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Theorem erlval 33245
Description: Value of the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
Hypotheses
Ref Expression
rlocval.1 𝐵 = (Base‘𝑅)
rlocval.2 0 = (0g𝑅)
rlocval.3 · = (.r𝑅)
rlocval.4 = (-g𝑅)
erlval.w 𝑊 = (𝐵 × 𝑆)
erlval.q = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ∃𝑡𝑆 (𝑡 · (((1st𝑎) · (2nd𝑏)) ((1st𝑏) · (2nd𝑎)))) = 0 )}
erlval.20 (𝜑𝑆𝐵)
Assertion
Ref Expression
erlval (𝜑 → (𝑅 ~RL 𝑆) = )
Distinct variable groups:   · ,𝑎,𝑏,𝑡   𝑅,𝑎,𝑏,𝑡   𝑆,𝑎,𝑏,𝑡   𝑊,𝑎,𝑏,𝑡   𝜑,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑡)   𝐵(𝑡,𝑎,𝑏)   (𝑡,𝑎,𝑏)   (𝑡,𝑎,𝑏)   0 (𝑡,𝑎,𝑏)

Proof of Theorem erlval
Dummy variables 𝑤 𝑟 𝑠 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 484 . . 3 ((𝜑𝑅 ∈ V) → 𝑅 ∈ V)
2 rlocval.1 . . . . . 6 𝐵 = (Base‘𝑅)
32fvexi 6921 . . . . 5 𝐵 ∈ V
43a1i 11 . . . 4 ((𝜑𝑅 ∈ V) → 𝐵 ∈ V)
5 erlval.20 . . . . 5 (𝜑𝑆𝐵)
65adantr 480 . . . 4 ((𝜑𝑅 ∈ V) → 𝑆𝐵)
74, 6ssexd 5330 . . 3 ((𝜑𝑅 ∈ V) → 𝑆 ∈ V)
8 erlval.q . . . 4 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ∃𝑡𝑆 (𝑡 · (((1st𝑎) · (2nd𝑏)) ((1st𝑏) · (2nd𝑎)))) = 0 )}
9 erlval.w . . . . . . 7 𝑊 = (𝐵 × 𝑆)
104, 7xpexd 7770 . . . . . . 7 ((𝜑𝑅 ∈ V) → (𝐵 × 𝑆) ∈ V)
119, 10eqeltrid 2843 . . . . . 6 ((𝜑𝑅 ∈ V) → 𝑊 ∈ V)
1211, 11xpexd 7770 . . . . 5 ((𝜑𝑅 ∈ V) → (𝑊 × 𝑊) ∈ V)
13 simprll 779 . . . . . . 7 ((𝜑 ∧ ((𝑎𝑊𝑏𝑊) ∧ ∃𝑡𝑆 (𝑡 · (((1st𝑎) · (2nd𝑏)) ((1st𝑏) · (2nd𝑎)))) = 0 )) → 𝑎𝑊)
14 simprlr 780 . . . . . . 7 ((𝜑 ∧ ((𝑎𝑊𝑏𝑊) ∧ ∃𝑡𝑆 (𝑡 · (((1st𝑎) · (2nd𝑏)) ((1st𝑏) · (2nd𝑎)))) = 0 )) → 𝑏𝑊)
1513, 14opabssxpd 5736 . . . . . 6 (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ∃𝑡𝑆 (𝑡 · (((1st𝑎) · (2nd𝑏)) ((1st𝑏) · (2nd𝑎)))) = 0 )} ⊆ (𝑊 × 𝑊))
1615adantr 480 . . . . 5 ((𝜑𝑅 ∈ V) → {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ∃𝑡𝑆 (𝑡 · (((1st𝑎) · (2nd𝑏)) ((1st𝑏) · (2nd𝑎)))) = 0 )} ⊆ (𝑊 × 𝑊))
1712, 16ssexd 5330 . . . 4 ((𝜑𝑅 ∈ V) → {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ∃𝑡𝑆 (𝑡 · (((1st𝑎) · (2nd𝑏)) ((1st𝑏) · (2nd𝑎)))) = 0 )} ∈ V)
188, 17eqeltrid 2843 . . 3 ((𝜑𝑅 ∈ V) → ∈ V)
19 fvexd 6922 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → (.r𝑟) ∈ V)
20 fveq2 6907 . . . . . . 7 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
