Theorem resmhm2b 18790

Description: Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.)

Hypothesis

Ref	Expression
resmhm2.u	⊢ 𝑈 = (𝑇 ↾s 𝑋)

Assertion

Ref	Expression
resmhm2b	⊢ ((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))

Proof of Theorem resmhm2b

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mhmrcl1 18755	. . . 4 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
2	1	adantl 481	. . 3 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Mnd)
3		resmhm2.u	. . . . 5 ⊢ 𝑈 = (𝑇 ↾s 𝑋)
4	3	submmnd 18781	. . . 4 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → 𝑈 ∈ Mnd)
5	4	ad2antrr 727	. . 3 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝑈 ∈ Mnd)
6		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
7		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝑇) = (Base‘𝑇)
8	6, 7	mhmf 18757	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
9	8	adantl 481	. . . . . . 7 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
10	9	ffnd 6669	. . . . . 6 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
11		simplr 769	. . . . . 6 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → ran 𝐹 ⊆ 𝑋)
12		df-f 6502	. . . . . 6 ⊢ (𝐹:(Base‘𝑆)⟶𝑋 ↔ (𝐹 Fn (Base‘𝑆) ∧ ran 𝐹 ⊆ 𝑋))
13	10, 11, 12	sylanbrc 584	. . . . 5 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶𝑋)
14	3	submbas 18782	. . . . . . 7 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → 𝑋 = (Base‘𝑈))
15	14	ad2antrr 727	. . . . . 6 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝑋 = (Base‘𝑈))
16	15	feq3d 6653	. . . . 5 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹:(Base‘𝑆)⟶𝑋 ↔ 𝐹:(Base‘𝑆)⟶(Base‘𝑈)))
17	13, 16	mpbid 232	. . . 4 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑈))
18		eqid 2736	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
19		eqid 2736	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
20	6, 18, 19	mhmlin 18761	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
21	20	3expb 1121	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
22	21	adantll 715	. . . . . 6 ⊢ ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
23	3, 19	ressplusg 17254	. . . . . . . 8 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → (+g‘𝑇) = (+g‘𝑈))
24	23	ad3antrrr 731	. . . . . . 7 ⊢ ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (+g‘𝑇) = (+g‘𝑈))
25	24	oveqd 7384	. . . . . 6 ⊢ ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)))
26	22, 25	eqtrd 2771	. . . . 5 ⊢ ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)))
27	26	ralrimivva 3180	. . . 4 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)))
28		eqid 2736	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
29		eqid 2736	. . . . . . 7 ⊢ (0g‘𝑇) = (0g‘𝑇)
30	28, 29	mhm0 18762	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
31	30	adantl 481	. . . . 5 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
32	3, 29	subm0 18783	. . . . . 6 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → (0g‘𝑇) = (0g‘𝑈))
33	32	ad2antrr 727	. . . . 5 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (0g‘𝑇) = (0g‘𝑈))
34	31, 33	eqtrd 2771	. . . 4 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g‘𝑆)) = (0g‘𝑈))
35	17, 27, 34	3jca 1129	. . 3 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹:(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑈)))
36		eqid 2736	. . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈)
37		eqid 2736	. . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈)
38		eqid 2736	. . . 4 ⊢ (0g‘𝑈) = (0g‘𝑈)
39	6, 36, 18, 37, 28, 38	ismhm 18753	. . 3 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑈) ↔ ((𝑆 ∈ Mnd ∧ 𝑈 ∈ Mnd) ∧ (𝐹:(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑈))))
40	2, 5, 35, 39	syl21anbrc 1346	. 2 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑈))
41	3	resmhm2 18789	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
42	41	ancoms 458	. . 3 ⊢ ((𝑋 ∈ (SubMnd‘𝑇) ∧ 𝐹 ∈ (𝑆 MndHom 𝑈)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
43	42	adantlr 716	. 2 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑈)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
44	40, 43	impbida 801	1 ⊢ ((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051   ⊆ wss 3889  ran crn 5632   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  Basecbs 17179   ↾_s cress 17200  +_gcplusg 17220  0_gc0g 17402  Mndcmnd 18702   MndHom cmhm 18749  SubMndcsubmnd 18750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-0g 17404  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752

This theorem is referenced by:  resghm2b  19209  resrhm2b  20579  m2cpmmhm  22710  dchrghm  27219  lgseisenlem4  27341