Theorem resmhm2b 19018

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 18982	. . . 4 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
2	1	adantl 487	. . 3 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Mnd)
3		resmhm2.u	. . . . 5 ⊢ 𝑈 = (𝑇 ↾s 𝑋)
4	3	submmnd 19009	. . . 4 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → 𝑈 ∈ Mnd)
5	4	ad2antrr 739	. . 3 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝑈 ∈ Mnd)
6		eqid 2761	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
7		eqid 2761	. . . . . . . . 9 ⊢ (Base‘𝑇) = (Base‘𝑇)
8	6, 7	mhmf 18984	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
9	8	adantl 487	. . . . . . 7 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
10	9	ffnd 6710	. . . . . 6 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
11		simplr 781	. . . . . 6 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → ran 𝐹 ⊆ 𝑋)
12		df-f 6542	. . . . . 6 ⊢ (𝐹:(Base‘𝑆)⟶𝑋 ↔ (𝐹 Fn (Base‘𝑆) ∧ ran 𝐹 ⊆ 𝑋))
13	10, 11, 12	sylanbrc 595	. . . . 5 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶𝑋)
14	3	submbas 19010	. . . . . . 7 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → 𝑋 = (Base‘𝑈))
15	14	ad2antrr 739	. . . . . 6 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝑋 = (Base‘𝑈))
16	15	feq3d 6694	. . . . 5 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹:(Base‘𝑆)⟶𝑋 ↔ 𝐹:(Base‘𝑆)⟶(Base‘𝑈)))
17	13, 16	mpbid 235	. . . 4 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑈))
18		eqid 2761	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
19		eqid 2761	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
20	6, 18, 19	mhmlin 18988	. . . . . . . 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 1138	. . . . . . 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 727	. . . . . 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 17462	. . . . . . . 8 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → (+g‘𝑇) = (+g‘𝑈))
24	23	ad3antrrr 743	. . . . . . 7 ⊢ ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (+g‘𝑇) = (+g‘𝑈))
25	24	oveqd 7437	. . . . . 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 2796	. . . . 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 3206	. . . 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 2761	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
29		eqid 2761	. . . . . . 7 ⊢ (0g‘𝑇) = (0g‘𝑇)
30	28, 29	mhm0 18989	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
31	30	adantl 487	. . . . 5 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g‘𝑆)) = (0g‘𝑇))
32	3, 29	subm0 19011	. . . . . 6 ⊢ (𝑋 ∈ (SubMnd‘𝑇) → (0g‘𝑇) = (0g‘𝑈))
33	32	ad2antrr 739	. . . . 5 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (0g‘𝑇) = (0g‘𝑈))
34	31, 33	eqtrd 2796	. . . 4 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g‘𝑆)) = (0g‘𝑈))
35	17, 27, 34	3jca 1146	. . 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 2761	. . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈)
37		eqid 2761	. . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈)
38		eqid 2761	. . . 4 ⊢ (0g‘𝑈) = (0g‘𝑈)
39	6, 36, 18, 37, 28, 38	ismhm 18980	. . 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 1363	. 2 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑈))
41	3	resmhm2 19017	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
42	41	ancoms 464	. . 3 ⊢ ((𝑋 ∈ (SubMnd‘𝑇) ∧ 𝐹 ∈ (𝑆 MndHom 𝑈)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
43	42	adantlr 728	. 2 ⊢ (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑈)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
44	40, 43	impbida 813	1 ⊢ ((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ⊆ wss 3899  ran crn 5652   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  Basecbs 17387   ↾_s cress 17408  +_gcplusg 17428  0_gc0g 17610  Mndcmnd 18923   MndHom cmhm 18976  SubMndcsubmnd 18977

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-0g 17612  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-mhm 18978  df-submnd 18979

This theorem is used by:  resghm2b  19448  resrhm2b  20854  m2cpmmhm  23063  dchrghm  27583  lgseisenlem4  27705