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Theorem srhmsubclem1 44712
 Description: Lemma 1 for srhmsubc 44715. (Contributed by AV, 19-Feb-2020.)
Hypotheses
Ref Expression
srhmsubc.s 𝑟𝑆 𝑟 ∈ Ring
srhmsubc.c 𝐶 = (𝑈𝑆)
Assertion
Ref Expression
srhmsubclem1 (𝑋𝐶𝑋 ∈ (𝑈 ∩ Ring))
Distinct variable groups:   𝑆,𝑟   𝑋,𝑟
Allowed substitution hints:   𝐶(𝑟)   𝑈(𝑟)

Proof of Theorem srhmsubclem1
StepHypRef Expression
1 eleq1 2877 . . . 4 (𝑟 = 𝑋 → (𝑟 ∈ Ring ↔ 𝑋 ∈ Ring))
2 srhmsubc.s . . . 4 𝑟𝑆 𝑟 ∈ Ring
31, 2vtoclri 3533 . . 3 (𝑋𝑆𝑋 ∈ Ring)
43anim2i 619 . 2 ((𝑋𝑈𝑋𝑆) → (𝑋𝑈𝑋 ∈ Ring))
5 srhmsubc.c . . 3 𝐶 = (𝑈𝑆)
65elin2 4124 . 2 (𝑋𝐶 ↔ (𝑋𝑈𝑋𝑆))
7 elin 3897 . 2 (𝑋 ∈ (𝑈 ∩ Ring) ↔ (𝑋𝑈𝑋 ∈ Ring))
84, 6, 73imtr4i 295 1 (𝑋𝐶𝑋 ∈ (𝑈 ∩ Ring))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ∩ cin 3880  Ringcrg 19293 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443  df-in 3888 This theorem is referenced by:  srhmsubclem2  44713  srhmsubcALTVlem1  44731
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