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Theorem supsrlem 9970
Description: Lemma for supremum theorem. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
supsrlem.1 𝐵 = {𝑤 ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}
supsrlem.2 𝐶R
Assertion
Ref Expression
supsrlem ((𝐶𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝐴   𝑥,𝐵,𝑦,𝑧,𝑤   𝑥,𝐶,𝑦,𝑧,𝑤

Proof of Theorem supsrlem
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supsrlem.2 . . . . . . 7 𝐶R
2 0idsr 9956 . . . . . . 7 (𝐶R → (𝐶 +R 0R) = 𝐶)
31, 2mp1i 13 . . . . . 6 ((𝐶𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → (𝐶 +R 0R) = 𝐶)
4 simpl 472 . . . . . 6 ((𝐶𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → 𝐶𝐴)
53, 4eqeltrd 2730 . . . . 5 ((𝐶𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → (𝐶 +R 0R) ∈ 𝐴)
6 1pr 9875 . . . . . . 7 1PP
76elexi 3244 . . . . . 6 1P ∈ V
8 opeq1 4433 . . . . . . . . . 10 (𝑤 = 1P → ⟨𝑤, 1P⟩ = ⟨1P, 1P⟩)
98eceq1d 7826 . . . . . . . . 9 (𝑤 = 1P → [⟨𝑤, 1P⟩] ~R = [⟨1P, 1P⟩] ~R )
10 df-0r 9920 . . . . . . . . 9 0R = [⟨1P, 1P⟩] ~R
119, 10syl6eqr 2703 . . . . . . . 8 (𝑤 = 1P → [⟨𝑤, 1P⟩] ~R = 0R)
1211oveq2d 6706 . . . . . . 7 (𝑤 = 1P → (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = (𝐶 +R 0R))
1312eleq1d 2715 . . . . . 6 (𝑤 = 1P → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴 ↔ (𝐶 +R 0R) ∈ 𝐴))
14 supsrlem.1 . . . . . 6 𝐵 = {𝑤 ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}
157, 13, 14elab2 3386 . . . . 5 (1P𝐵 ↔ (𝐶 +R 0R) ∈ 𝐴)
165, 15sylibr 224 . . . 4 ((𝐶𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → 1P𝐵)
17 ne0i 3954 . . . 4 (1P𝐵𝐵 ≠ ∅)
1816, 17syl 17 . . 3 ((𝐶𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → 𝐵 ≠ ∅)
19 breq1 4688 . . . . . . . 8 (𝑦 = 𝐶 → (𝑦 <R 𝑥𝐶 <R 𝑥))
2019rspccv 3337 . . . . . . 7 (∀𝑦𝐴 𝑦 <R 𝑥 → (𝐶𝐴𝐶 <R 𝑥))
21 0lt1sr 9954 . . . . . . . . . . . . 13 0R <R 1R
22 m1r 9941 . . . . . . . . . . . . . 14 -1RR
23 ltasr 9959 . . . . . . . . . . . . . 14 (-1RR → (0R <R 1R ↔ (-1R +R 0R) <R (-1R +R 1R)))
2422, 23ax-mp 5 . . . . . . . . . . . . 13 (0R <R 1R ↔ (-1R +R 0R) <R (-1R +R 1R))
2521, 24mpbi 220 . . . . . . . . . . . 12 (-1R +R 0R) <R (-1R +R 1R)
26 0idsr 9956 . . . . . . . . . . . . 13 (-1RR → (-1R +R 0R) = -1R)
2722, 26ax-mp 5 . . . . . . . . . . . 12 (-1R +R 0R) = -1R
28 m1p1sr 9951 . . . . . . . . . . . 12 (-1R +R 1R) = 0R
2925, 27, 283brtr3i 4714 . . . . . . . . . . 11 -1R <R 0R
30 ltasr 9959 . . . . . . . . . . . 12 (𝐶R → (-1R <R 0R ↔ (𝐶 +R -1R) <R (𝐶 +R 0R)))
311, 30ax-mp 5 . . . . . . . . . . 11 (-1R <R 0R ↔ (𝐶 +R -1R) <R (𝐶 +R 0R))
3229, 31mpbi 220 . . . . . . . . . 10 (𝐶 +R -1R) <R (𝐶 +R 0R)
331, 2ax-mp 5 . . . . . . . . . 10 (𝐶 +R 0R) = 𝐶
3432, 33breqtri 4710 . . . . . . . . 9 (𝐶 +R -1R) <R 𝐶
35 ltsosr 9953 . . . . . . . . . 10 <R Or R
36 ltrelsr 9927 . . . . . . . . . 10 <R ⊆ (R × R)
3735, 36sotri 5558 . . . . . . . . 9 (((𝐶 +R -1R) <R 𝐶𝐶 <R 𝑥) → (𝐶 +R -1R) <R 𝑥)
3834, 37mpan 706 . . . . . . . 8 (𝐶 <R 𝑥 → (𝐶 +R -1R) <R 𝑥)
391map2psrpr 9969 . . . . . . . 8 ((𝐶 +R -1R) <R 𝑥 ↔ ∃𝑣P (𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥)
4038, 39sylib 208 . . . . . . 7 (𝐶 <R 𝑥 → ∃𝑣P (𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥)
4120, 40syl6 35 . . . . . 6 (∀𝑦𝐴 𝑦 <R 𝑥 → (𝐶𝐴 → ∃𝑣P (𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥))
42 breq2 4689 . . . . . . . . . 10 ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥 → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ 𝑦 <R 𝑥))
4342ralbidv 3015 . . . . . . . . 9 ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥 → (∀𝑦𝐴 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ ∀𝑦𝐴 𝑦 <R 𝑥))
4414abeq2i 2764 . . . . . . . . . . 11 (𝑤𝐵 ↔ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴)
45 breq1 4688 . . . . . . . . . . . . 13 (𝑦 = (𝐶 +R [⟨𝑤, 1P⟩] ~R ) → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R )))
