erdszelem8 - Mathbox for Mario Carneiro

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Theorem erdszelem8 30434

Description: Lemma for erdsze 30438. (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.k	⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
erdszelem.o	⊢ 𝑂 Or ℝ
erdszelem.a	⊢ (𝜑 → 𝐴 ∈ (1...𝑁))
erdszelem.b	⊢ (𝜑 → 𝐵 ∈ (1...𝑁))
erdszelem.l	⊢ (𝜑 → 𝐴 < 𝐵)

**Assertion**
Ref	Expression
erdszelem8	⊢ (𝜑 → ((𝐾‘𝐴) = (𝐾‘𝐵) → ¬ (𝐹‘𝐴)𝑂(𝐹‘𝐵)))

Distinct variable groups: 𝑥,𝑦,𝐵 𝑥,𝐹,𝑦 𝑥,𝐴,𝑦 𝑥,𝑂,𝑦 𝑥,𝑁,𝑦 𝜑,𝑥,𝑦 Allowed substitution hints: 𝐾(𝑥,𝑦)

Proof of Theorem erdszelem8 Dummy variables 𝑤 𝑓 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hashf 12987	. . . . 5 ⊢ #:V⟶(ℕ₀ ∪ {+∞})
2		ffun 5961	. . . . 5 ⊢ (#:V⟶(ℕ₀ ∪ {+∞}) → Fun #)
3	1, 2	ax-mp 5	. . . 4 ⊢ Fun #
4		erdszelem.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (1...𝑁))
5		erdsze.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6		erdsze.f	. . . . . 6 ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ)
7		erdszelem.k	. . . . . 6 ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((# “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))
8		erdszelem.o	. . . . . 6 ⊢ 𝑂 Or ℝ
9	5, 6, 7, 8	erdszelem5 30431	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (1...𝑁)) → (𝐾‘𝐴) ∈ (# “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}))
10	4, 9	mpdan 699	. . . 4 ⊢ (𝜑 → (𝐾‘𝐴) ∈ (# “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}))
11		fvelima 6158	. . . 4 ⊢ ((Fun # ∧ (𝐾‘𝐴) ∈ (# “ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)})) → ∃𝑓 ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} (#‘𝑓) = (𝐾‘𝐴))
12	3, 10, 11	sylancr 694	. . 3 ⊢ (𝜑 → ∃𝑓 ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} (#‘𝑓) = (𝐾‘𝐴))
13		eqid 2610	. . . . . 6 ⊢ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}
14	13	erdszelem1 30427	. . . . 5 ⊢ (𝑓 ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} ↔ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓))
15		fzfid 12634	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (1...𝐴) ∈ Fin)
16		simplr1 1096	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → 𝑓 ⊆ (1...𝐴))
17		ssfi 8065	. . . . . . . . . . 11 ⊢ (((1...𝐴) ∈ Fin ∧ 𝑓 ⊆ (1...𝐴)) → 𝑓 ∈ Fin)
18	15, 16, 17	syl2anc 691	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → 𝑓 ∈ Fin)
19		hashcl 13009	. . . . . . . . . 10 ⊢ (𝑓 ∈ Fin → (#‘𝑓) ∈ ℕ₀)
20	18, 19	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (#‘𝑓) ∈ ℕ₀)
21	20	nn0red 11229	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (#‘𝑓) ∈ ℝ)
22		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)} = {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}
23	22	erdszelem2 30428	. . . . . . . . . . . . . 14 ⊢ ((# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}) ∈ Fin ∧ (# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}) ⊆ ℕ)
24	23	simpri 477	. . . . . . . . . . . . 13 ⊢ (# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}) ⊆ ℕ
25		nnssre 10901	. . . . . . . . . . . . 13 ⊢ ℕ ⊆ ℝ
26	24, 25	sstri 3577	. . . . . . . . . . . 12 ⊢ (# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}) ⊆ ℝ
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}) ⊆ ℝ)
28		erdszelem.l	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 < 𝐵)
29		elfznn 12241	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ (1...𝑁) → 𝐴 ∈ ℕ)
30	4, 29	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐴 ∈ ℕ)
