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Theorem erov 7887
Description: The value 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)
Assertion
Ref Expression
erov ((𝜑𝑃𝐴𝑄𝐵) → ([𝑃]𝑅 [𝑄]𝑆) = [(𝑃 + 𝑄)]𝑇)
Distinct variable groups:   𝑞,𝑝,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧,𝐴   𝐵,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝐽,𝑝,𝑞,𝑥,𝑦,𝑧   𝑃,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝑅,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝐾,𝑝,𝑞,𝑥,𝑦,𝑧   𝑄,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,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)   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 erov
StepHypRef Expression
1 eropr.1 . . . . 5 𝐽 = (𝐴 / 𝑅)
2 eropr.2 . . . . 5 𝐾 = (𝐵 / 𝑆)
3 eropr.3 . . . . 5 (𝜑𝑇𝑍)
4 eropr.4 . . . . 5 (𝜑𝑅 Er 𝑈)
5 eropr.5 . . . . 5 (𝜑𝑆 Er 𝑉)
6 eropr.6 . . . . 5 (𝜑𝑇 Er 𝑊)
7 eropr.7 . . . . 5 (𝜑𝐴𝑈)
8 eropr.8 . . . . 5 (𝜑𝐵𝑉)
9 eropr.9 . . . . 5 (𝜑𝐶𝑊)
10 eropr.10 . . . . 5 (𝜑+ :(𝐴 × 𝐵)⟶𝐶)
11 eropr.11 . . . . 5 ((𝜑 ∧ ((𝑟𝐴𝑠𝐴) ∧ (𝑡𝐵𝑢𝐵))) → ((𝑟𝑅𝑠𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))
12 eropr.12 . . . . 5 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12erovlem 7886 . . . 4 (𝜑 = (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))
14133ad2ant1 1102 . . 3 ((𝜑𝑃𝐴𝑄𝐵) → = (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))
15 simprl 809 . . . . . . . 8 (((𝜑𝑃𝐴𝑄𝐵) ∧ (𝑥 = [𝑃]𝑅𝑦 = [𝑄]𝑆)) → 𝑥 = [𝑃]𝑅)
1615eqeq1d 2653 . . . . . . 7 (((𝜑𝑃𝐴𝑄𝐵) ∧ (𝑥 = [𝑃]𝑅𝑦 = [𝑄]𝑆)) → (𝑥 = [𝑝]𝑅 ↔ [𝑃]𝑅 = [𝑝]𝑅))
17 simprr 811 . . . . . . . 8 (((𝜑𝑃𝐴𝑄𝐵) ∧ (𝑥 = [𝑃]𝑅𝑦 = [𝑄]𝑆)) → 𝑦 = [𝑄]𝑆)
1817eqeq1d 2653 . . . . . . 7 (((𝜑𝑃𝐴𝑄𝐵) ∧ (𝑥 = [𝑃]𝑅𝑦 = [𝑄]𝑆)) → (𝑦 = [𝑞]𝑆 ↔ [𝑄]𝑆 = [𝑞]𝑆))
1916, 18anbi12d 747 . . . . . 6 (((𝜑𝑃𝐴𝑄𝐵) ∧ (𝑥 = [𝑃]𝑅𝑦 = [𝑄]𝑆)) → ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ↔ ([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆)))
2019anbi1d 741 . . . . 5 (((𝜑𝑃𝐴𝑄𝐵) ∧ (𝑥 = [𝑃]𝑅𝑦 = [𝑄]𝑆)) → (((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
21202rexbidv 3086 . . . 4 (((𝜑𝑃𝐴𝑄𝐵) ∧ (𝑥 = [𝑃]𝑅𝑦 = [𝑄]𝑆)) → (∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ∃𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
2221iotabidv 5910 . . 3 (((𝜑𝑃𝐴𝑄𝐵) ∧ (𝑥 = [𝑃]𝑅𝑦 = [𝑄]𝑆)) → (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) = (℩𝑧𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