4645rspccv 3337 . . . . . . . . . . . 12 (∀𝑦𝐴 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴 → (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R )))
471ltpsrpr 9968 . . . . . . . . . . . 12 ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ 𝑤<P 𝑣)
4846, 47syl6ib 241 . . . . . . . . . . 11 (∀𝑦𝐴 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴𝑤<P 𝑣))
4944, 48syl5bi 232 . . . . . . . . . 10 (∀𝑦𝐴 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (𝑤𝐵𝑤<P 𝑣))
5049ralrimiv 2994 . . . . . . . . 9 (∀𝑦𝐴 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∀𝑤𝐵 𝑤<P 𝑣)
5143, 50syl6bir 244 . . . . . . . 8 ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥 → (∀𝑦𝐴 𝑦 <R 𝑥 → ∀𝑤𝐵 𝑤<P 𝑣))
5251com12 32 . . . . . . 7 (∀𝑦𝐴 𝑦 <R 𝑥 → ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥 → ∀𝑤𝐵 𝑤<P 𝑣))
5352reximdv 3045 . . . . . 6 (∀𝑦𝐴 𝑦 <R 𝑥 → (∃𝑣P (𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥 → ∃𝑣P𝑤𝐵 𝑤<P 𝑣))
5441, 53syld 47 . . . . 5 (∀𝑦𝐴 𝑦 <R 𝑥 → (𝐶𝐴 → ∃𝑣P𝑤𝐵 𝑤<P 𝑣))
5554rexlimivw 3058 . . . 4 (∃𝑥R𝑦𝐴 𝑦 <R 𝑥 → (𝐶𝐴 → ∃𝑣P𝑤𝐵 𝑤<P 𝑣))
5655impcom 445 . . 3 ((𝐶𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → ∃𝑣P𝑤𝐵 𝑤<P 𝑣)
57 supexpr 9914 . . 3 ((𝐵 ≠ ∅ ∧ ∃𝑣P𝑤𝐵 𝑤<P 𝑣) → ∃𝑣P (∀𝑤𝐵 ¬ 𝑣<P 𝑤 ∧ ∀𝑤P (𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢)))
5818, 56, 57syl2anc 694 . 2 ((𝐶𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → ∃𝑣P (∀𝑤𝐵 ¬ 𝑣<P 𝑤 ∧ ∀𝑤P (𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢)))
591mappsrpr 9967 . . . . . . 7 ((𝐶 +R -1R) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ 𝑣P)
6036brel 5202 . . . . . . 7 ((𝐶 +R -1R) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ((𝐶 +R -1R) ∈ R ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∈ R))
6159, 60sylbir 225 . . . . . 6 (𝑣P → ((𝐶 +R -1R) ∈ R ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∈ R))
6261simprd 478 . . . . 5 (𝑣P → (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∈ R)
6362adantl 481 . . . 4 (((𝐶𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) ∧ 𝑣P) → (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∈ R)
6435, 36sotri 5558 . . . . . . . . . . . . . . 15 (((𝐶 +R -1R) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦) → (𝐶 +R -1R) <R 𝑦)
6559, 64sylanbr 489 . . . . . . . . . . . . . 14 ((𝑣P ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦) → (𝐶 +R -1R) <R 𝑦)
661map2psrpr 9969 . . . . . . . . . . . . . 14 ((𝐶 +R -1R) <R 𝑦 ↔ ∃𝑤P (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦)
6765, 66sylib 208 . . . . . . . . . . . . 13 ((𝑣P ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦) → ∃𝑤P (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦)
68 rexex 3031 . . . . . . . . . . . . 13 (∃𝑤P (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ∃𝑤(𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦)
69 df-ral 2946 . . . . . . . . . . . . . . 15 (∀𝑤𝐵 ¬ 𝑣<P 𝑤 ↔ ∀𝑤(𝑤𝐵 → ¬ 𝑣<P 𝑤))
70 19.29 1841 . . . . . . . . . . . . . . . 16 ((∀𝑤(𝑤𝐵 → ¬ 𝑣<P 𝑤) ∧ ∃𝑤(𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦) → ∃𝑤((𝑤𝐵 → ¬ 𝑣<P 𝑤) ∧ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦))
71 eleq1 2718 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴𝑦𝐴))
7244, 71syl5bb 272 . . . . . . . . . . . . . . . . . . 19 ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (𝑤𝐵𝑦𝐴))
731ltpsrpr 9968 . . . . . . . . . . . . . . . . . . . . 21 ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ↔ 𝑣<P 𝑤)
74 breq2 4689 . . . . . . . . . . . . . . . . . . . . 21 ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ↔ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
7573, 74syl5bbr 274 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (𝑣<P 𝑤 ↔ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
7675notbid 307 . . . . . . . . . . . . . . . . . . 19 ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (¬ 𝑣<P 𝑤 ↔ ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
