erdszelem11 - Mathbox for Mario Carneiro

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Theorem erdszelem11 32664

Description: Lemma for erdsze 32665. (Contributed by Mario Carneiro, 22-Jan-2015.)

**Hypotheses**
Ref	Expression
erdsze.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
erdsze.f	⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)
erdszelem.i	⊢ 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
erdszelem.j	⊢ 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ^◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
erdszelem.t	⊢ 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩)
erdszelem.r	⊢ (𝜑 → 𝑅 ∈ ℕ)
erdszelem.s	⊢ (𝜑 → 𝑆 ∈ ℕ)
erdszelem.m	⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁)

**Assertion**
Ref	Expression
erdszelem11	⊢ (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))))

Distinct variable groups: 𝑥,𝑦 𝑛,𝑠,𝑥,𝑦,𝐹 𝑛,𝐼,𝑠,𝑥,𝑦 𝑛,𝐽,𝑠,𝑥,𝑦 𝑅,𝑠,𝑥,𝑦 𝑛,𝑁,𝑠,𝑥,𝑦 𝜑,𝑛,𝑠,𝑥,𝑦 𝑆,𝑠,𝑥,𝑦 𝑇,𝑠 Allowed substitution hints: 𝑅(𝑛) 𝑆(𝑛) 𝑇(𝑥,𝑦,𝑛)

Proof of Theorem erdszelem11 Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		erdsze.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		erdsze.f	. . . 4 ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)
3		erdszelem.i	. . . 4 ⊢ 𝐼 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
4		erdszelem.j	. . . 4 ⊢ 𝐽 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , ^◡ < (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
5		erdszelem.t	. . . 4 ⊢ 𝑇 = (𝑛 ∈ (1...𝑁) ↦ ⟨(𝐼‘𝑛), (𝐽‘𝑛)⟩)
6		erdszelem.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℕ)
7		erdszelem.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℕ)
8		erdszelem.m	. . . 4 ⊢ (𝜑 → ((𝑅 − 1) · (𝑆 − 1)) < 𝑁)
9	1, 2, 3, 4, 5, 6, 7, 8	erdszelem10 32663	. . 3 ⊢ (𝜑 → ∃𝑚 ∈ (1...𝑁)(¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1))))
10	1	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)))) → 𝑁 ∈ ℕ)
11	2	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)))) → 𝐹:(1...𝑁)–1-1→ℝ)
12		ltso 10744	. . . . . . 7 ⊢ < Or ℝ
13		simprl 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)))) → 𝑚 ∈ (1...𝑁))
14	6	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)))) → 𝑅 ∈ ℕ)
15		simprr 773	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)))) → ¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)))
16	10, 11, 3, 12, 13, 14, 15	erdszelem7 32660	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)))) → ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))))
17	16	expr 461	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → (¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)) → ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠)))))
18	1	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))) → 𝑁 ∈ ℕ)
19	2	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))) → 𝐹:(1...𝑁)–1-1→ℝ)
20		gtso 10745	. . . . . . 7 ⊢ ^◡ < Or ℝ
21		simprl 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))) → 𝑚 ∈ (1...𝑁))
22	7	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))) → 𝑆 ∈ ℕ)
23		simprr 773	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))) → ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))
24	18, 19, 4, 20, 21, 22, 23	erdszelem7 32660	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ (1...𝑁) ∧ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)))) → ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠))))
25	24	expr 461	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → (¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1)) → ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))))
26	17, 25	orim12d 963	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → ((¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1))) → (∃𝑠 ∈ 𝒫 (1...𝑁)(𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠))))))
27	26	rexlimdva 3206	. . 3 ⊢ (𝜑 → (∃𝑚 ∈ (1...𝑁)(¬ (𝐼‘𝑚) ∈ (1...(𝑅 − 1)) ∨ ¬ (𝐽‘𝑚) ∈ (1...(𝑆 − 1))) → (∃𝑠 ∈ 𝒫 (1...𝑁)(𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠))))))
28	9, 27	mpd 15	. 2 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 (1...𝑁)(𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))))
29		r19.43 3267	. 2 ⊢ (∃𝑠 ∈ 𝒫 (1...𝑁)((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))) ↔ (∃𝑠 ∈ 𝒫 (1...𝑁)(𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ ∃𝑠 ∈ 𝒫 (1...𝑁)(𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))))
30	28, 29	sylibr 237	1 ⊢ (𝜑 → ∃𝑠 ∈ 𝒫 (1...𝑁)((𝑅 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , < (𝑠, (𝐹 “ 𝑠))) ∨ (𝑆 ≤ (♯‘𝑠) ∧ (𝐹 ↾ 𝑠) Isom < , ^◡ < (𝑠, (𝐹 “ 𝑠)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 ∨ wo 845 = wceq 1539 ∈ wcel 2112 ∃wrex 3069 {crab 3072 𝒫 cpw 4487 ⟨cop 4521 class class class wbr 5025 ↦ cmpt 5105 ^◡ccnv 5516 ↾ cres 5519 “ cima 5520 –1-1→wf1 6325 ‘cfv 6328 Isom wiso 6329 (class class class)co 7143 supcsup 8922 ℝcr 10559 1c1 10561 · cmul 10565 < clt 10698 ≤ cle 10699 − cmin 10893 ℕcn 11659 ...cfz 12924 ♯chash 13725

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 ax-pre-sup 10638

This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-isom 6337 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-sup 8924 df-dju 9348 df-card 9386 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-n0 11920 df-xnn0 11992 df-z 12006 df-uz 12268 df-fz 12925 df-hash 13726

This theorem is referenced by: erdsze 32665

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