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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnghmsubcsetclem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for rnghmsubcsetc 46247. (Contributed by AV, 9-Mar-2020.) |
Ref | Expression |
---|---|
rnghmsubcsetc.c | ⊢ 𝐶 = (ExtStrCat‘𝑈) |
rnghmsubcsetc.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rnghmsubcsetc.b | ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈)) |
rnghmsubcsetc.h | ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) |
Ref | Expression |
---|---|
rnghmsubcsetclem1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnghmsubcsetc.b | . . . . . 6 ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈)) | |
2 | 1 | eleq2d 2823 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Rng ∩ 𝑈))) |
3 | elin 3926 | . . . . . 6 ⊢ (𝑥 ∈ (Rng ∩ 𝑈) ↔ (𝑥 ∈ Rng ∧ 𝑥 ∈ 𝑈)) | |
4 | 3 | simplbi 498 | . . . . 5 ⊢ (𝑥 ∈ (Rng ∩ 𝑈) → 𝑥 ∈ Rng) |
5 | 2, 4 | syl6bi 252 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ Rng)) |
6 | 5 | imp 407 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ Rng) |
7 | eqid 2736 | . . . 4 ⊢ (Base‘𝑥) = (Base‘𝑥) | |
8 | 7 | idrnghm 46178 | . . 3 ⊢ (𝑥 ∈ Rng → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥)) |
9 | 6, 8 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RngHomo 𝑥)) |
10 | rnghmsubcsetc.c | . . 3 ⊢ 𝐶 = (ExtStrCat‘𝑈) | |
11 | eqid 2736 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
12 | rnghmsubcsetc.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
13 | 12 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑈 ∈ 𝑉) |
14 | 3 | simprbi 497 | . . . . 5 ⊢ (𝑥 ∈ (Rng ∩ 𝑈) → 𝑥 ∈ 𝑈) |
15 | 2, 14 | syl6bi 252 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝑈)) |
16 | 15 | imp 407 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝑈) |
17 | 10, 11, 13, 16 | estrcid 18020 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) = ( I ↾ (Base‘𝑥))) |
18 | rnghmsubcsetc.h | . . . 4 ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) | |
19 | 18 | oveqdr 7384 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝐻𝑥) = (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑥)) |
20 | eqid 2736 | . . . . . . . 8 ⊢ (RngCat‘𝑈) = (RngCat‘𝑈) | |
21 | eqid 2736 | . . . . . . . 8 ⊢ (Base‘(RngCat‘𝑈)) = (Base‘(RngCat‘𝑈)) | |
22 | eqid 2736 | . . . . . . . 8 ⊢ (Hom ‘(RngCat‘𝑈)) = (Hom ‘(RngCat‘𝑈)) | |
23 | 20, 21, 12, 22 | rngchomfval 46236 | . . . . . . 7 ⊢ (𝜑 → (Hom ‘(RngCat‘𝑈)) = ( RngHomo ↾ ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈))))) |
24 | 20, 21, 12 | rngcbas 46235 | . . . . . . . . . 10 ⊢ (𝜑 → (Base‘(RngCat‘𝑈)) = (𝑈 ∩ Rng)) |
25 | incom 4161 | . . . . . . . . . . . 12 ⊢ (Rng ∩ 𝑈) = (𝑈 ∩ Rng) | |
26 | 1, 25 | eqtrdi 2792 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) |
27 | 26 | eqcomd 2742 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑈 ∩ Rng) = 𝐵) |
28 | 24, 27 | eqtrd 2776 | . . . . . . . . 9 ⊢ (𝜑 → (Base‘(RngCat‘𝑈)) = 𝐵) |
29 | 28 | sqxpeqd 5665 | . . . . . . . 8 ⊢ (𝜑 → ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈))) = (𝐵 × 𝐵)) |
30 | 29 | reseq2d 5937 | . . . . . . 7 ⊢ (𝜑 → ( RngHomo ↾ ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈)))) = ( RngHomo ↾ (𝐵 × 𝐵))) |
31 | 23, 30 | eqtrd 2776 | . . . . . 6 ⊢ (𝜑 → (Hom ‘(RngCat‘𝑈)) = ( RngHomo ↾ (𝐵 × 𝐵))) |
32 | 31 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (Hom ‘(RngCat‘𝑈)) = ( RngHomo ↾ (𝐵 × 𝐵))) |
33 | 32 | eqcomd 2742 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( RngHomo ↾ (𝐵 × 𝐵)) = (Hom ‘(RngCat‘𝑈))) |
34 | 33 | oveqd 7373 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑥) = (𝑥(Hom ‘(RngCat‘𝑈))𝑥)) |
35 | 26 | eleq2d 2823 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (𝑈 ∩ Rng))) |
36 | 35 | biimpa 477 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝑈 ∩ Rng)) |
37 | 24 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (Base‘(RngCat‘𝑈)) = (𝑈 ∩ Rng)) |
38 | 36, 37 | eleqtrrd 2841 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (Base‘(RngCat‘𝑈))) |
39 | 20, 21, 13, 22, 38, 38 | rngchom 46237 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥(Hom ‘(RngCat‘𝑈))𝑥) = (𝑥 RngHomo 𝑥)) |
40 | 19, 34, 39 | 3eqtrd 2780 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝐻𝑥) = (𝑥 RngHomo 𝑥)) |
41 | 9, 17, 40 | 3eltr4d 2853 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∩ cin 3909 I cid 5530 × cxp 5631 ↾ cres 5635 ‘cfv 6496 (class class class)co 7356 Basecbs 17082 Hom chom 17143 Idccid 17544 ExtStrCatcestrc 18008 Rngcrng 46144 RngHomo crngh 46155 RngCatcrngc 46227 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-map 8766 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-z 12499 df-dec 12618 df-uz 12763 df-fz 13424 df-struct 17018 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-ress 17112 df-plusg 17145 df-hom 17156 df-cco 17157 df-cat 17547 df-cid 17548 df-resc 17693 df-estrc 18009 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-grp 18750 df-ghm 19004 df-abl 19563 df-mgp 19895 df-mgmhm 46045 df-rng 46145 df-rnghomo 46157 df-rngc 46229 |
This theorem is referenced by: rnghmsubcsetc 46247 |
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