23 eropr.13 . . . . 5 (𝜑𝑅𝑋)
24 ecelqsg 7845 . . . . . 6 ((𝑅𝑋𝑃𝐴) → [𝑃]𝑅 ∈ (𝐴 / 𝑅))
2524, 1syl6eleqr 2741 . . . . 5 ((𝑅𝑋𝑃𝐴) → [𝑃]𝑅𝐽)
2623, 25sylan 487 . . . 4 ((𝜑𝑃𝐴) → [𝑃]𝑅𝐽)
27263adant3 1101 . . 3 ((𝜑𝑃𝐴𝑄𝐵) → [𝑃]𝑅𝐽)
28 eropr.14 . . . . 5 (𝜑𝑆𝑌)
29 ecelqsg 7845 . . . . . 6 ((𝑆𝑌𝑄𝐵) → [𝑄]𝑆 ∈ (𝐵 / 𝑆))
3029, 2syl6eleqr 2741 . . . . 5 ((𝑆𝑌𝑄𝐵) → [𝑄]𝑆𝐾)
3128, 30sylan 487 . . . 4 ((𝜑𝑄𝐵) → [𝑄]𝑆𝐾)
32313adant2 1100 . . 3 ((𝜑𝑃𝐴𝑄𝐵) → [𝑄]𝑆𝐾)
33 iotaex 5906 . . . 4 (℩𝑧𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ V
3433a1i 11 . . 3 ((𝜑𝑃𝐴𝑄𝐵) → (℩𝑧𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ V)
3514, 22, 27, 32, 34ovmpt2d 6830 . 2 ((𝜑𝑃𝐴𝑄𝐵) → ([𝑃]𝑅 [𝑄]𝑆) = (℩𝑧𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
36 eqid 2651 . . . . . . 7 [𝑃]𝑅 = [𝑃]𝑅
37 eqid 2651 . . . . . . 7 [𝑄]𝑆 = [𝑄]𝑆
3836, 37pm3.2i 470 . . . . . 6 ([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆)
39 eqid 2651 . . . . . 6 [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇
4038, 39pm3.2i 470 . . . . 5 (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇)
41 eceq1 7825 . . . . . . . . 9 (𝑝 = 𝑃 → [𝑝]𝑅 = [𝑃]𝑅)
4241eqeq2d 2661 . . . . . . . 8 (𝑝 = 𝑃 → ([𝑃]𝑅 = [𝑝]𝑅 ↔ [𝑃]𝑅 = [𝑃]𝑅))
4342anbi1d 741 . . . . . . 7 (𝑝 = 𝑃 → (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ↔ ([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆)))
44 oveq1 6697 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝 + 𝑞) = (𝑃 + 𝑞))
4544eceq1d 7826 . . . . . . . 8 (𝑝 = 𝑃 → [(𝑝 + 𝑞)]𝑇 = [(𝑃 + 𝑞)]𝑇)
4645eqeq2d 2661 . . . . . . 7 (𝑝 = 𝑃 → ([(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇 ↔ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑞)]𝑇))
4743, 46anbi12d 747 . . . . . 6 (𝑝 = 𝑃 → ((([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇) ↔ (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑞)]𝑇)))
48 eceq1 7825 . . . . . . . . 9 (𝑞 = 𝑄 → [𝑞]𝑆 = [𝑄]𝑆)
4948eqeq2d 2661 . . . . . . . 8 (𝑞 = 𝑄 → ([𝑄]𝑆 = [𝑞]𝑆 ↔ [𝑄]𝑆 = [𝑄]𝑆))
5049anbi2d 740 . . . . . . 7 (𝑞 = 𝑄 → (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ↔ ([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆)))
51 oveq2 6698 . . . . . . . . 9 (𝑞 = 𝑄 → (𝑃 + 𝑞) = (𝑃 + 𝑄))
5251eceq1d 7826 . . . . . . . 8 (𝑞 = 𝑄 → [(𝑃 + 𝑞)]𝑇 = [(𝑃 + 𝑄)]𝑇)
5352eqeq2d 2661 . . . . . . 7 (𝑞 = 𝑄 → ([(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑞)]𝑇 ↔ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇))