31	30	nnred 10912	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 ∈ ℝ)
32		erdszelem.b	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐵 ∈ (1...𝑁))
33		elfznn 12241	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 ∈ (1...𝑁) → 𝐵 ∈ ℕ)
34	32, 33	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐵 ∈ ℕ)
35	34	nnred 10912	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐵 ∈ ℝ)
36	31, 35	ltnled 10063	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
37	28, 36	mpbid 221	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ¬ 𝐵 ≤ 𝐴)
38		elfzle2 12216	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ (1...𝐴) → 𝐵 ≤ 𝐴)
39	37, 38	nsyl 134	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ 𝐵 ∈ (1...𝐴))
40	39	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → ¬ 𝐵 ∈ (1...𝐴))
41	16, 40	ssneldd 3571	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → ¬ 𝐵 ∈ 𝑓)
42	32	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → 𝐵 ∈ (1...𝑁))
43		hashunsng 13042	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (1...𝑁) → ((𝑓 ∈ Fin ∧ ¬ 𝐵 ∈ 𝑓) → (#‘(𝑓 ∪ {𝐵})) = ((#‘𝑓) + 1)))
44	42, 43	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → ((𝑓 ∈ Fin ∧ ¬ 𝐵 ∈ 𝑓) → (#‘(𝑓 ∪ {𝐵})) = ((#‘𝑓) + 1)))
45	18, 41, 44	mp2and 711	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (#‘(𝑓 ∪ {𝐵})) = ((#‘𝑓) + 1))
46		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ (1...𝑁) → 𝐴 ∈ ℤ)
47	4, 46	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐴 ∈ ℤ)
48		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐵 ∈ (1...𝑁) → 𝐵 ∈ ℤ)
49	32, 48	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐵 ∈ ℤ)
50	31, 35, 28	ltled 10064	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
51		eluz2 11569	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐵 ∈ (ℤ_≥‘𝐴) ↔ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵))
52	47, 49, 50, 51	syl3anbrc 1239	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐵 ∈ (ℤ_≥‘𝐴))
53		fzss2 12252	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ (ℤ_≥‘𝐴) → (1...𝐴) ⊆ (1...𝐵))
54	52, 53	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...𝐴) ⊆ (1...𝐵))
55	54	ad2antrr 758	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (1...𝐴) ⊆ (1...𝐵))
56	16, 55	sstrd 3578	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → 𝑓 ⊆ (1...𝐵))
57		elfz1end 12242	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ ℕ ↔ 𝐵 ∈ (1...𝐵))
58	34, 57	sylib 207	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ (1...𝐵))
59	58	ad2antrr 758	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → 𝐵 ∈ (1...𝐵))
60	59	snssd 4281	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → {𝐵} ⊆ (1...𝐵))
61	56, 60	unssd 3751	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (𝑓 ∪ {𝐵}) ⊆ (1...𝐵))
62		simplr2 1097	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)))
63		f1f 6014	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐹:(1...𝑁)–1-1→ℝ → 𝐹:(1...𝑁)⟶ℝ)
64	6, 63	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐹:(1...𝑁)⟶ℝ)
65	64	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → 𝐹:(1...𝑁)⟶ℝ)
66		elfzuz3 12210	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∈ (1...𝑁) → 𝑁 ∈ (ℤ_≥‘𝐴))
67		fzss2 12252	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ (ℤ_≥‘𝐴) → (1...𝐴) ⊆ (1...𝑁))
68	4, 66, 67	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (1...𝐴) ⊆ (1...𝑁))