7772, 76imbi12d 333 . . . . . . . . . . . . . . . . . 18 ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ((𝑤𝐵 → ¬ 𝑣<P 𝑤) ↔ (𝑦𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦)))
7877biimpac 502 . . . . . . . . . . . . . . . . 17 (((𝑤𝐵 → ¬ 𝑣<P 𝑤) ∧ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦) → (𝑦𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
7978exlimiv 1898 . . . . . . . . . . . . . . . 16 (∃𝑤((𝑤𝐵 → ¬ 𝑣<P 𝑤) ∧ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦) → (𝑦𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
8070, 79syl 17 . . . . . . . . . . . . . . 15 ((∀𝑤(𝑤𝐵 → ¬ 𝑣<P 𝑤) ∧ ∃𝑤(𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦) → (𝑦𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
8169, 80sylanb 488 . . . . . . . . . . . . . 14 ((∀𝑤𝐵 ¬ 𝑣<P 𝑤 ∧ ∃𝑤(𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦) → (𝑦𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
8281expcom 450 . . . . . . . . . . . . 13 (∃𝑤(𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (∀𝑤𝐵 ¬ 𝑣<P 𝑤 → (𝑦𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦)))
8367, 68, 823syl 18 . . . . . . . . . . . 12 ((𝑣P ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦) → (∀𝑤𝐵 ¬ 𝑣<P 𝑤 → (𝑦𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦)))
8483impd 446 . . . . . . . . . . 11 ((𝑣P ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦) → ((∀𝑤𝐵 ¬ 𝑣<P 𝑤𝑦𝐴) → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
8584impancom 455 . . . . . . . . . 10 ((𝑣P ∧ (∀𝑤𝐵 ¬ 𝑣<P 𝑤𝑦𝐴)) → ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
8685pm2.01d 181 . . . . . . . . 9 ((𝑣P ∧ (∀𝑤𝐵 ¬ 𝑣<P 𝑤𝑦𝐴)) → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦)
8786expr 642 . . . . . . . 8 ((𝑣P ∧ ∀𝑤𝐵 ¬ 𝑣<P 𝑤) → (𝑦𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
8887ralrimiv 2994 . . . . . . 7 ((𝑣P ∧ ∀𝑤𝐵 ¬ 𝑣<P 𝑤) → ∀𝑦𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦)
8988ex 449 . . . . . 6 (𝑣P → (∀𝑤𝐵 ¬ 𝑣<P 𝑤 → ∀𝑦𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
9089adantl 481 . . . . 5 (((𝐶𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) ∧ 𝑣P) → (∀𝑤𝐵 ¬ 𝑣<P 𝑤 → ∀𝑦𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
91 r19.29 3101 . . . . . . . . . . . . . 14 ((∀𝑤P (𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢) ∧ ∃𝑤P (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦) → ∃𝑤P ((𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢) ∧ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦))
92 breq1 4688 . . . . . . . . . . . . . . . . . . 19 ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R )))
9347, 92syl5bbr 274 . . . . . . . . . . . . . . . . . 18 ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (𝑤<P 𝑣𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R )))
9493biimprd 238 . . . . . . . . . . . . . . . . 17 ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → 𝑤<P 𝑣))
95 vex 3234 . . . . . . . . . . . . . . . . . . . . 21 𝑢 ∈ V
96 opeq1 4433 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = 𝑢 → ⟨𝑤, 1P⟩ = ⟨𝑢, 1P⟩)
9796eceq1d 7826 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 = 𝑢 → [⟨𝑤, 1P⟩] ~R = [⟨𝑢, 1P⟩] ~R )
9897oveq2d 6706 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = 𝑢 → (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = (𝐶 +R [⟨𝑢, 1P⟩] ~R ))
9998eleq1d 2715 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = 𝑢 → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴 ↔ (𝐶 +R [⟨𝑢, 1P⟩] ~R ) ∈ 𝐴))
10095, 99, 14elab2 3386 . . . . . . . . . . . . . . . . . . . 20 (𝑢𝐵 ↔ (𝐶 +R [⟨𝑢, 1P⟩] ~R ) ∈ 𝐴)
101 breq2 4689 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = (𝐶 +R [⟨𝑢, 1P⟩] ~R ) → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧 ↔ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑢, 1P⟩] ~R )))
1021ltpsrpr 9968 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑢, 1P⟩] ~R ) ↔ 𝑤<P 𝑢)