5450, 53anbi12d 747 . . . . . 6 (𝑞 = 𝑄 → ((([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑞)]𝑇) ↔ (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇)))
5547, 54rspc2ev 3355 . . . . 5 ((𝑃𝐴𝑄𝐵 ∧ (([𝑃]𝑅 = [𝑃]𝑅 ∧ [𝑄]𝑆 = [𝑄]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑃 + 𝑄)]𝑇)) → ∃𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇))
5640, 55mp3an3 1453 . . . 4 ((𝑃𝐴𝑄𝐵) → ∃𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇))
57563adant1 1099 . . 3 ((𝜑𝑃𝐴𝑄𝐵) → ∃𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇))
58 ecexg 7791 . . . . . 6 (𝑇𝑍 → [(𝑃 + 𝑄)]𝑇 ∈ V)
593, 58syl 17 . . . . 5 (𝜑 → [(𝑃 + 𝑄)]𝑇 ∈ V)
60593ad2ant1 1102 . . . 4 ((𝜑𝑃𝐴𝑄𝐵) → [(𝑃 + 𝑄)]𝑇 ∈ V)
61 simp1 1081 . . . . 5 ((𝜑𝑃𝐴𝑄𝐵) → 𝜑)
621, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11eroveu 7885 . . . . 5 ((𝜑 ∧ ([𝑃]𝑅𝐽 ∧ [𝑄]𝑆𝐾)) → ∃!𝑧𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
6361, 27, 32, 62syl12anc 1364 . . . 4 ((𝜑𝑃𝐴𝑄𝐵) → ∃!𝑧𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
64 simpr 476 . . . . . . 7 (((𝜑𝑃𝐴𝑄𝐵) ∧ 𝑧 = [(𝑃 + 𝑄)]𝑇) → 𝑧 = [(𝑃 + 𝑄)]𝑇)
6564eqeq1d 2653 . . . . . 6 (((𝜑𝑃𝐴𝑄𝐵) ∧ 𝑧 = [(𝑃 + 𝑄)]𝑇) → (𝑧 = [(𝑝 + 𝑞)]𝑇 ↔ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇))
6665anbi2d 740 . . . . 5 (((𝜑𝑃𝐴𝑄𝐵) ∧ 𝑧 = [(𝑃 + 𝑄)]𝑇) → ((([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇)))
67662rexbidv 3086 . . . 4 (((𝜑𝑃𝐴𝑄𝐵) ∧ 𝑧 = [(𝑃 + 𝑄)]𝑇) → (∃𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) ↔ ∃𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇)))
6860, 63, 67iota2d 5914 . . 3 ((𝜑𝑃𝐴𝑄𝐵) → (∃𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ [(𝑃 + 𝑄)]𝑇 = [(𝑝 + 𝑞)]𝑇) ↔ (℩𝑧𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) = [(𝑃 + 𝑄)]𝑇))
6957, 68mpbid 222 . 2 ((𝜑𝑃𝐴𝑄𝐵) → (℩𝑧𝑝𝐴𝑞𝐵 (([𝑃]𝑅 = [𝑝]𝑅 ∧ [𝑄]𝑆 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) = [(𝑃 + 𝑄)]𝑇)
7035, 69eqtrd 2685 1 ((𝜑𝑃𝐴𝑄𝐵) → ([𝑃]𝑅 [𝑄]𝑆) = [(𝑃 + 𝑄)]𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  ∃!weu 2498  wrex 2942  Vcvv 3231  wss 3607   class class class wbr 4685   × cxp 5141  cio 5887  wf 5922  (class class class)co 6690  {coprab 6691  cmpt2 6692   Er wer 7784  [cec 7785   / cqs 7786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-er 7787  df-ec 7789  df-qs 7793
This theorem is referenced by:  erov2  7889
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