69	68	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (1...𝐴) ⊆ (1...𝑁))
70	16, 69	sstrd 3578	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → 𝑓 ⊆ (1...𝑁))
71		fzssuz 12253	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1...𝑁) ⊆ (ℤ_≥‘1)
72		uzssz 11583	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (ℤ_≥‘1) ⊆ ℤ
73		zssre 11261	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ℤ ⊆ ℝ
74	72, 73	sstri 3577	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℤ_≥‘1) ⊆ ℝ
75	71, 74	sstri 3577	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1...𝑁) ⊆ ℝ
76		ltso 9997	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ < Or ℝ
77		soss 4977	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1...𝑁) ⊆ ℝ → ( < Or ℝ → < Or (1...𝑁)))
78	75, 76, 77	mp2 9	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ < Or (1...𝑁)
79		soisores 6477	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((( < Or (1...𝑁) ∧ 𝑂 Or ℝ) ∧ (𝐹:(1...𝑁)⟶ℝ ∧ 𝑓 ⊆ (1...𝑁))) → ((𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ↔ ∀𝑧 ∈ 𝑓 ∀𝑤 ∈ 𝑓 (𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤))))
80	78, 8, 79	mpanl12 714	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹:(1...𝑁)⟶ℝ ∧ 𝑓 ⊆ (1...𝑁)) → ((𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ↔ ∀𝑧 ∈ 𝑓 ∀𝑤 ∈ 𝑓 (𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤))))
81	65, 70, 80	syl2anc 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → ((𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ↔ ∀𝑧 ∈ 𝑓 ∀𝑤 ∈ 𝑓 (𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤))))
82	62, 81	mpbid 221	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → ∀𝑧 ∈ 𝑓 ∀𝑤 ∈ 𝑓 (𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤)))
83	82	r19.21bi 2916	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → ∀𝑤 ∈ 𝑓 (𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤)))
84	16	sselda 3568	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → 𝑧 ∈ (1...𝐴))
85		elfzle2 12216	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 ∈ (1...𝐴) → 𝑧 ≤ 𝐴)
86	84, 85	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → 𝑧 ≤ 𝐴)
87	70	sselda 3568	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → 𝑧 ∈ (1...𝑁))
88	75, 87	sseldi 3566	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → 𝑧 ∈ ℝ)
89	4	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → 𝐴 ∈ (1...𝑁))
90	89, 29	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → 𝐴 ∈ ℕ)
91	90	nnred 10912	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → 𝐴 ∈ ℝ)
92	88, 91	lenltd 10062	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → (𝑧 ≤ 𝐴 ↔ ¬ 𝐴 < 𝑧))
93	86, 92	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → ¬ 𝐴 < 𝑧)
94	62	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)))
95		simplr3 1098	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → 𝐴 ∈ 𝑓)
96	95	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → 𝐴 ∈ 𝑓)
97		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → 𝑧 ∈ 𝑓)
98		isorel 6476	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ (𝐴 ∈ 𝑓 ∧ 𝑧 ∈ 𝑓)) → (𝐴 < 𝑧 ↔ ((𝐹 ↾ 𝑓)‘𝐴)𝑂((𝐹 ↾ 𝑓)‘𝑧)))
99		fvres 6117	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐴 ∈ 𝑓 → ((𝐹 ↾ 𝑓)‘𝐴) = (𝐹‘𝐴))
100		fvres 6117	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ 𝑓 → ((𝐹 ↾ 𝑓)‘𝑧) = (𝐹‘𝑧))
101	99, 100	breqan12d 4599	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝐴 ∈ 𝑓 ∧ 𝑧 ∈ 𝑓) → (((𝐹 ↾ 𝑓)‘𝐴)𝑂((𝐹 ↾ 𝑓)‘𝑧) ↔ (𝐹‘𝐴)𝑂(𝐹‘𝑧)))
