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Theorem eroprf 7732
 Description: Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
eropr.1 𝐽 = (𝐴 / 𝑅)
eropr.2 𝐾 = (𝐵 / 𝑆)
eropr.3 (𝜑𝑇𝑍)
eropr.4 (𝜑𝑅 Er 𝑈)
eropr.5 (𝜑𝑆 Er 𝑉)
eropr.6 (𝜑𝑇 Er 𝑊)
eropr.7 (𝜑𝐴𝑈)
eropr.8 (𝜑𝐵𝑉)
eropr.9 (𝜑𝐶𝑊)
eropr.10 (𝜑+ :(𝐴 × 𝐵)⟶𝐶)
eropr.11 ((𝜑 ∧ ((𝑟𝐴𝑠𝐴) ∧ (𝑡𝐵𝑢𝐵))) → ((𝑟𝑅𝑠𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))
eropr.12 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}
eropr.13 (𝜑𝑅𝑋)
eropr.14 (𝜑𝑆𝑌)
eropr.15 𝐿 = (𝐶 / 𝑇)
Assertion
Ref Expression
eroprf (𝜑 :(𝐽 × 𝐾)⟶𝐿)
Distinct variable groups:   𝑞,𝑝,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧,𝐴   𝐵,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝐿,𝑝,𝑞,𝑥,𝑦,𝑧   𝐽,𝑝,𝑞,𝑥,𝑦,𝑧   𝑅,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝐾,𝑝,𝑞,𝑥,𝑦,𝑧   𝑆,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   + ,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝜑,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝑇,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝑋,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑧   𝑌,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑧
Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   (𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝑈(𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝐽(𝑢,𝑡,𝑠,𝑟)   𝐾(𝑢,𝑡,𝑠,𝑟)   𝐿(𝑢,𝑡,𝑠,𝑟)   𝑉(𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝑊(𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)   𝑍(𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)

Proof of Theorem eroprf
StepHypRef Expression
1 eropr.3 . . . . . . . . . . . 12 (𝜑𝑇𝑍)
21ad2antrr 758 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ (𝑝𝐴𝑞𝐵)) → 𝑇𝑍)
3 eropr.10 . . . . . . . . . . . . 13 (𝜑+ :(𝐴 × 𝐵)⟶𝐶)
43adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → + :(𝐴 × 𝐵)⟶𝐶)
54fovrnda 6703 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ (𝑝𝐴𝑞𝐵)) → (𝑝 + 𝑞) ∈ 𝐶)
6 ecelqsg 7689 . . . . . . . . . . 11 ((𝑇𝑍 ∧ (𝑝 + 𝑞) ∈ 𝐶) → [(𝑝 + 𝑞)]𝑇 ∈ (𝐶 / 𝑇))
72, 5, 6syl2anc 691 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ (𝑝𝐴𝑞𝐵)) → [(𝑝 + 𝑞)]𝑇 ∈ (𝐶 / 𝑇))
8 eropr.15 . . . . . . . . . 10 𝐿 = (𝐶 / 𝑇)
97, 8syl6eleqr 2699 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ (𝑝𝐴𝑞𝐵)) → [(𝑝 + 𝑞)]𝑇𝐿)
10 eleq1a 2683 . . . . . . . . 9 ([(𝑝 + 𝑞)]𝑇𝐿 → (𝑧 = [(𝑝 + 𝑞)]𝑇𝑧𝐿))
119, 10syl 17 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ (𝑝𝐴𝑞𝐵)) → (𝑧 = [(𝑝 + 𝑞)]𝑇𝑧𝐿))
1211adantld 482 . . . . . . 7 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ (𝑝𝐴𝑞𝐵)) → (((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → 𝑧𝐿))
1312rexlimdvva 3020 . . . . . 6 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → 𝑧𝐿))
1413abssdv 3639 . . . . 5 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → {𝑧 ∣ ∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)} ⊆ 𝐿)
15 eropr.1 . . . . . . 7 𝐽 = (𝐴 / 𝑅)
16 eropr.2 . . . . . . 7 𝐾 = (𝐵 / 𝑆)
17 eropr.4 . . . . . . 7 (𝜑𝑅 Er 𝑈)
18 eropr.5 . . . . . . 7 (𝜑𝑆 Er 𝑉)
19 eropr.6 . . . . . . 7 (𝜑𝑇 Er 𝑊)
20 eropr.7 . . . . . . 7 (𝜑𝐴𝑈)
21 eropr.8 . . . . . . 7 (𝜑𝐵𝑉)
22 eropr.9 . . . . . . 7 (𝜑𝐶𝑊)
23 eropr.11 . . . . . . 7 ((𝜑 ∧ ((𝑟𝐴𝑠𝐴) ∧ (𝑡𝐵𝑢𝐵))) → ((𝑟𝑅𝑠𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))
2415, 16, 1, 17, 18, 19, 20, 21, 22, 3, 23eroveu 7729 . . . . . 6 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → ∃!𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
25 iotacl 5791 . . . . . 6 (∃!𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ {𝑧 ∣ ∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)})
2624, 25syl 17 . . . . 5 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ {𝑧 ∣ ∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)})
2714, 26sseldd 3569 . . . 4 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ 𝐿)
2827ralrimivva 2954 . . 3 (𝜑 → ∀𝑥𝐽𝑦𝐾 (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ 𝐿)
29 eqid 2610 . . . 4 (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))) = (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
3029fmpt2 7126 . . 3 (∀𝑥𝐽𝑦𝐾 (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ 𝐿 ↔ (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))):(𝐽 × 𝐾)⟶𝐿)
3128, 30sylib 207 . 2 (𝜑 → (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))):(𝐽 × 𝐾)⟶𝐿)
32 eropr.12 . . . 4 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}
3315, 16, 1, 17, 18, 19, 20, 21, 22, 3, 23, 32erovlem 7730 . . 3 (𝜑 = (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))
3433feq1d 5943 . 2 (𝜑 → ( :(𝐽 × 𝐾)⟶𝐿 ↔ (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))):(𝐽 × 𝐾)⟶𝐿))
3531, 34mpbird 246 1 (𝜑 :(𝐽 × 𝐾)⟶𝐿)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃!weu 2458  {cab 2596  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540   class class class wbr 4583   × cxp 5036  ℩cio 5766  ⟶wf 5800  (class class class)co 6549  {coprab 6550   ↦ cmpt2 6551   Er wer 7626  [cec 7627   / cqs 7628 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-ec 7631  df-qs 7635 This theorem is referenced by:  eroprf2  7734
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