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Theorem rhmimasubrnglem 20457
Description: Lemma for rhmimasubrng 20458: Modified part of mhmima 18745. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 16-Feb-2025.)
Hypothesis
Ref Expression
rhmimasubrnglem.b 𝑀 = (mulGrp‘𝑅)
Assertion
Ref Expression
rhmimasubrnglem ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋))
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝑥,𝑅   𝑥,𝑋,𝑦
Allowed substitution hint:   𝑅(𝑦)

Proof of Theorem rhmimasubrnglem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 simpll 764 . . . . . . . 8 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧𝑋𝑥𝑋)) → 𝐹 ∈ (𝑀 MndHom 𝑁))
2 eqid 2731 . . . . . . . . . . . . 13 (Base‘𝑅) = (Base‘𝑅)
32subrngss 20440 . . . . . . . . . . . 12 (𝑋 ∈ (SubRng‘𝑅) → 𝑋 ⊆ (Base‘𝑅))
4 rhmimasubrnglem.b . . . . . . . . . . . . 13 𝑀 = (mulGrp‘𝑅)
54, 2mgpbas 20038 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑀)
63, 5sseqtrdi 4032 . . . . . . . . . . 11 (𝑋 ∈ (SubRng‘𝑅) → 𝑋 ⊆ (Base‘𝑀))
76adantl 481 . . . . . . . . . 10 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → 𝑋 ⊆ (Base‘𝑀))
87adantr 480 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑋 ⊆ (Base‘𝑀))
9 simprl 768 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑧𝑋)
108, 9sseldd 3983 . . . . . . . 8 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑧 ∈ (Base‘𝑀))
11 simprr 770 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑥𝑋)
128, 11sseldd 3983 . . . . . . . 8 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑥 ∈ (Base‘𝑀))
13 eqid 2731 . . . . . . . . 9 (Base‘𝑀) = (Base‘𝑀)
14 eqid 2731 . . . . . . . . 9 (+g𝑀) = (+g𝑀)
15 eqid 2731 . . . . . . . . 9 (+g𝑁) = (+g𝑁)
1613, 14, 15mhmlin 18718 . . . . . . . 8 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑧 ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐹‘(𝑧(+g𝑀)𝑥)) = ((𝐹𝑧)(+g𝑁)(𝐹𝑥)))
171, 10, 12, 16syl3anc 1370 . . . . . . 7 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧𝑋𝑥𝑋)) → (𝐹‘(𝑧(+g𝑀)𝑥)) = ((𝐹𝑧)(+g𝑁)(𝐹𝑥)))
18 eqid 2731 . . . . . . . . . . . 12 (Base‘𝑁) = (Base‘𝑁)
1913, 18mhmf 18714 . . . . . . . . . . 11 (𝐹 ∈ (𝑀 MndHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
2019adantr 480 . . . . . . . . . 10 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
2120ffnd 6718 . . . . . . . . 9 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → 𝐹 Fn (Base‘𝑀))
2221adantr 480 . . . . . . . 8 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧𝑋𝑥𝑋)) → 𝐹 Fn (Base‘𝑀))
23 eqid 2731 . . . . . . . . . . . . 13 (.r𝑅) = (.r𝑅)
244, 23mgpplusg 20036 . . . . . . . . . . . 12 (.r𝑅) = (+g𝑀)
2524eqcomi 2740 . . . . . . . . . . 11 (+g𝑀) = (.r𝑅)
2625subrngmcl 20449 . . . . . . . . . 10 ((𝑋 ∈ (SubRng‘𝑅) ∧ 𝑧𝑋𝑥𝑋) → (𝑧(+g𝑀)𝑥) ∈ 𝑋)
27263expb 1119 . . . . . . . . 9 ((𝑋 ∈ (SubRng‘𝑅) ∧ (𝑧𝑋𝑥𝑋)) → (𝑧(+g𝑀)𝑥) ∈ 𝑋)
2827adantll 711 . . . . . . . 8 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧𝑋𝑥𝑋)) → (𝑧(+g𝑀)𝑥) ∈ 𝑋)
29 fnfvima 7237 . . . . . . . 8 ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀) ∧ (𝑧(+g𝑀)𝑥) ∈ 𝑋) → (𝐹‘(𝑧(+g𝑀)𝑥)) ∈ (𝐹𝑋))
3022, 8, 28, 29syl3anc 1370 . . . . . . 7 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧𝑋𝑥𝑋)) → (𝐹‘(𝑧(+g𝑀)𝑥)) ∈ (𝐹𝑋))
3117, 30eqeltrrd 2833 . . . . . 6 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ (𝑧𝑋𝑥𝑋)) → ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋))
3231anassrs 467 . . . . 5 ((((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ 𝑧𝑋) ∧ 𝑥𝑋) → ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋))
3332ralrimiva 3145 . . . 4 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ 𝑧𝑋) → ∀𝑥𝑋 ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋))
34 oveq2 7420 . . . . . . . 8 (𝑦 = (𝐹𝑥) → ((𝐹𝑧)(+g𝑁)𝑦) = ((𝐹𝑧)(+g𝑁)(𝐹𝑥)))
3534eleq1d 2817 . . . . . . 7 (𝑦 = (𝐹𝑥) → (((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋)))
3635ralima 7242 . . . . . 6 ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀)) → (∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑥𝑋 ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋)))
3721, 7, 36syl2anc 583 . . . . 5 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → (∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑥𝑋 ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋)))
3837adantr 480 . . . 4 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ 𝑧𝑋) → (∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑥𝑋 ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋)))
3933, 38mpbird 257 . . 3 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) ∧ 𝑧𝑋) → ∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋))
4039ralrimiva 3145 . 2 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → ∀𝑧𝑋𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋))
41 oveq1 7419 . . . . . 6 (𝑥 = (𝐹𝑧) → (𝑥(+g𝑁)𝑦) = ((𝐹𝑧)(+g𝑁)𝑦))
4241eleq1d 2817 . . . . 5 (𝑥 = (𝐹𝑧) → ((𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋)))
4342ralbidv 3176 . . . 4 (𝑥 = (𝐹𝑧) → (∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋)))
4443ralima 7242 . . 3 ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀)) → (∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑧𝑋𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋)))
4521, 7, 44syl2anc 583 . 2 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → (∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑧𝑋𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋)))
4640, 45mpbird 257 1 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = 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  .rcmulr 17205   MndHom cmhm 18706  mulGrpcmgp 20032  SubRngcsubrng 20437
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  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  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-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  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-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-er 8709  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-3 12283  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-ress 17181  df-plusg 17217  df-mulr 17218  df-mgm 18568  df-sgrp 18647  df-mhm 18708  df-subg 19043  df-abl 19696  df-mgp 20033  df-rng 20051  df-subrng 20438
This theorem is referenced by: (None)
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