102	101	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ (𝐴 ∈ 𝑓 ∧ 𝑧 ∈ 𝑓)) → (((𝐹 ↾ 𝑓)‘𝐴)𝑂((𝐹 ↾ 𝑓)‘𝑧) ↔ (𝐹‘𝐴)𝑂(𝐹‘𝑧)))
103	98, 102	bitrd 267	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ (𝐴 ∈ 𝑓 ∧ 𝑧 ∈ 𝑓)) → (𝐴 < 𝑧 ↔ (𝐹‘𝐴)𝑂(𝐹‘𝑧)))
104	94, 96, 97, 103	syl12anc 1316	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → (𝐴 < 𝑧 ↔ (𝐹‘𝐴)𝑂(𝐹‘𝑧)))
105	93, 104	mtbid 313	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → ¬ (𝐹‘𝐴)𝑂(𝐹‘𝑧))
106		simplr 788	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → (𝐹‘𝐴)𝑂(𝐹‘𝐵))
107	65	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → 𝐹:(1...𝑁)⟶ℝ)
108	107, 87	ffvelrnd 6268	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → (𝐹‘𝑧) ∈ ℝ)
109	107, 89	ffvelrnd 6268	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → (𝐹‘𝐴) ∈ ℝ)
110	42	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → 𝐵 ∈ (1...𝑁))
111	107, 110	ffvelrnd 6268	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → (𝐹‘𝐵) ∈ ℝ)
112		sotr2 4988	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑂 Or ℝ ∧ ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝐴) ∈ ℝ ∧ (𝐹‘𝐵) ∈ ℝ)) → ((¬ (𝐹‘𝐴)𝑂(𝐹‘𝑧) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (𝐹‘𝑧)𝑂(𝐹‘𝐵)))
113	8, 112	mpan 702	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝐴) ∈ ℝ ∧ (𝐹‘𝐵) ∈ ℝ) → ((¬ (𝐹‘𝐴)𝑂(𝐹‘𝑧) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (𝐹‘𝑧)𝑂(𝐹‘𝐵)))
114	108, 109, 111, 113	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → ((¬ (𝐹‘𝐴)𝑂(𝐹‘𝑧) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (𝐹‘𝑧)𝑂(𝐹‘𝐵)))
115	105, 106, 114	mp2and 711	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → (𝐹‘𝑧)𝑂(𝐹‘𝐵))
116	115	a1d 25	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → (𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝐵)))
117		elsni 4142	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 ∈ {𝐵} → 𝑤 = 𝐵)
118	117	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ {𝐵} → (𝐹‘𝑤) = (𝐹‘𝐵))
119	118	breq2d 4595	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ {𝐵} → ((𝐹‘𝑧)𝑂(𝐹‘𝑤) ↔ (𝐹‘𝑧)𝑂(𝐹‘𝐵)))
120	119	imbi2d 329	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 ∈ {𝐵} → ((𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤)) ↔ (𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝐵))))
121	116, 120	syl5ibrcom 236	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → (𝑤 ∈ {𝐵} → (𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤))))
122	121	ralrimiv 2948	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → ∀𝑤 ∈ {𝐵} (𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤)))
123		ralunb 3756	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑤 ∈ (𝑓 ∪ {𝐵})(𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤)) ↔ (∀𝑤 ∈ 𝑓 (𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤)) ∧ ∀𝑤 ∈ {𝐵} (𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤))))
124	83, 122, 123	sylanbrc 695	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑧 ∈ 𝑓) → ∀𝑤 ∈ (𝑓 ∪ {𝐵})(𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤)))
125	124	ralrimiva 2949	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → ∀𝑧 ∈ 𝑓 ∀𝑤 ∈ (𝑓 ∪ {𝐵})(𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤)))
126	61	sselda 3568	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑤 ∈ (𝑓 ∪ {𝐵})) → 𝑤 ∈ (1...𝐵))
