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Theorem mhmimalem 18744
Description: Lemma for mhmima 18745 and similar theorems, formerly part of proof for mhmima 18745. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 16-Feb-2025.)
Hypotheses
Ref Expression
mhmimalem.f (𝜑𝐹 ∈ (𝑀 MndHom 𝑁))
mhmimalem.s (𝜑𝑋 ⊆ (Base‘𝑀))
mhmimalem.a (𝜑 = (+g𝑀))
mhmimalem.p (𝜑+ = (+g𝑁))
mhmimalem.c ((𝜑𝑧𝑋𝑥𝑋) → (𝑧 𝑥) ∈ 𝑋)
Assertion
Ref Expression
mhmimalem (𝜑 → ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥 + 𝑦) ∈ (𝐹𝑋))
Distinct variable groups:   𝑥,𝐹,𝑦,𝑧   𝑥,𝑀,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑦, + ,𝑥,𝑧   𝜑,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑦)   (𝑥,𝑦,𝑧)   𝑁(𝑥,𝑦,𝑧)

Proof of Theorem mhmimalem
StepHypRef Expression
1 mhmimalem.f . . . . . . . . . 10 (𝜑𝐹 ∈ (𝑀 MndHom 𝑁))
21adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝑋𝑥𝑋)) → 𝐹 ∈ (𝑀 MndHom 𝑁))
3 mhmimalem.s . . . . . . . . . . 11 (𝜑𝑋 ⊆ (Base‘𝑀))
43adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝑋𝑥𝑋)) → 𝑋 ⊆ (Base‘𝑀))
5 simprl 768 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝑋𝑥𝑋)) → 𝑧𝑋)
64, 5sseldd 3983 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝑋𝑥𝑋)) → 𝑧 ∈ (Base‘𝑀))
7 simprr 770 . . . . . . . . . 10 ((𝜑 ∧ (𝑧𝑋𝑥𝑋)) → 𝑥𝑋)
84, 7sseldd 3983 . . . . . . . . 9 ((𝜑 ∧ (𝑧𝑋𝑥𝑋)) → 𝑥 ∈ (Base‘𝑀))
9 eqid 2731 . . . . . . . . . 10 (Base‘𝑀) = (Base‘𝑀)
10 eqid 2731 . . . . . . . . . 10 (+g𝑀) = (+g𝑀)
11 eqid 2731 . . . . . . . . . 10 (+g𝑁) = (+g𝑁)
129, 10, 11mhmlin 18718 . . . . . . . . 9 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑧 ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐹‘(𝑧(+g𝑀)𝑥)) = ((𝐹𝑧)(+g𝑁)(𝐹𝑥)))
132, 6, 8, 12syl3anc 1370 . . . . . . . 8 ((𝜑 ∧ (𝑧𝑋𝑥𝑋)) → (𝐹‘(𝑧(+g𝑀)𝑥)) = ((𝐹𝑧)(+g𝑁)(𝐹𝑥)))
14 mhmimalem.a . . . . . . . . . . . 12 (𝜑 = (+g𝑀))
1514oveqd 7429 . . . . . . . . . . 11 (𝜑 → (𝑧 𝑥) = (𝑧(+g𝑀)𝑥))
1615fveq2d 6895 . . . . . . . . . 10 (𝜑 → (𝐹‘(𝑧 𝑥)) = (𝐹‘(𝑧(+g𝑀)𝑥)))
17 mhmimalem.p . . . . . . . . . . 11 (𝜑+ = (+g𝑁))
1817oveqd 7429 . . . . . . . . . 10 (𝜑 → ((𝐹𝑧) + (𝐹𝑥)) = ((𝐹𝑧)(+g𝑁)(𝐹𝑥)))
1916, 18eqeq12d 2747 . . . . . . . . 9 (𝜑 → ((𝐹‘(𝑧 𝑥)) = ((𝐹𝑧) + (𝐹𝑥)) ↔ (𝐹‘(𝑧(+g𝑀)𝑥)) = ((𝐹𝑧)(+g𝑁)(𝐹𝑥))))
2019adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑧𝑋𝑥𝑋)) → ((𝐹‘(𝑧 𝑥)) = ((𝐹𝑧) + (𝐹𝑥)) ↔ (𝐹‘(𝑧(+g𝑀)𝑥)) = ((𝐹𝑧)(+g𝑁)(𝐹𝑥))))
2113, 20mpbird 257 . . . . . . 7 ((𝜑 ∧ (𝑧𝑋𝑥𝑋)) → (𝐹‘(𝑧 𝑥)) = ((𝐹𝑧) + (𝐹𝑥)))
22 eqid 2731 . . . . . . . . . . . 12 (Base‘𝑁) = (Base‘𝑁)
239, 22mhmf 18714 . . . . . . . . . . 11 (𝐹 ∈ (𝑀 MndHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