2120adantr 480 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (.r𝑟) = (.r𝑅))
22 rlocval.3 . . . . . 6 · = (.r𝑅)
2321, 22eqtr4di 2793 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → (.r𝑟) = · )
24 fvexd 6922 . . . . . . 7 (((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) → (Base‘𝑟) ∈ V)
25 vex 3482 . . . . . . . 8 𝑠 ∈ V
2625a1i 11 . . . . . . 7 (((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) → 𝑠 ∈ V)
2724, 26xpexd 7770 . . . . . 6 (((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) → ((Base‘𝑟) × 𝑠) ∈ V)
28 fveq2 6907 . . . . . . . . . 10 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
2928ad2antrr 726 . . . . . . . . 9 (((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) → (Base‘𝑟) = (Base‘𝑅))
3029, 2eqtr4di 2793 . . . . . . . 8 (((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) → (Base‘𝑟) = 𝐵)
31 simplr 769 . . . . . . . 8 (((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) → 𝑠 = 𝑆)
3230, 31xpeq12d 5720 . . . . . . 7 (((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) → ((Base‘𝑟) × 𝑠) = (𝐵 × 𝑆))
3332, 9eqtr4di 2793 . . . . . 6 (((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) → ((Base‘𝑟) × 𝑠) = 𝑊)
34 simpr 484 . . . . . . . . . . 11 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → 𝑤 = 𝑊)
3534eleq2d 2825 . . . . . . . . . 10 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (𝑎𝑤𝑎𝑊))
3634eleq2d 2825 . . . . . . . . . 10 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (𝑏𝑤𝑏𝑊))
3735, 36anbi12d 632 . . . . . . . . 9 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → ((𝑎𝑤𝑏𝑤) ↔ (𝑎𝑊𝑏𝑊)))
3831adantr 480 . . . . . . . . . 10 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → 𝑠 = 𝑆)
39 simplr 769 . . . . . . . . . . . 12 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → 𝑥 = · )
40 eqidd 2736 . . . . . . . . . . . 12 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → 𝑡 = 𝑡)
41 fveq2 6907 . . . . . . . . . . . . . . 15 (𝑟 = 𝑅 → (-g𝑟) = (-g𝑅))
4241ad3antrrr 730 . . . . . . . . . . . . . 14 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (-g𝑟) = (-g𝑅))
43 rlocval.4 . . . . . . . . . . . . . 14 = (-g𝑅)
4442, 43eqtr4di 2793 . . . . . . . . . . . . 13 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (-g𝑟) = )
