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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnghmsubcsetclem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for rnghmsubcsetc 44255. (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 2900 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Rng ∩ 𝑈))) |
3 | elin 4171 | . . . . . 6 ⊢ (𝑥 ∈ (Rng ∩ 𝑈) ↔ (𝑥 ∈ Rng ∧ 𝑥 ∈ 𝑈)) | |
4 | 3 | simplbi 500 | . . . . 5 ⊢ (𝑥 ∈ (Rng ∩ 𝑈) → 𝑥 ∈ Rng) |
5 | 2, 4 | syl6bi 255 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ Rng)) |
6 | 5 | imp 409 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ Rng) |
7 | eqid 2823 | . . . 4 ⊢ (Base‘𝑥) = (Base‘𝑥) | |
8 | 7 | idrnghm 44186 | . . 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 2823 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
12 | rnghmsubcsetc.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
13 | 12 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑈 ∈ 𝑉) |
14 | 3 | simprbi 499 | . . . . 5 ⊢ (𝑥 ∈ (Rng ∩ 𝑈) → 𝑥 ∈ 𝑈) |
15 | 2, 14 | syl6bi 255 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝑈)) |
16 | 15 | imp 409 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝑈) |
17 | 10, 11, 13, 16 | estrcid 17386 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) = ( I ↾ (Base‘𝑥))) |
18 | rnghmsubcsetc.h | . . . 4 ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) | |
19 | 18 | oveqdr 7186 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝐻𝑥) = (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑥)) |
20 | eqid 2823 | . . . . . . . 8 ⊢ (RngCat‘𝑈) = (RngCat‘𝑈) | |
21 | eqid 2823 | . . . . . . . 8 ⊢ (Base‘(RngCat‘𝑈)) = (Base‘(RngCat‘𝑈)) | |
22 | eqid 2823 | . . . . . . . 8 ⊢ (Hom ‘(RngCat‘𝑈)) = (Hom ‘(RngCat‘𝑈)) | |
23 | 20, 21, 12, 22 | rngchomfval 44244 | . . . . . . 7 ⊢ (𝜑 → (Hom ‘(RngCat‘𝑈)) = ( RngHomo ↾ ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈))))) |
24 | 20, 21, 12 | rngcbas 44243 | . . . . . . . . . 10 ⊢ (𝜑 → (Base‘(RngCat‘𝑈)) = (𝑈 ∩ Rng)) |
25 | incom 4180 | . . . . . . . . . . . 12 ⊢ (Rng ∩ 𝑈) = (𝑈 ∩ Rng) | |
26 | 1, 25 | syl6eq 2874 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) |
27 | 26 | eqcomd 2829 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑈 ∩ Rng) = 𝐵) |
28 | 24, 27 | eqtrd 2858 | . . . . . . . . 9 ⊢ (𝜑 → (Base‘(RngCat‘𝑈)) = 𝐵) |
29 | 28 | sqxpeqd 5589 | . . . . . . . 8 ⊢ (𝜑 → ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈))) = (𝐵 × 𝐵)) |
30 | 29 | reseq2d 5855 | . . . . . . 7 ⊢ (𝜑 → ( RngHomo ↾ ((Base‘(RngCat‘𝑈)) × (Base‘(RngCat‘𝑈)))) = ( RngHomo ↾ (𝐵 × 𝐵))) |
31 | 23, 30 | eqtrd 2858 | . . . . . 6 ⊢ (𝜑 → (Hom ‘(RngCat‘𝑈)) = ( RngHomo ↾ (𝐵 × 𝐵))) |
32 | 31 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (Hom ‘(RngCat‘𝑈)) = ( RngHomo ↾ (𝐵 × 𝐵))) |
33 | 32 | eqcomd 2829 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( RngHomo ↾ (𝐵 × 𝐵)) = (Hom ‘(RngCat‘𝑈))) |
34 | 33 | oveqd 7175 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥( RngHomo ↾ (𝐵 × 𝐵))𝑥) = (𝑥(Hom ‘(RngCat‘𝑈))𝑥)) |
35 | 26 | eleq2d 2900 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (𝑈 ∩ Rng))) |
36 | 35 | biimpa 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝑈 ∩ Rng)) |
37 | 24 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (Base‘(RngCat‘𝑈)) = (𝑈 ∩ Rng)) |
38 | 36, 37 | eleqtrrd 2918 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (Base‘(RngCat‘𝑈))) |
39 | 20, 21, 13, 22, 38, 38 | rngchom 44245 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥(Hom ‘(RngCat‘𝑈))𝑥) = (𝑥 RngHomo 𝑥)) |
40 | 19, 34, 39 | 3eqtrd 2862 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝐻𝑥) = (𝑥 RngHomo 𝑥)) |
41 | 9, 17, 40 | 3eltr4d 2930 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 I cid 5461 × cxp 5555 ↾ cres 5559 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 Hom chom 16578 Idccid 16938 ExtStrCatcestrc 17374 Rngcrng 44152 RngHomo crngh 44163 RngCatcrngc 44235 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-hom 16591 df-cco 16592 df-cat 16941 df-cid 16942 df-resc 17083 df-estrc 17375 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-ghm 18358 df-abl 18911 df-mgp 19242 df-mgmhm 44053 df-rng0 44153 df-rnghomo 44165 df-rngc 44237 |
This theorem is referenced by: rnghmsubcsetc 44255 |
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