Theorem mhmimalem 18758

Description: Lemma for mhmima 18759 and similar theorems, formerly part of proof for mhmima 18759. (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 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝐹 ∈ (𝑀 MndHom 𝑁))
3		mhmimalem.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝑀))
4	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑋 ⊆ (Base‘𝑀))
5		simprl 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
6	4, 5	sseldd 3950	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑧 ∈ (Base‘𝑀))
7		simprr 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
8	4, 7	sseldd 3950	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ (Base‘𝑀))
9		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘𝑀) = (Base‘𝑀)
10		eqid 2730	. . . . . . . . . 10 ⊢ (+g‘𝑀) = (+g‘𝑀)
11		eqid 2730	. . . . . . . . . 10 ⊢ (+g‘𝑁) = (+g‘𝑁)
12	9, 10, 11	mhmlin 18727	. . . . . . . . 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 1373	. . . . . . . 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 7407	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑧 ⊕ 𝑥) = (𝑧(+g‘𝑀)𝑥))
16	15	fveq2d 6865	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘(𝑧 ⊕ 𝑥)) = (𝐹‘(𝑧(+g‘𝑀)𝑥)))
17		mhmimalem.p	. . . . . . . . . . 11 ⊢ (𝜑 → + = (+g‘𝑁))
18	17	oveqd 7407	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐹‘𝑧) + (𝐹‘𝑥)) = ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)))
19	16, 18	eqeq12d 2746	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹‘(𝑧 ⊕ 𝑥)) = ((𝐹‘𝑧) + (𝐹‘𝑥)) ↔ (𝐹‘(𝑧(+g‘𝑀)𝑥)) = ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥))))
20	19	adantr 480	. . . . . . . 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 257	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑧 ⊕ 𝑥)) = ((𝐹‘𝑧) + (𝐹‘𝑥)))
22		eqid 2730	. . . . . . . . . . . 12 ⊢ (Base‘𝑁) = (Base‘𝑁)
23	9, 22	mhmf 18723	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑀 MndHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
24	1, 23	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
25	24	ffnd 6692	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn (Base‘𝑀))
26	25	adantr 480	. . . . . . . 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 1120	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑧 ⊕ 𝑥) ∈ 𝑋)
29		fnfvima 7210	. . . . . . . 8 ⊢ ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀) ∧ (𝑧 ⊕ 𝑥) ∈ 𝑋) → (𝐹‘(𝑧 ⊕ 𝑥)) ∈ (𝐹 “ 𝑋))
30	26, 4, 28, 29	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑧 ⊕ 𝑥)) ∈ (𝐹 “ 𝑋))
31	21, 30	eqeltrrd 2830	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → ((𝐹‘𝑧) + (𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
32	31	anassrs 467	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑧) + (𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
33	32	ralrimiva 3126	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧) + (𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
34		oveq2 7398	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝑧) + 𝑦) = ((𝐹‘𝑧) + (𝐹‘𝑥)))
35	34	eleq1d 2814	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋) ↔ ((𝐹‘𝑧) + (𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
36	35	ralima 7214	. . . . . 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 584	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧) + (𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
38	37	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧) + (𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
39	33, 38	mpbird 257	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋))
40	39	ralrimiva 3126	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋))
41		oveq1 7397	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥 + 𝑦) = ((𝐹‘𝑧) + 𝑦))
42	41	eleq1d 2814	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝑥 + 𝑦) ∈ (𝐹 “ 𝑋) ↔ ((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋)))
43	42	ralbidv 3157	. . . 4 ⊢ (𝑥 = (𝐹‘𝑧) → (∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥 + 𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋)))
44	43	ralima 7214	. . 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 584	. 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 257	1 ⊢ (𝜑 → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥 + 𝑦) ∈ (𝐹 “ 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045   ⊆ wss 3917   “ cima 5644   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227   MndHom cmhm 18715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-mhm 18717

This theorem is referenced by:  mhmima  18759  rhmimasubrng  20482