4539oveqd 7448 . . . . . . . . . . . . 13 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → ((1st𝑎)𝑥(2nd𝑏)) = ((1st𝑎) · (2nd𝑏)))
4639oveqd 7448 . . . . . . . . . . . . 13 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → ((1st𝑏)𝑥(2nd𝑎)) = ((1st𝑏) · (2nd𝑎)))
4744, 45, 46oveq123d 7452 . . . . . . . . . . . 12 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (((1st𝑎)𝑥(2nd𝑏))(-g𝑟)((1st𝑏)𝑥(2nd𝑎))) = (((1st𝑎) · (2nd𝑏)) ((1st𝑏) · (2nd𝑎))))
4839, 40, 47oveq123d 7452 . . . . . . . . . . 11 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (𝑡𝑥(((1st𝑎)𝑥(2nd𝑏))(-g𝑟)((1st𝑏)𝑥(2nd𝑎)))) = (𝑡 · (((1st𝑎) · (2nd𝑏)) ((1st𝑏) · (2nd𝑎)))))
49 fveq2 6907 . . . . . . . . . . . . 13 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
5049ad3antrrr 730 . . . . . . . . . . . 12 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (0g𝑟) = (0g𝑅))
51 rlocval.2 . . . . . . . . . . . 12 0 = (0g𝑅)
5250, 51eqtr4di 2793 . . . . . . . . . . 11 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (0g𝑟) = 0 )
5348, 52eqeq12d 2751 . . . . . . . . . 10 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → ((𝑡𝑥(((1st𝑎)𝑥(2nd𝑏))(-g𝑟)((1st𝑏)𝑥(2nd𝑎)))) = (0g𝑟) ↔ (𝑡 · (((1st𝑎) · (2nd𝑏)) ((1st𝑏) · (2nd𝑎)))) = 0 ))
5438, 53rexeqbidv 3345 . . . . . . . . 9 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (∃𝑡𝑠 (𝑡𝑥(((1st𝑎)𝑥(2nd𝑏))(-g𝑟)((1st𝑏)𝑥(2nd𝑎)))) = (0g𝑟) ↔ ∃𝑡𝑆 (𝑡 · (((1st𝑎) · (2nd𝑏)) ((1st𝑏) · (2nd𝑎)))) = 0 ))
5537, 54anbi12d 632 . . . . . . . 8 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → (((𝑎𝑤𝑏𝑤) ∧ ∃𝑡𝑠 (𝑡𝑥(((1st𝑎)𝑥(2nd𝑏))(-g𝑟)((1st𝑏)𝑥(2nd𝑎)))) = (0g𝑟)) ↔ ((𝑎𝑊𝑏𝑊) ∧ ∃𝑡𝑆 (𝑡 · (((1st𝑎) · (2nd𝑏)) ((1st𝑏) · (2nd𝑎)))) = 0 )))
5655opabbidv 5214 . . . . . . 7 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑤𝑏𝑤) ∧ ∃𝑡𝑠 (𝑡𝑥(((1st𝑎)𝑥(2nd𝑏))(-g𝑟)((1st𝑏)𝑥(2nd𝑎)))) = (0g𝑟))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑊𝑏𝑊) ∧ ∃𝑡𝑆 (𝑡 · (((1st𝑎) · (2nd𝑏)) ((1st𝑏) · (2nd𝑎)))) = 0 )})
5756, 8eqtr4di 2793 . . . . . 6 ((((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) ∧ 𝑤 = 𝑊) → {⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑤𝑏𝑤) ∧ ∃𝑡𝑠 (𝑡𝑥(((1st𝑎)𝑥(2nd𝑏))(-g𝑟)((1st𝑏)𝑥(2nd𝑎)))) = (0g𝑟))} = )
