Theorem mhmimalem 19025

Description: Lemma for mhmima 19026 and similar theorems, formerly part of proof for mhmima 19026. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 16-Feb-2025.)

Hypotheses

Ref	Expression
mhmimalem.f	⊢ (𝜑 → 𝐹 ∈ (𝑀 MndHom 𝑁))
mhmimalem.s	⊢ (𝜑 → 𝑋 ⊆ (Base‘𝑀))
mhmimalem.a	⊢ (𝜑 → ⊕ = (+g‘𝑀))
mhmimalem.p	⊢ (𝜑 → + = (+g‘𝑁))
mhmimalem.c	⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧 ⊕ 𝑥) ∈ 𝑋)

Assertion

Ref	Expression
mhmimalem	⊢ (𝜑 → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥 + 𝑦) ∈ (𝐹 “ 𝑋))

Distinct variable groups:   𝑥,𝐹,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑦, + ,𝑥,𝑧   𝜑,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑦)   ⊕ (𝑥, 𝑦, 𝑧)   𝑀(𝑥, 𝑦, 𝑧)   𝑁(𝑥, 𝑦, 𝑧)

Proof of Theorem mhmimalem

Step	Hyp	Ref	Expression
1		mhmimalem.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ (𝑀 MndHom 𝑁))
2	1	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝐹 ∈ (𝑀 MndHom 𝑁))
3		mhmimalem.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝑀))
4	3	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑋 ⊆ (Base‘𝑀))
5		simprl 783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
6	4, 5	sseldd 3909	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑧 ∈ (Base‘𝑀))
7		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
8	4, 7	sseldd 3909	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ (Base‘𝑀))
9		eqid 2760	. . . . . . . . . 10 ⊢ (Base‘𝑀) = (Base‘𝑀)
10		eqid 2760	. . . . . . . . . 10 ⊢ (+g‘𝑀) = (+g‘𝑀)
11		eqid 2760	. . . . . . . . . 10 ⊢ (+g‘𝑁) = (+g‘𝑁)
12	9, 10, 11	mhmlin 18993	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑧 ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐹‘(𝑧(+g‘𝑀)𝑥)) = ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)))
13	2, 6, 8, 12	syl3anc 1398	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑧(+g‘𝑀)𝑥)) = ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)))
14		mhmimalem.a	. . . . . . . . . . . 12 ⊢ (𝜑 → ⊕ = (+g‘𝑀))
15	14	oveqd 7437	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑧 ⊕ 𝑥) = (𝑧(+g‘𝑀)𝑥))
16	15	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘(𝑧 ⊕ 𝑥)) = (𝐹‘(𝑧(+g‘𝑀)𝑥)))
17		mhmimalem.p	. . . . . . . . . . 11 ⊢ (𝜑 → + = (+g‘𝑁))
18	17	oveqd 7437	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐹‘𝑧) + (𝐹‘𝑥)) = ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)))
19	16, 18	eqeq12d 2776	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹‘(𝑧 ⊕ 𝑥)) = ((𝐹‘𝑧) + (𝐹‘𝑥)) ↔ (𝐹‘(𝑧(+g‘𝑀)𝑥)) = ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥))))
20	19	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → ((𝐹‘(𝑧 ⊕ 𝑥)) = ((𝐹‘𝑧) + (𝐹‘𝑥)) ↔ (𝐹‘(𝑧(+g‘𝑀)𝑥)) = ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥))))
21	13, 20	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑧 ⊕ 𝑥)) = ((𝐹‘𝑧) + (𝐹‘𝑥)))
22		eqid 2760	. . . . . . . . . . . 12 ⊢ (Base‘𝑁) = (Base‘𝑁)
23	9, 22	mhmf 18989	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑀 MndHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
24	1, 23	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
25	24	ffnd 6710	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn (Base‘𝑀))
26	25	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝐹 Fn (Base‘𝑀))
27		mhmimalem.c	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧 ⊕ 𝑥) ∈ 𝑋)
28	27	3expb 1138	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑧 ⊕ 𝑥) ∈ 𝑋)
29		fnfvima 7239	. . . . . . . 8 ⊢ ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀) ∧ (𝑧 ⊕ 𝑥) ∈ 𝑋) → (𝐹‘(𝑧 ⊕ 𝑥)) ∈ (𝐹 “ 𝑋))
30	26, 4, 28, 29	syl3anc 1398	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑧 ⊕ 𝑥)) ∈ (𝐹 “ 𝑋))
31	21, 30	eqeltrrd 2861	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → ((𝐹‘𝑧) + (𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
32	31	anassrs 473	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑧) + (𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
33	32	ralrimiva 3154	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧) + (𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
34		oveq2 7428	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝑧) + 𝑦) = ((𝐹‘𝑧) + (𝐹‘𝑥)))
35	34	eleq1d 2845	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋) ↔ ((𝐹‘𝑧) + (𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
36	35	ralima 7243	. . . . . 6 ⊢ ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀)) → (∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧) + (𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
37	25, 3, 36	syl2anc 596	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧) + (𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
38	37	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧) + (𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
39	33, 38	mpbird 260	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋))
40	39	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋))
41		oveq1 7427	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥 + 𝑦) = ((𝐹‘𝑧) + 𝑦))
42	41	eleq1d 2845	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝑥 + 𝑦) ∈ (𝐹 “ 𝑋) ↔ ((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋)))
43	42	ralbidv 3185	. . . 4 ⊢ (𝑥 = (𝐹‘𝑧) → (∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥 + 𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋)))
44	43	ralima 7243	. . 3 ⊢ ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀)) → (∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥 + 𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋)))
45	25, 3, 44	syl2anc 596	. 2 ⊢ (𝜑 → (∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥 + 𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋)))
46	40, 45	mpbird 260	1 ⊢ (𝜑 → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥 + 𝑦) ∈ (𝐹 “ 𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895   “ cima 5653   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  Basecbs 17392  +_gcplusg 17433   MndHom cmhm 18981

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 7751

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-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  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 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8852  df-mhm 18983

This theorem is used by:  mhmima  19026  rhmimasubrng  20823