103101, 102syl6bb 276 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = (𝐶 +R [⟨𝑢, 1P⟩] ~R ) → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧𝑤<P 𝑢))
104103rspcev 3340 . . . . . . . . . . . . . . . . . . . 20 (((𝐶 +R [⟨𝑢, 1P⟩] ~R ) ∈ 𝐴𝑤<P 𝑢) → ∃𝑧𝐴 (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧)
105100, 104sylanb 488 . . . . . . . . . . . . . . . . . . 19 ((𝑢𝐵𝑤<P 𝑢) → ∃𝑧𝐴 (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧)
106105rexlimiva 3057 . . . . . . . . . . . . . . . . . 18 (∃𝑢𝐵 𝑤<P 𝑢 → ∃𝑧𝐴 (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧)
107 breq1 4688 . . . . . . . . . . . . . . . . . . 19 ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧𝑦 <R 𝑧))
108107rexbidv 3081 . . . . . . . . . . . . . . . . . 18 ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (∃𝑧𝐴 (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧 ↔ ∃𝑧𝐴 𝑦 <R 𝑧))
109106, 108syl5ib 234 . . . . . . . . . . . . . . . . 17 ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (∃𝑢𝐵 𝑤<P 𝑢 → ∃𝑧𝐴 𝑦 <R 𝑧))
11094, 109imim12d 81 . . . . . . . . . . . . . . . 16 ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ((𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢) → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧)))
111110impcom 445 . . . . . . . . . . . . . . 15 (((𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢) ∧ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦) → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧))
112111rexlimivw 3058 . . . . . . . . . . . . . 14 (∃𝑤P ((𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢) ∧ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦) → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧))
11391, 112syl 17 . . . . . . . . . . . . 13 ((∀𝑤P (𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢) ∧ ∃𝑤P (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦) → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧))
11466, 113sylan2b 491 . . . . . . . . . . . 12 ((∀𝑤P (𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢) ∧ (𝐶 +R -1R) <R 𝑦) → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧))
115114ex 449 . . . . . . . . . . 11 (∀𝑤P (𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢) → ((𝐶 +R -1R) <R 𝑦 → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧)))
116115adantl 481 . . . . . . . . . 10 (((𝐶𝐴𝑣P) ∧ ∀𝑤P (𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢)) → ((𝐶 +R -1R) <R 𝑦 → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧)))
117116a1dd 50 . . . . . . . . 9 (((𝐶𝐴𝑣P) ∧ ∀𝑤P (𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢)) → ((𝐶 +R -1R) <R 𝑦 → (𝑦R → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧))))
11835, 36sotri2 5560 . . . . . . . . . . . . 13 ((𝑦R ∧ ¬ (𝐶 +R -1R) <R 𝑦 ∧ (𝐶 +R -1R) <R 𝐶) → 𝑦 <R 𝐶)
11934, 118mp3an3 1453 . . . . . . . . . . . 12 ((𝑦R ∧ ¬ (𝐶 +R -1R) <R 𝑦) → 𝑦 <R 𝐶)
120 breq2 4689 . . . . . . . . . . . . . . 15 (𝑧 = 𝐶 → (𝑦 <R 𝑧𝑦 <R 𝐶))
121120rspcev 3340 . . . . . . . . . . . . . 14 ((𝐶𝐴𝑦 <R 𝐶) → ∃𝑧𝐴 𝑦 <R 𝑧)
122121ex 449 . . . . . . . . . . . . 13 (𝐶𝐴 → (𝑦 <R 𝐶 → ∃𝑧𝐴 𝑦 <R 𝑧))
123122a1dd 50 . . . . . . . . . . . 12 (𝐶𝐴 → (𝑦 <R 𝐶 → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧)))
124119, 123syl5 34 . . . . . . . . . . 11 (𝐶𝐴 → ((𝑦R ∧ ¬ (𝐶 +R -1R) <R 𝑦) → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧)))
125124expcomd 453 . . . . . . . . . 10 (𝐶𝐴 → (¬ (𝐶 +R -1R) <R 𝑦 → (𝑦R → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧))))
126125ad2antrr 762 . . . . . . . . 9 (((𝐶𝐴𝑣P) ∧ ∀𝑤P (𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢)) → (¬ (𝐶 +R -1R) <R 𝑦 → (𝑦R → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧))))
127117, 126pm2.61d 170 . . . . . . . 8 (((𝐶𝐴𝑣P) ∧ ∀𝑤P (𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢)) → (𝑦R → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧)))
