Theorem mhmimalem 18790

Description: Lemma for mhmima 18791 and similar theorems, formerly part of proof for mhmima 18791. (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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝐹 ∈ (𝑀 MndHom 𝑁))
3		mhmimalem.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝑀))
4	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑋 ⊆ (Base‘𝑀))
5		simprl 776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
6	4, 5	sseldd 3923	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑧 ∈ (Base‘𝑀))
7		simprr 778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
8	4, 7	sseldd 3923	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ (Base‘𝑀))
9		eqid 2740	. . . . . . . . . 10 ⊢ (Base‘𝑀) = (Base‘𝑀)
10		eqid 2740	. . . . . . . . . 10 ⊢ (+g‘𝑀) = (+g‘𝑀)
11		eqid 2740	. . . . . . . . . 10 ⊢ (+g‘𝑁) = (+g‘𝑁)
12	9, 10, 11	mhmlin 18759	. . . . . . . . 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 1379	. . . . . . . 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 7380	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑧 ⊕ 𝑥) = (𝑧(+g‘𝑀)𝑥))
16	15	fveq2d 6838	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘(𝑧 ⊕ 𝑥)) = (𝐹‘(𝑧(+g‘𝑀)𝑥)))
17		mhmimalem.p	. . . . . . . . . . 11 ⊢ (𝜑 → + = (+g‘𝑁))
18	17	oveqd 7380	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐹‘𝑧) + (𝐹‘𝑥)) = ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)))
19	16, 18	eqeq12d 2756	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹‘(𝑧 ⊕ 𝑥)) = ((𝐹‘𝑧) + (𝐹‘𝑥)) ↔ (𝐹‘(𝑧(+g‘𝑀)𝑥)) = ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥))))
20	19	adantr 481	. . . . . . . 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 258	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑧 ⊕ 𝑥)) = ((𝐹‘𝑧) + (𝐹‘𝑥)))
22		eqid 2740	. . . . . . . . . . . 12 ⊢ (Base‘𝑁) = (Base‘𝑁)
23	9, 22	mhmf 18755	. . . . . . . . . . 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 6663	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn (Base‘𝑀))
26	25	adantr 481	. . . . . . . 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 1126	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑧 ⊕ 𝑥) ∈ 𝑋)
29		fnfvima 7184	. . . . . . . 8 ⊢ ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀) ∧ (𝑧 ⊕ 𝑥) ∈ 𝑋) → (𝐹‘(𝑧 ⊕ 𝑥)) ∈ (𝐹 “ 𝑋))
30	26, 4, 28, 29	syl3anc 1379	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑧 ⊕ 𝑥)) ∈ (𝐹 “ 𝑋))
31	21, 30	eqeltrrd 2841	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → ((𝐹‘𝑧) + (𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
32	31	anassrs 468	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑧) + (𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
33	32	ralrimiva 3132	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧) + (𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
34		oveq2 7371	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝑧) + 𝑦) = ((𝐹‘𝑧) + (𝐹‘𝑥)))
35	34	eleq1d 2825	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋) ↔ ((𝐹‘𝑧) + (𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
36	35	ralima 7188	. . . . . 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 590	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧) + (𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
38	37	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧) + (𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
39	33, 38	mpbird 258	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋))
40	39	ralrimiva 3132	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋))
41		oveq1 7370	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥 + 𝑦) = ((𝐹‘𝑧) + 𝑦))
42	41	eleq1d 2825	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝑥 + 𝑦) ∈ (𝐹 “ 𝑋) ↔ ((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋)))
43	42	ralbidv 3163	. . . 4 ⊢ (𝑥 = (𝐹‘𝑧) → (∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥 + 𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧) + 𝑦) ∈ (𝐹 “ 𝑋)))
44	43	ralima 7188	. . 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 590	. 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 258	1 ⊢ (𝜑 → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥 + 𝑦) ∈ (𝐹 “ 𝑋))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054   ⊆ wss 3890   “ cima 5628   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  +_gcplusg 17218   MndHom cmhm 18747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-map 8772  df-mhm 18749

This theorem is referenced by:  mhmima  18791  rhmimasubrng  20545