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Mirrors > Home > MPE Home > Th. List > Mathboxes > rhmsubclem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for rhmsubc 47089. (Contributed by AV, 2-Mar-2020.) |
Ref | Expression |
---|---|
rngcrescrhm.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
rngcrescrhm.c | ⊢ 𝐶 = (RngCat‘𝑈) |
rngcrescrhm.r | ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) |
rngcrescrhm.h | ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) |
Ref | Expression |
---|---|
rhmsubclem3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngcrescrhm.r | . . . . . 6 ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) | |
2 | 1 | eleq2d 2818 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑅 ↔ 𝑥 ∈ (Ring ∩ 𝑈))) |
3 | elinel1 4195 | . . . . 5 ⊢ (𝑥 ∈ (Ring ∩ 𝑈) → 𝑥 ∈ Ring) | |
4 | 2, 3 | syl6bi 253 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑅 → 𝑥 ∈ Ring)) |
5 | 4 | imp 406 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → 𝑥 ∈ Ring) |
6 | eqid 2731 | . . . 4 ⊢ (Base‘𝑥) = (Base‘𝑥) | |
7 | 6 | idrhm 20385 | . . 3 ⊢ (𝑥 ∈ Ring → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥)) |
8 | 5, 7 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ( I ↾ (Base‘𝑥)) ∈ (𝑥 RingHom 𝑥)) |
9 | rngcrescrhm.c | . . 3 ⊢ 𝐶 = (RngCat‘𝑈) | |
10 | eqid 2731 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
11 | 9 | eqcomi 2740 | . . . 4 ⊢ (RngCat‘𝑈) = 𝐶 |
12 | 11 | fveq2i 6894 | . . 3 ⊢ (Id‘(RngCat‘𝑈)) = (Id‘𝐶) |
13 | rngcrescrhm.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
14 | 13 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → 𝑈 ∈ 𝑉) |
15 | incom 4201 | . . . . . 6 ⊢ (Ring ∩ 𝑈) = (𝑈 ∩ Ring) | |
16 | ringssrng 20178 | . . . . . . 7 ⊢ Ring ⊆ Rng | |
17 | sslin 4234 | . . . . . . 7 ⊢ (Ring ⊆ Rng → (𝑈 ∩ Ring) ⊆ (𝑈 ∩ Rng)) | |
18 | 16, 17 | mp1i 13 | . . . . . 6 ⊢ (𝜑 → (𝑈 ∩ Ring) ⊆ (𝑈 ∩ Rng)) |
19 | 15, 18 | eqsstrid 4030 | . . . . 5 ⊢ (𝜑 → (Ring ∩ 𝑈) ⊆ (𝑈 ∩ Rng)) |
20 | 9, 10, 13 | rngcbas 46964 | . . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Rng)) |
21 | 19, 1, 20 | 3sstr4d 4029 | . . . 4 ⊢ (𝜑 → 𝑅 ⊆ (Base‘𝐶)) |
22 | 21 | sselda 3982 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → 𝑥 ∈ (Base‘𝐶)) |
23 | 9, 10, 12, 14, 22, 6 | rngcid 46978 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ((Id‘(RngCat‘𝑈))‘𝑥) = ( I ↾ (Base‘𝑥))) |
24 | rngcrescrhm.h | . . . 4 ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) | |
25 | 13, 9, 1, 24 | rhmsubclem2 47086 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅 ∧ 𝑥 ∈ 𝑅) → (𝑥𝐻𝑥) = (𝑥 RingHom 𝑥)) |
26 | 25 | 3anidm23 1420 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → (𝑥𝐻𝑥) = (𝑥 RingHom 𝑥)) |
27 | 8, 23, 26 | 3eltr4d 2847 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∩ cin 3947 ⊆ wss 3948 I cid 5573 × cxp 5674 ↾ cres 5678 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 Idccid 17616 Rngcrng 20050 Ringcrg 20131 RingHom crh 20364 RngCatcrngc 46956 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-hom 17228 df-cco 17229 df-0g 17394 df-cat 17619 df-cid 17620 df-homf 17621 df-ssc 17764 df-resc 17765 df-subc 17766 df-estrc 18081 df-mgm 18568 df-mgmhm 18620 df-sgrp 18647 df-mnd 18663 df-mhm 18708 df-grp 18861 df-minusg 18862 df-ghm 19132 df-cmn 19695 df-abl 19696 df-mgp 20033 df-rng 20051 df-ur 20080 df-ring 20133 df-rnghm 20331 df-rhm 20367 df-rngc 46958 |
This theorem is referenced by: rhmsubc 47089 |
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