4623  df-rel 4624  df-cnv 4625  df-co 4626  df-dm 4627  df-rn 4628  df-res 4629  df-iota 5167  df-fun 5207  df-fn 5208  df-f 5209  df-fo 5211  df-fv 5213  df-riota 5818  df-ov 5865  df-inn 8888  df-2 8946  df-ndx 12428  df-slot 12429  df-base 12431  df-plusg 12502  df-0g 12625  df-mgm 12637  df-sgrp 12670  df-mnd 12680

This theorem is referenced by:  mhmfmhm  12837  ghmgrp  12838