127		elfzle2 12216	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 ∈ (1...𝐵) → 𝑤 ≤ 𝐵)
128	127	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑤 ∈ (1...𝐵)) → 𝑤 ≤ 𝐵)
129		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑤 ∈ (1...𝐵) → 𝑤 ∈ ℤ)
130	129	zred 11358	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 ∈ (1...𝐵) → 𝑤 ∈ ℝ)
131	130	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑤 ∈ (1...𝐵)) → 𝑤 ∈ ℝ)
132	35	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑤 ∈ (1...𝐵)) → 𝐵 ∈ ℝ)
133	131, 132	lenltd 10062	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑤 ∈ (1...𝐵)) → (𝑤 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑤))
134	128, 133	mpbid 221	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑤 ∈ (1...𝐵)) → ¬ 𝐵 < 𝑤)
135	126, 134	syldan 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑤 ∈ (𝑓 ∪ {𝐵})) → ¬ 𝐵 < 𝑤)
136	135	pm2.21d 117	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) ∧ 𝑤 ∈ (𝑓 ∪ {𝐵})) → (𝐵 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤)))
137	136	ralrimiva 2949	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → ∀𝑤 ∈ (𝑓 ∪ {𝐵})(𝐵 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤)))
138		elsni 4142	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ {𝐵} → 𝑧 = 𝐵)
139	138	breq1d 4593	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ {𝐵} → (𝑧 < 𝑤 ↔ 𝐵 < 𝑤))
140	139	imbi1d 330	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ {𝐵} → ((𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤)) ↔ (𝐵 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤))))
141	140	ralbidv 2969	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ {𝐵} → (∀𝑤 ∈ (𝑓 ∪ {𝐵})(𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤)) ↔ ∀𝑤 ∈ (𝑓 ∪ {𝐵})(𝐵 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤))))
142	137, 141	syl5ibrcom 236	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (𝑧 ∈ {𝐵} → ∀𝑤 ∈ (𝑓 ∪ {𝐵})(𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤))))
143	142	ralrimiv 2948	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → ∀𝑧 ∈ {𝐵}∀𝑤 ∈ (𝑓 ∪ {𝐵})(𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤)))
144		ralunb 3756	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑧 ∈ (𝑓 ∪ {𝐵})∀𝑤 ∈ (𝑓 ∪ {𝐵})(𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤)) ↔ (∀𝑧 ∈ 𝑓 ∀𝑤 ∈ (𝑓 ∪ {𝐵})(𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤)) ∧ ∀𝑧 ∈ {𝐵}∀𝑤 ∈ (𝑓 ∪ {𝐵})(𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤))))
145	125, 143, 144	sylanbrc 695	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → ∀𝑧 ∈ (𝑓 ∪ {𝐵})∀𝑤 ∈ (𝑓 ∪ {𝐵})(𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤)))
146	42	snssd 4281	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → {𝐵} ⊆ (1...𝑁))
147	70, 146	unssd 3751	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (𝑓 ∪ {𝐵}) ⊆ (1...𝑁))
148		soisores 6477	. . . . . . . . . . . . . . . . . 18 ⊢ ((( < Or (1...𝑁) ∧ 𝑂 Or ℝ) ∧ (𝐹:(1...𝑁)⟶ℝ ∧ (𝑓 ∪ {𝐵}) ⊆ (1...𝑁))) → ((𝐹 ↾ (𝑓 ∪ {𝐵})) Isom < , 𝑂 ((𝑓 ∪ {𝐵}), (𝐹 “ (𝑓 ∪ {𝐵}))) ↔ ∀𝑧 ∈ (𝑓 ∪ {𝐵})∀𝑤 ∈ (𝑓 ∪ {𝐵})(𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤))))
149	78, 8, 148	mpanl12 714	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:(1...𝑁)⟶ℝ ∧ (𝑓 ∪ {𝐵}) ⊆ (1...𝑁)) → ((𝐹 ↾ (𝑓 ∪ {𝐵})) Isom < , 𝑂 ((𝑓 ∪ {𝐵}), (𝐹 “ (𝑓 ∪ {𝐵}))) ↔ ∀𝑧 ∈ (𝑓 ∪ {𝐵})∀𝑤 ∈ (𝑓 ∪ {𝐵})(𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤))))
