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Theorem abvpropd 18836
Description: If two structures have the same ring components, they have the same collection of absolute values. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
abvpropd.1 (𝜑𝐵 = (Base‘𝐾))
abvpropd.2 (𝜑𝐵 = (Base‘𝐿))
abvpropd.3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
abvpropd.4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
Assertion
Ref Expression
abvpropd (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Proof of Theorem abvpropd
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 abvpropd.1 . . . . 5 (𝜑𝐵 = (Base‘𝐾))
2 abvpropd.2 . . . . 5 (𝜑𝐵 = (Base‘𝐿))
3 abvpropd.3 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
4 abvpropd.4 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
51, 2, 3, 4ringpropd 18576 . . . 4 (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
61, 2eqtr3d 2657 . . . . . 6 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
76feq2d 6029 . . . . 5 (𝜑 → (𝑓:(Base‘𝐾)⟶(0[,)+∞) ↔ 𝑓:(Base‘𝐿)⟶(0[,)+∞)))
81, 2, 3grpidpropd 17255 . . . . . . . . . . 11 (𝜑 → (0g𝐾) = (0g𝐿))
98adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝐵) → (0g𝐾) = (0g𝐿))
109eqeq2d 2631 . . . . . . . . 9 ((𝜑𝑥𝐵) → (𝑥 = (0g𝐾) ↔ 𝑥 = (0g𝐿)))
1110bibi2d 332 . . . . . . . 8 ((𝜑𝑥𝐵) → (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ↔ ((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿))))
124fveq2d 6193 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑓‘(𝑥(.r𝐾)𝑦)) = (𝑓‘(𝑥(.r𝐿)𝑦)))
1312eqeq1d 2623 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ↔ (𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦))))
143fveq2d 6193 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑓‘(𝑥(+g𝐾)𝑦)) = (𝑓‘(𝑥(+g𝐿)𝑦)))
1514breq1d 4661 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)) ↔ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))
1613, 15anbi12d 747 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))) ↔ ((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))
1716anassrs 680 . . . . . . . . 9 (((𝜑𝑥𝐵) ∧ 𝑦𝐵) → (((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))) ↔ ((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))
1817ralbidva 2984 . . . . . . . 8 ((𝜑𝑥𝐵) → (∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))) ↔ ∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))
1911, 18anbi12d 747 . . . . . . 7 ((𝜑𝑥𝐵) → ((((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))))
2019ralbidva 2984 . . . . . 6 (𝜑 → (∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ ∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))))
211raleqdv 3142 . . . . . . . 8 (𝜑 → (∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))) ↔ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))
2221anbi2d 740 . . . . . . 7 (𝜑 → ((((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))))
231, 22raleqbidv 3150 . . . . . 6 (𝜑 → (∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ ∀𝑥 ∈ (Base‘𝐾)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))))
242raleqdv 3142 . . . . . . . 8 (𝜑 → (∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))) ↔ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))
2524anbi2d 740 . . . . . . 7 (𝜑 → ((((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))))
262, 25raleqbidv 3150 . . . . . 6 (𝜑 → (∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ ∀𝑥 ∈ (Base‘𝐿)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))))
2720, 23, 263bitr3d 298 . . . . 5 (𝜑 → (∀𝑥 ∈ (Base‘𝐾)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ ∀𝑥 ∈ (Base‘𝐿)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))))
287, 27anbi12d 747 . . . 4 (𝜑 → ((𝑓:(Base‘𝐾)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐾)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))) ↔ (𝑓:(Base‘𝐿)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐿)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))))
295, 28anbi12d 747 . . 3 (𝜑 → ((𝐾 ∈ Ring ∧ (𝑓:(Base‘𝐾)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐾)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))) ↔ (𝐿 ∈ Ring ∧ (𝑓:(Base‘𝐿)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐿)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))))))
30 eqid 2621 . . . . 5 (AbsVal‘𝐾) = (AbsVal‘𝐾)
3130abvrcl 18815 . . . 4 (𝑓 ∈ (AbsVal‘𝐾) → 𝐾 ∈ Ring)
32 eqid 2621 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
33 eqid 2621 . . . . 5 (+g𝐾) = (+g𝐾)
34 eqid 2621 . . . . 5 (.r𝐾) = (.r𝐾)
35 eqid 2621 . . . . 5 (0g𝐾) = (0g𝐾)
3630, 32, 33, 34, 35isabv 18813 . . . 4 (𝐾 ∈ Ring → (𝑓 ∈ (AbsVal‘𝐾) ↔ (𝑓:(Base‘𝐾)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐾)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))))
3731, 36biadan2 674 . . 3 (𝑓 ∈ (AbsVal‘𝐾) ↔ (𝐾 ∈ Ring ∧ (𝑓:(Base‘𝐾)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐾)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐾)) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑓‘(𝑥(.r𝐾)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐾)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))))
38 eqid 2621 . . . . 5 (AbsVal‘𝐿) = (AbsVal‘𝐿)
3938abvrcl 18815 . . . 4 (𝑓 ∈ (AbsVal‘𝐿) → 𝐿 ∈ Ring)
40 eqid 2621 . . . . 5 (Base‘𝐿) = (Base‘𝐿)
41 eqid 2621 . . . . 5 (+g𝐿) = (+g𝐿)
42 eqid 2621 . . . . 5 (.r𝐿) = (.r𝐿)
43 eqid 2621 . . . . 5 (0g𝐿) = (0g𝐿)
4438, 40, 41, 42, 43isabv 18813 . . . 4 (𝐿 ∈ Ring → (𝑓 ∈ (AbsVal‘𝐿) ↔ (𝑓:(Base‘𝐿)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐿)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))))
4539, 44biadan2 674 . . 3 (𝑓 ∈ (AbsVal‘𝐿) ↔ (𝐿 ∈ Ring ∧ (𝑓:(Base‘𝐿)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝐿)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝐿)) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑓‘(𝑥(.r𝐿)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝐿)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))))
4629, 37, 453bitr4g 303 . 2 (𝜑 → (𝑓 ∈ (AbsVal‘𝐾) ↔ 𝑓 ∈ (AbsVal‘𝐿)))
4746eqrdv 2619 1 (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  wral 2911   class class class wbr 4651  wf 5882  cfv 5886  (class class class)co 6647  0cc0 9933   + caddc 9936   · cmul 9938  +∞cpnf 10068  cle 10072  [,)cico 12174  Basecbs 15851  +gcplusg 15935  .rcmulr 15936  0gc0g 16094  Ringcrg 18541  AbsValcabv 18810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-er 7739  df-map 7856  df-en 7953  df-dom 7954  df-sdom 7955  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-nn 11018  df-2 11076  df-ndx 15854  df-slot 15855  df-base 15857  df-sets 15858  df-plusg 15948  df-0g 16096  df-mgm 17236  df-sgrp 17278  df-mnd 17289  df-grp 17419  df-mgp 18484  df-ring 18543  df-abv 18811
This theorem is referenced by:  tngnrg  22472  abvpropd2  29637
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