128127ralrimiv 2994 . . . . . . 7 (((𝐶𝐴𝑣P) ∧ ∀𝑤P (𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢)) → ∀𝑦R (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧))
129128ex 449 . . . . . 6 ((𝐶𝐴𝑣P) → (∀𝑤P (𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢) → ∀𝑦R (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧)))
130129adantlr 751 . . . . 5 (((𝐶𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) ∧ 𝑣P) → (∀𝑤P (𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢) → ∀𝑦R (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧)))
13190, 130anim12d 585 . . . 4 (((𝐶𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) ∧ 𝑣P) → ((∀𝑤𝐵 ¬ 𝑣<P 𝑤 ∧ ∀𝑤P (𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢)) → (∀𝑦𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦 ∧ ∀𝑦R (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧))))
132 breq1 4688 . . . . . . . 8 (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (𝑥 <R 𝑦 ↔ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
133132notbid 307 . . . . . . 7 (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (¬ 𝑥 <R 𝑦 ↔ ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
134133ralbidv 3015 . . . . . 6 (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ↔ ∀𝑦𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
135 breq2 4689 . . . . . . . 8 (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (𝑦 <R 𝑥𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R )))
136135imbi1d 330 . . . . . . 7 (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ((𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧) ↔ (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧)))
137136ralbidv 3015 . . . . . 6 (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧) ↔ ∀𝑦R (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧)))
138134, 137anbi12d 747 . . . . 5 (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ((∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)) ↔ (∀𝑦𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦 ∧ ∀𝑦R (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧))))
139138rspcev 3340 . . . 4 (((𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∈ R ∧ (∀𝑦𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦 ∧ ∀𝑦R (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧𝐴 𝑦 <R 𝑧))) → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)))
14063, 131, 139syl6an 567 . . 3 (((𝐶𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) ∧ 𝑣P) → ((∀𝑤𝐵 ¬ 𝑣<P 𝑤 ∧ ∀𝑤P (𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢)) → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧))))
141140rexlimdva 3060 . 2 ((𝐶𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → (∃𝑣P (∀𝑤𝐵 ¬ 𝑣<P 𝑤 ∧ ∀𝑤P (𝑤<P 𝑣 → ∃𝑢𝐵 𝑤<P 𝑢)) → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧))))
14258, 141mpd 15 1 ((𝐶𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  c0 3948  cop 4216   class class class wbr 4685  (class class class)co 6690  [cec 7785  Pcnp 9719  1Pc1p 9720  <P cltp 9723   ~R cer 9724  Rcnr 9725  0Rc0r 9726  1Rc1r 9727  -1Rcm1r 9728   +R cplr 9729   <R cltr 9731
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-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  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-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  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-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ec 7789  df-qs 7793  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-plpq 9768  df-mpq 9769  df-ltpq 9770  df-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776  df-rq 9777  df-ltnq 9778  df-np 9841  df-1p 9842  df-plp 9843  df-mp 9844  df-ltp 9845  df-enr 9915  df-nr 9916  df-plr 9917  df-mr 9918  df-ltr 9919  df-0r 9920  df-1r 9921  df-m1r 9922
This theorem is referenced by:  supsr  9971
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