150	65, 147, 149	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → ((𝐹 ↾ (𝑓 ∪ {𝐵})) Isom < , 𝑂 ((𝑓 ∪ {𝐵}), (𝐹 “ (𝑓 ∪ {𝐵}))) ↔ ∀𝑧 ∈ (𝑓 ∪ {𝐵})∀𝑤 ∈ (𝑓 ∪ {𝐵})(𝑧 < 𝑤 → (𝐹‘𝑧)𝑂(𝐹‘𝑤))))
151	145, 150	mpbird 246	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (𝐹 ↾ (𝑓 ∪ {𝐵})) Isom < , 𝑂 ((𝑓 ∪ {𝐵}), (𝐹 “ (𝑓 ∪ {𝐵}))))
152		ssun2 3739	. . . . . . . . . . . . . . . 16 ⊢ {𝐵} ⊆ (𝑓 ∪ {𝐵})
153		snssg 4268	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ (1...𝐵) → (𝐵 ∈ (𝑓 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝑓 ∪ {𝐵})))
154	59, 153	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (𝐵 ∈ (𝑓 ∪ {𝐵}) ↔ {𝐵} ⊆ (𝑓 ∪ {𝐵})))
155	152, 154	mpbiri 247	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → 𝐵 ∈ (𝑓 ∪ {𝐵}))
156	22	erdszelem1 30427	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∪ {𝐵}) ∈ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)} ↔ ((𝑓 ∪ {𝐵}) ⊆ (1...𝐵) ∧ (𝐹 ↾ (𝑓 ∪ {𝐵})) Isom < , 𝑂 ((𝑓 ∪ {𝐵}), (𝐹 “ (𝑓 ∪ {𝐵}))) ∧ 𝐵 ∈ (𝑓 ∪ {𝐵})))
157	61, 151, 155, 156	syl3anbrc 1239	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (𝑓 ∪ {𝐵}) ∈ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)})
158		vex 3176	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
159		snex 4835	. . . . . . . . . . . . . . . . 17 ⊢ {𝐵} ∈ V
160	158, 159	unex 6854	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 ∪ {𝐵}) ∈ V
161	1	fdmi 5965	. . . . . . . . . . . . . . . 16 ⊢ dom # = V
162	160, 161	eleqtrri 2687	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∪ {𝐵}) ∈ dom #
163		funfvima 6396	. . . . . . . . . . . . . . 15 ⊢ ((Fun # ∧ (𝑓 ∪ {𝐵}) ∈ dom #) → ((𝑓 ∪ {𝐵}) ∈ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)} → (#‘(𝑓 ∪ {𝐵})) ∈ (# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)})))
164	3, 162, 163	mp2an 704	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∪ {𝐵}) ∈ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)} → (#‘(𝑓 ∪ {𝐵})) ∈ (# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}))
165	157, 164	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (#‘(𝑓 ∪ {𝐵})) ∈ (# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}))
166	45, 165	eqeltrrd 2689	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → ((#‘𝑓) + 1) ∈ (# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}))
167		ne0i 3880	. . . . . . . . . . . 12 ⊢ (((#‘𝑓) + 1) ∈ (# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}) → (# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}) ≠ ∅)
168	166, 167	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}) ≠ ∅)
169	23	simpli 473	. . . . . . . . . . . 12 ⊢ (# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}) ∈ Fin
170		fimaxre2 10848	. . . . . . . . . . . 12 ⊢ (((# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}) ⊆ ℝ ∧ (# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}) ∈ Fin) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ (# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)})𝑤 ≤ 𝑧)
171	27, 169, 170	sylancl 693	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ (# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)})𝑤 ≤ 𝑧)
172		suprub 10863	. . . . . . . . . . 11 ⊢ ((((# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}) ⊆ ℝ ∧ (# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}) ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ (# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)})𝑤 ≤ 𝑧) ∧ ((#‘𝑓) + 1) ∈ (# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)})) → ((#‘𝑓) + 1) ≤ sup((# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}), ℝ, < ))
