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Mirrors > Home > MPE Home > Th. List > Mathboxes > srhmsubcALTVlem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for srhmsubcALTV 46832. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
srhmsubcALTV.s | ⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring |
srhmsubcALTV.c | ⊢ 𝐶 = (𝑈 ∩ 𝑆) |
srhmsubcALTV.j | ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) |
Ref | Expression |
---|---|
srhmsubcALTVlem2 | ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋𝐽𝑌) = (𝑋(Hom ‘(RingCatALTV‘𝑈))𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srhmsubcALTV.j | . . . 4 ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))) |
3 | oveq12 7405 | . . . 4 ⊢ ((𝑟 = 𝑋 ∧ 𝑠 = 𝑌) → (𝑟 RingHom 𝑠) = (𝑋 RingHom 𝑌)) | |
4 | 3 | adantl 483 | . . 3 ⊢ (((𝑈 ∈ 𝑉 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) ∧ (𝑟 = 𝑋 ∧ 𝑠 = 𝑌)) → (𝑟 RingHom 𝑠) = (𝑋 RingHom 𝑌)) |
5 | simpl 484 | . . . 4 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) → 𝑋 ∈ 𝐶) | |
6 | 5 | adantl 483 | . . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → 𝑋 ∈ 𝐶) |
7 | simpr 486 | . . . 4 ⊢ ((𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) → 𝑌 ∈ 𝐶) | |
8 | 7 | adantl 483 | . . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → 𝑌 ∈ 𝐶) |
9 | ovexd 7431 | . . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋 RingHom 𝑌) ∈ V) | |
10 | 2, 4, 6, 8, 9 | ovmpod 7547 | . 2 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋𝐽𝑌) = (𝑋 RingHom 𝑌)) |
11 | eqid 2733 | . . 3 ⊢ (RingCatALTV‘𝑈) = (RingCatALTV‘𝑈) | |
12 | eqid 2733 | . . 3 ⊢ (Base‘(RingCatALTV‘𝑈)) = (Base‘(RingCatALTV‘𝑈)) | |
13 | simpl 484 | . . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → 𝑈 ∈ 𝑉) | |
14 | eqid 2733 | . . 3 ⊢ (Hom ‘(RingCatALTV‘𝑈)) = (Hom ‘(RingCatALTV‘𝑈)) | |
15 | srhmsubcALTV.s | . . . . 5 ⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring | |
16 | srhmsubcALTV.c | . . . . 5 ⊢ 𝐶 = (𝑈 ∩ 𝑆) | |
17 | 15, 16 | srhmsubcALTVlem1 46830 | . . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ (Base‘(RingCatALTV‘𝑈))) |
18 | 5, 17 | sylan2 594 | . . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → 𝑋 ∈ (Base‘(RingCatALTV‘𝑈))) |
19 | 15, 16 | srhmsubcALTVlem1 46830 | . . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑌 ∈ 𝐶) → 𝑌 ∈ (Base‘(RingCatALTV‘𝑈))) |
20 | 7, 19 | sylan2 594 | . . 3 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → 𝑌 ∈ (Base‘(RingCatALTV‘𝑈))) |
21 | 11, 12, 13, 14, 18, 20 | ringchomALTV 46786 | . 2 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋(Hom ‘(RingCatALTV‘𝑈))𝑌) = (𝑋 RingHom 𝑌)) |
22 | 10, 21 | eqtr4d 2776 | 1 ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋𝐽𝑌) = (𝑋(Hom ‘(RingCatALTV‘𝑈))𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 ∩ cin 3945 ‘cfv 6535 (class class class)co 7396 ∈ cmpo 7398 Basecbs 17131 Hom chom 17195 Ringcrg 20038 RingHom crh 20226 RingCatALTVcringcALTV 46742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-dec 12665 df-uz 12810 df-fz 13472 df-struct 17067 df-slot 17102 df-ndx 17114 df-base 17132 df-hom 17208 df-cco 17209 df-ringcALTV 46744 |
This theorem is referenced by: srhmsubcALTV 46832 |
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