241, 23syl 17 . . . . . . . . . 10 (𝜑𝐹:(Base‘𝑀)⟶(Base‘𝑁))
2524ffnd 6718 . . . . . . . . 9 (𝜑𝐹 Fn (Base‘𝑀))
2625adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑧𝑋𝑥𝑋)) → 𝐹 Fn (Base‘𝑀))
27 mhmimalem.c . . . . . . . . 9 ((𝜑𝑧𝑋𝑥𝑋) → (𝑧 𝑥) ∈ 𝑋)
28273expb 1119 . . . . . . . 8 ((𝜑 ∧ (𝑧𝑋𝑥𝑋)) → (𝑧 𝑥) ∈ 𝑋)
29 fnfvima 7237 . . . . . . . 8 ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀) ∧ (𝑧 𝑥) ∈ 𝑋) → (𝐹‘(𝑧 𝑥)) ∈ (𝐹𝑋))
3026, 4, 28, 29syl3anc 1370 . . . . . . 7 ((𝜑 ∧ (𝑧𝑋𝑥𝑋)) → (𝐹‘(𝑧 𝑥)) ∈ (𝐹𝑋))
3121, 30eqeltrrd 2833 . . . . . 6 ((𝜑 ∧ (𝑧𝑋𝑥𝑋)) → ((𝐹𝑧) + (𝐹𝑥)) ∈ (𝐹𝑋))
3231anassrs 467 . . . . 5 (((𝜑𝑧𝑋) ∧ 𝑥𝑋) → ((𝐹𝑧) + (𝐹𝑥)) ∈ (𝐹𝑋))
3332ralrimiva 3145 . . . 4 ((𝜑𝑧𝑋) → ∀𝑥𝑋 ((𝐹𝑧) + (𝐹𝑥)) ∈ (𝐹𝑋))
34 oveq2 7420 . . . . . . . 8 (𝑦 = (𝐹𝑥) → ((𝐹𝑧) + 𝑦) = ((𝐹𝑧) + (𝐹𝑥)))
3534eleq1d 2817 . . . . . . 7 (𝑦 = (𝐹𝑥) → (((𝐹𝑧) + 𝑦) ∈ (𝐹𝑋) ↔ ((𝐹𝑧) + (𝐹𝑥)) ∈ (𝐹𝑋)))
3635ralima 7242 . . . . . 6 ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀)) → (∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧) + 𝑦) ∈ (𝐹𝑋) ↔ ∀𝑥𝑋 ((𝐹𝑧) + (𝐹𝑥)) ∈ (𝐹𝑋)))
3725, 3, 36syl2anc 583 . . . . 5 (𝜑 → (∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧) + 𝑦) ∈ (𝐹𝑋) ↔ ∀𝑥𝑋 ((𝐹𝑧) + (𝐹𝑥)) ∈ (𝐹𝑋)))
3837adantr 480 . . . 4 ((𝜑𝑧𝑋) → (∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧) + 𝑦) ∈ (𝐹𝑋) ↔ ∀𝑥𝑋 ((𝐹𝑧) + (𝐹𝑥)) ∈ (𝐹𝑋)))
3933, 38mpbird 257 . . 3 ((𝜑𝑧𝑋) → ∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧) + 𝑦) ∈ (𝐹𝑋))
4039ralrimiva 3145 . 2 (𝜑 → ∀𝑧𝑋𝑦 ∈ (𝐹𝑋)((𝐹𝑧) + 𝑦) ∈ (𝐹𝑋))
41 oveq1 7419 . . . . . 6 (𝑥 = (𝐹𝑧) → (𝑥 + 𝑦) = ((𝐹𝑧) + 𝑦))
4241eleq1d 2817 . . . . 5 (𝑥 = (𝐹𝑧) → ((𝑥 + 𝑦) ∈ (𝐹𝑋) ↔ ((𝐹𝑧) + 𝑦) ∈ (𝐹𝑋)))
4342ralbidv 3176 . . . 4 (𝑥 = (𝐹𝑧) → (∀𝑦 ∈ (𝐹𝑋)(𝑥 + 𝑦) ∈ (𝐹𝑋) ↔ ∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧) + 𝑦) ∈ (𝐹𝑋)))
4443ralima 7242 . . 3 ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀)) → (∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥 + 𝑦) ∈ (𝐹𝑋) ↔ ∀𝑧𝑋𝑦 ∈ (𝐹𝑋)((𝐹𝑧) + 𝑦) ∈ (𝐹𝑋)))
4525, 3, 44syl2anc 583 . 2 (𝜑 → (∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥 + 𝑦) ∈ (𝐹𝑋) ↔ ∀𝑧𝑋𝑦 ∈ (𝐹𝑋)((𝐹𝑧) + 𝑦) ∈ (𝐹𝑋)))
4640, 45mpbird 257 1 (𝜑 → ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥 + 𝑦) ∈ (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  wss 3948  cima 5679   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7412  Basecbs 17151  +gcplusg 17204   MndHom cmhm 18706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8828  df-mhm 18708
This theorem is referenced by:  mhmima  18745  rhmimasubrng  20458
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