5827, 33, 57csbied2 3948 . . . . 5 (((𝑟 = 𝑅𝑠 = 𝑆) ∧ 𝑥 = · ) → ((Base‘𝑟) × 𝑠) / 𝑤{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑤𝑏𝑤) ∧ ∃𝑡𝑠 (𝑡𝑥(((1st𝑎)𝑥(2nd𝑏))(-g𝑟)((1st𝑏)𝑥(2nd𝑎)))) = (0g𝑟))} = )
5919, 23, 58csbied2 3948 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → (.r𝑟) / 𝑥((Base‘𝑟) × 𝑠) / 𝑤{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑤𝑏𝑤) ∧ ∃𝑡𝑠 (𝑡𝑥(((1st𝑎)𝑥(2nd𝑏))(-g𝑟)((1st𝑏)𝑥(2nd𝑎)))) = (0g𝑟))} = )
60 df-erl 33242 . . . 4 ~RL = (𝑟 ∈ V, 𝑠 ∈ V ↦ (.r𝑟) / 𝑥((Base‘𝑟) × 𝑠) / 𝑤{⟨𝑎, 𝑏⟩ ∣ ((𝑎𝑤𝑏𝑤) ∧ ∃𝑡𝑠 (𝑡𝑥(((1st𝑎)𝑥(2nd𝑏))(-g𝑟)((1st𝑏)𝑥(2nd𝑎)))) = (0g𝑟))})
6159, 60ovmpoga 7587 . . 3 ((𝑅 ∈ V ∧ 𝑆 ∈ V ∧ ∈ V) → (𝑅 ~RL 𝑆) = )
621, 7, 18, 61syl3anc 1370 . 2 ((𝜑𝑅 ∈ V) → (𝑅 ~RL 𝑆) = )
6360reldmmpo 7567 . . . . 5 Rel dom ~RL
6463ovprc1 7470 . . . 4 𝑅 ∈ V → (𝑅 ~RL 𝑆) = ∅)
6564adantl 481 . . 3 ((𝜑 ∧ ¬ 𝑅 ∈ V) → (𝑅 ~RL 𝑆) = ∅)
668, 15eqsstrid 4044 . . . . . 6 (𝜑 ⊆ (𝑊 × 𝑊))
6766adantr 480 . . . . 5 ((𝜑 ∧ ¬ 𝑅 ∈ V) → ⊆ (𝑊 × 𝑊))
68 fvprc 6899 . . . . . . . . . . 11 𝑅 ∈ V → (Base‘𝑅) = ∅)
692, 68eqtrid 2787 . . . . . . . . . 10 𝑅 ∈ V → 𝐵 = ∅)
7069xpeq1d 5718 . . . . . . . . 9 𝑅 ∈ V → (𝐵 × 𝑆) = (∅ × 𝑆))
71 0xp 5787 . . . . . . . . 9 (∅ × 𝑆) = ∅
7270, 71eqtrdi 2791 . . . . . . . 8 𝑅 ∈ V → (𝐵 × 𝑆) = ∅)
739, 72eqtrid 2787 . . . . . . 7 𝑅 ∈ V → 𝑊 = ∅)
74 id 22 . . . . . . . . 9 (𝑊 = ∅ → 𝑊 = ∅)
7574, 74xpeq12d 5720 . . . . . . . 8 (𝑊 = ∅ → (𝑊 × 𝑊) = (∅ × ∅))
76 0xp 5787 . . . . . . . 8 (∅ × ∅) = ∅
7775, 76eqtrdi 2791 . . . . . . 7 (𝑊 = ∅ → (𝑊 × 𝑊) = ∅)
7873, 77syl 17 . . . . . 6 𝑅 ∈ V → (𝑊 × 𝑊) = ∅)
7978adantl 481 . . . . 5 ((𝜑 ∧ ¬ 𝑅 ∈ V) → (𝑊 × 𝑊) = ∅)
8067, 79sseqtrd 4036 . . . 4 ((𝜑 ∧ ¬ 𝑅 ∈ V) → ⊆ ∅)
81 ss0 4408 . . . 4 ( ⊆ ∅ → = ∅)
8280, 81syl 17 . . 3 ((𝜑 ∧ ¬ 𝑅 ∈ V) → = ∅)
8365, 82eqtr4d 2778 . 2 ((𝜑 ∧ ¬ 𝑅 ∈ V) → (𝑅 ~RL 𝑆) = )
8462, 83pm2.61dan 813 1 (𝜑 → (𝑅 ~RL 𝑆) = )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478  csb 3908  wss 3963  c0 4339  {copab 5210   × cxp 5687  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  Basecbs 17245  .rcmulr 17299  0gc0g 17486  -gcsg 18966   ~RL cerl 33240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-erl 33242
This theorem is referenced by:  erlcl1  33247  erlcl2  33248  erldi  33249  erlbrd  33250  erler  33252  fracerl  33288
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