173	27, 168, 171, 166, 172	syl31anc 1321	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → ((#‘𝑓) + 1) ≤ sup((# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}), ℝ, < ))
174	5, 6, 7	erdszelem3 30429	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ (1...𝑁) → (𝐾‘𝐵) = sup((# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}), ℝ, < ))
175	32, 174	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘𝐵) = sup((# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}), ℝ, < ))
176	175	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (𝐾‘𝐵) = sup((# “ {𝑦 ∈ 𝒫 (1...𝐵) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐵 ∈ 𝑦)}), ℝ, < ))
177	173, 176	breqtrrd 4611	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → ((#‘𝑓) + 1) ≤ (𝐾‘𝐵))
178	5, 6, 7, 8	erdszelem6 30432	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾:(1...𝑁)⟶ℕ)
179	178, 32	ffvelrnd 6268	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾‘𝐵) ∈ ℕ)
180	179	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (𝐾‘𝐵) ∈ ℕ)
181	180	nnnn0d 11228	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (𝐾‘𝐵) ∈ ℕ₀)
182		nn0ltp1le 11312	. . . . . . . . . 10 ⊢ (((#‘𝑓) ∈ ℕ₀ ∧ (𝐾‘𝐵) ∈ ℕ₀) → ((#‘𝑓) < (𝐾‘𝐵) ↔ ((#‘𝑓) + 1) ≤ (𝐾‘𝐵)))
183	20, 181, 182	syl2anc 691	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → ((#‘𝑓) < (𝐾‘𝐵) ↔ ((#‘𝑓) + 1) ≤ (𝐾‘𝐵)))
184	177, 183	mpbird 246	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (#‘𝑓) < (𝐾‘𝐵))
185	21, 184	ltned 10052	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) ∧ (𝐹‘𝐴)𝑂(𝐹‘𝐵)) → (#‘𝑓) ≠ (𝐾‘𝐵))
186	185	ex 449	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) → ((𝐹‘𝐴)𝑂(𝐹‘𝐵) → (#‘𝑓) ≠ (𝐾‘𝐵)))
187		neeq1 2844	. . . . . . 7 ⊢ ((#‘𝑓) = (𝐾‘𝐴) → ((#‘𝑓) ≠ (𝐾‘𝐵) ↔ (𝐾‘𝐴) ≠ (𝐾‘𝐵)))
188	187	imbi2d 329	. . . . . 6 ⊢ ((#‘𝑓) = (𝐾‘𝐴) → (((𝐹‘𝐴)𝑂(𝐹‘𝐵) → (#‘𝑓) ≠ (𝐾‘𝐵)) ↔ ((𝐹‘𝐴)𝑂(𝐹‘𝐵) → (𝐾‘𝐴) ≠ (𝐾‘𝐵))))
189	186, 188	syl5ibcom 234	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ⊆ (1...𝐴) ∧ (𝐹 ↾ 𝑓) Isom < , 𝑂 (𝑓, (𝐹 “ 𝑓)) ∧ 𝐴 ∈ 𝑓)) → ((#‘𝑓) = (𝐾‘𝐴) → ((𝐹‘𝐴)𝑂(𝐹‘𝐵) → (𝐾‘𝐴) ≠ (𝐾‘𝐵))))
190	14, 189	sylan2b 491	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)}) → ((#‘𝑓) = (𝐾‘𝐴) → ((𝐹‘𝐴)𝑂(𝐹‘𝐵) → (𝐾‘𝐴) ≠ (𝐾‘𝐵))))
191	190	rexlimdva 3013	. . 3 ⊢ (𝜑 → (∃𝑓 ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝐴 ∈ 𝑦)} (#‘𝑓) = (𝐾‘𝐴) → ((𝐹‘𝐴)𝑂(𝐹‘𝐵) → (𝐾‘𝐴) ≠ (𝐾‘𝐵))))
192	12, 191	mpd 15	. 2 ⊢ (𝜑 → ((𝐹‘𝐴)𝑂(𝐹‘𝐵) → (𝐾‘𝐴) ≠ (𝐾‘𝐵)))
193	192	necon2bd 2798	1 ⊢ (𝜑 → ((𝐾‘𝐴) = (𝐾‘𝐵) → ¬ (𝐹‘𝐴)𝑂(𝐹‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ∪ cun 3538 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 Or wor 4958 dom cdm 5038 ↾ cres 5040 “ cima 5041 Fun wfun 5798 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804 Isom wiso 5805 (class class class)co 6549 Fincfn 7841 supcsup 8229 ℝcr 9814 1c1 9816 + caddc 9818 +∞cpnf 9950 < clt 9953 ≤ cle 9954 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197 #chash 12979

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-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 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-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 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-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-hash 12980

This theorem is referenced by: erdszelem9 30435

W3C validator