Theorem rnghmsubcsetclem1 20539

Description: Lemma 1 for rnghmsubcsetc 20541. (Contributed by AV, 9-Mar-2020.)

Hypotheses

Ref	Expression
rnghmsubcsetc.c	⊢ 𝐶 = (ExtStrCat‘𝑈)
rnghmsubcsetc.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
rnghmsubcsetc.b	⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈))
rnghmsubcsetc.h	⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))

Assertion

Ref	Expression
rnghmsubcsetclem1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))

Proof of Theorem rnghmsubcsetclem1

Step	Hyp	Ref	Expression
1		rnghmsubcsetc.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈))
2	1	eleq2d 2815	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Rng ∩ 𝑈)))
3		elin 3916	. . . . . 6 ⊢ (𝑥 ∈ (Rng ∩ 𝑈) ↔ (𝑥 ∈ Rng ∧ 𝑥 ∈ 𝑈))
4	3	simplbi 497	. . . . 5 ⊢ (𝑥 ∈ (Rng ∩ 𝑈) → 𝑥 ∈ Rng)
5	2, 4	biimtrdi 253	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ Rng))
6	5	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ Rng)
7		eqid 2730	. . . 4 ⊢ (Base‘𝑥) = (Base‘𝑥)
8	7	idrnghm 20369	. . 3 ⊢ (𝑥 ∈ Rng → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHom 𝑥))
9	6, 8	syl 17	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHom 𝑥))
10		rnghmsubcsetc.c	. . 3 ⊢ 𝐶 = (ExtStrCat‘𝑈)
11		eqid 2730	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
12		rnghmsubcsetc.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
13	12	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑈 ∈ 𝑉)
14	3	simprbi 496	. . . . 5 ⊢ (𝑥 ∈ (Rng ∩ 𝑈) → 𝑥 ∈ 𝑈)
15	2, 14	biimtrdi 253	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝑈))
16	15	imp 406	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝑈)
17	10, 11, 13, 16	estrcid 18032	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) = ( I ↾ (Base‘𝑥)))
18		rnghmsubcsetc.h	. . . 4 ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))
19	18	oveqdr 7369	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝐻𝑥) = (𝑥( RngHom ↾ (𝐵 × 𝐵))𝑥))
20		eqid 2730	. . . . . . . 8 ⊢ (RngCat‘𝑈) = (RngCat‘𝑈)
21		eqid 2730	. . . . . . . 8 ⊢ (Base‘(RngCat‘𝑈)) = (Base‘(RngCat‘𝑈))
22		eqid 2730	. . . . . . . 8 ⊢ (Hom ‘(RngCat‘𝑈)) = (Hom ‘(RngCat‘𝑈))
23	20, 21, 12, 22	rngchomfval 20530	. . . . . . 7 ⊢ (𝜑 → (Hom ‘(RngCat‘𝑈)) = ( RngHom ↾ ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈)))))
24	20, 21, 12	rngcbas 20529	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘(RngCat‘𝑈)) = (𝑈 ∩ Rng))
25		incom 4157	. . . . . . . . . . . 12 ⊢ (Rng ∩ 𝑈) = (𝑈 ∩ Rng)
26	1, 25	eqtrdi 2781	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
27	26	eqcomd 2736	. . . . . . . . . 10 ⊢ (𝜑 → (𝑈 ∩ Rng) = 𝐵)
28	24, 27	eqtrd 2765	. . . . . . . . 9 ⊢ (𝜑 → (Base‘(RngCat‘𝑈)) = 𝐵)
29	28	sqxpeqd 5646	. . . . . . . 8 ⊢ (𝜑 → ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈))) = (𝐵 × 𝐵))
30	29	reseq2d 5925	. . . . . . 7 ⊢ (𝜑 → ( RngHom ↾ ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈)))) = ( RngHom ↾ (𝐵 × 𝐵)))
31	23, 30	eqtrd 2765	. . . . . 6 ⊢ (𝜑 → (Hom ‘(RngCat‘𝑈)) = ( RngHom ↾ (𝐵 × 𝐵)))
32	31	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (Hom ‘(RngCat‘𝑈)) = ( RngHom ↾ (𝐵 × 𝐵)))
33	32	eqcomd 2736	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( RngHom ↾ (𝐵 × 𝐵)) = (Hom ‘(RngCat‘𝑈)))
34	33	oveqd 7358	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥( RngHom ↾ (𝐵 × 𝐵))𝑥) = (𝑥(Hom ‘(RngCat‘𝑈))𝑥))
35	26	eleq2d 2815	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (𝑈 ∩ Rng)))
36	35	biimpa 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝑈 ∩ Rng))
37	24	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (Base‘(RngCat‘𝑈)) = (𝑈 ∩ Rng))
38	36, 37	eleqtrrd 2832	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (Base‘(RngCat‘𝑈)))
39	20, 21, 13, 22, 38, 38	rngchom 20531	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥(Hom ‘(RngCat‘𝑈))𝑥) = (𝑥 RngHom 𝑥))
40	19, 34, 39	3eqtrd 2769	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝐻𝑥) = (𝑥 RngHom 𝑥))
41	9, 17, 40	3eltr4d 2844	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2110   ∩ cin 3899   I cid 5508   × cxp 5612   ↾ cres 5616  ‘cfv 6477  (class class class)co 7341  Basecbs 17112  Hom chom 17164  Idccid 17563  ExtStrCatcestrc 18020  Rngcrng 20063   RngHom crnghm 20345  RngCatcrngc 20524

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 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-er 8617  df-map 8747  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-nn 12118  df-2 12180  df-3 12181  df-4 12182  df-5 12183  df-6 12184  df-7 12185  df-8 12186  df-9 12187  df-n0 12374  df-z 12461  df-dec 12581  df-uz 12725  df-fz 13400  df-struct 17050  df-sets 17067  df-slot 17085  df-ndx 17097  df-base 17113  df-ress 17134  df-plusg 17166  df-hom 17177  df-cco 17178  df-cat 17566  df-cid 17567  df-resc 17710  df-estrc 18021  df-mgm 18540  df-mgmhm 18592  df-sgrp 18619  df-mnd 18635  df-grp 18841  df-ghm 19118  df-abl 19688  df-mgp 20052  df-rng 20064  df-rnghm 20347  df-rngc 20525

This theorem is referenced by:  rnghmsubcsetc  20541