gtiso - Mathbox for Thierry Arnoux

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Theorem gtiso 32989

Description: Two ways to write a strictly decreasing function on the reals. (Contributed by Thierry Arnoux, 6-Apr-2017.)

**Assertion**
Ref	Expression
gtiso	⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom < , ^◡ < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ^◡ ≤ (𝐴, 𝐵)))

**Proof of Theorem gtiso**
Step	Hyp	Ref	Expression
1		eqid 2769	. . . . 5 ⊢ ((𝐴 × 𝐴) ∖ < ) = ((𝐴 × 𝐴) ∖ < )
2		eqid 2769	. . . . 5 ⊢ ((𝐵 × 𝐵) ∖ ^◡ < ) = ((𝐵 × 𝐵) ∖ ^◡ < )
3	1, 2	isocnv3 7333	. . . 4 ⊢ (𝐹 Isom < , ^◡ < (𝐴, 𝐵) ↔ 𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵))
4	3	a1i 11	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom < , ^◡ < (𝐴, 𝐵) ↔ 𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵)))
5		df-le 11251	. . . . . . . . . 10 ⊢ ≤ = ((ℝ^* × ℝ^*) ∖ ^◡ < )
6	5	cnveqi 5863	. . . . . . . . 9 ⊢ ^◡ ≤ = ^◡((ℝ^* × ℝ^*) ∖ ^◡ < )
7		cnvdif 6143	. . . . . . . . 9 ⊢ ^◡((ℝ^* × ℝ^) ∖ ^◡ < ) = (^◡(ℝ^ × ℝ^*) ∖ ^◡^◡ < )
8		cnvxp 6157	. . . . . . . . . 10 ⊢ ^◡(ℝ^* × ℝ^) = (ℝ^ × ℝ^*)
9		ltrel 11273	. . . . . . . . . . 11 ⊢ Rel <
10		dfrel2 6190	. . . . . . . . . . 11 ⊢ (Rel < ↔ ^◡^◡ < = < )
11	9, 10	mpbi 233	. . . . . . . . . 10 ⊢ ^◡^◡ < = <
12	8, 11	difeq12i 4087	. . . . . . . . 9 ⊢ (^◡(ℝ^* × ℝ^) ∖ ^◡^◡ < ) = ((ℝ^ × ℝ^*) ∖ < )
13	6, 7, 12	3eqtri 2796	. . . . . . . 8 ⊢ ^◡ ≤ = ((ℝ^* × ℝ^*) ∖ < )
14	13	ineq1i 4177	. . . . . . 7 ⊢ (^◡ ≤ ∩ (𝐴 × 𝐴)) = (((ℝ^* × ℝ^*) ∖ < ) ∩ (𝐴 × 𝐴))
15		indif1 4243	. . . . . . 7 ⊢ (((ℝ^* × ℝ^) ∖ < ) ∩ (𝐴 × 𝐴)) = (((ℝ^ × ℝ^*) ∩ (𝐴 × 𝐴)) ∖ < )
16	14, 15	eqtri 2792	. . . . . 6 ⊢ (^◡ ≤ ∩ (𝐴 × 𝐴)) = (((ℝ^* × ℝ^*) ∩ (𝐴 × 𝐴)) ∖ < )
17		xpss12 5679	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐴 ⊆ ℝ^) → (𝐴 × 𝐴) ⊆ (ℝ^ × ℝ^*))
18	17	anidms 576	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → (𝐴 × 𝐴) ⊆ (ℝ^* × ℝ^*))
19		sseqin2 4184	. . . . . . . 8 ⊢ ((𝐴 × 𝐴) ⊆ (ℝ^* × ℝ^) ↔ ((ℝ^ × ℝ^*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
20	18, 19	sylib 221	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → ((ℝ^* × ℝ^*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
21	20	difeq1d 4088	. . . . . 6 ⊢ (𝐴 ⊆ ℝ^* → (((ℝ^* × ℝ^*) ∩ (𝐴 × 𝐴)) ∖ < ) = ((𝐴 × 𝐴) ∖ < ))
22	16, 21	eqtr2id 2817	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → ((𝐴 × 𝐴) ∖ < ) = (^◡ ≤ ∩ (𝐴 × 𝐴)))
23	22	adantr 485	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → ((𝐴 × 𝐴) ∖ < ) = (^◡ ≤ ∩ (𝐴 × 𝐴)))
24		isoeq2 7319	. . . 4 ⊢ (((𝐴 × 𝐴) ∖ < ) = (^◡ ≤ ∩ (𝐴 × 𝐴)) → (𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵)))
25	23, 24	syl 18	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵)))
26	5	ineq1i 4177	. . . . . . 7 ⊢ ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ^* × ℝ^*) ∖ ^◡ < ) ∩ (𝐵 × 𝐵))
27		indif1 4243	. . . . . . 7 ⊢ (((ℝ^* × ℝ^) ∖ ^◡ < ) ∩ (𝐵 × 𝐵)) = (((ℝ^ × ℝ^*) ∩ (𝐵 × 𝐵)) ∖ ^◡ < )
28	26, 27	eqtri 2792	. . . . . 6 ⊢ ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ^* × ℝ^*) ∩ (𝐵 × 𝐵)) ∖ ^◡ < )
29		xpss12 5679	. . . . . . . . 9 ⊢ ((𝐵 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) → (𝐵 × 𝐵) ⊆ (ℝ^ × ℝ^*))
30	29	anidms 576	. . . . . . . 8 ⊢ (𝐵 ⊆ ℝ^* → (𝐵 × 𝐵) ⊆ (ℝ^* × ℝ^*))
31		sseqin2 4184	. . . . . . . 8 ⊢ ((𝐵 × 𝐵) ⊆ (ℝ^* × ℝ^) ↔ ((ℝ^ × ℝ^*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
32	30, 31	sylib 221	. . . . . . 7 ⊢ (𝐵 ⊆ ℝ^* → ((ℝ^* × ℝ^*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
33	32	difeq1d 4088	. . . . . 6 ⊢ (𝐵 ⊆ ℝ^* → (((ℝ^* × ℝ^*) ∩ (𝐵 × 𝐵)) ∖ ^◡ < ) = ((𝐵 × 𝐵) ∖ ^◡ < ))
34	28, 33	eqtr2id 2817	. . . . 5 ⊢ (𝐵 ⊆ ℝ^* → ((𝐵 × 𝐵) ∖ ^◡ < ) = ( ≤ ∩ (𝐵 × 𝐵)))
35	34	adantl 486	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → ((𝐵 × 𝐵) ∖ ^◡ < ) = ( ≤ ∩ (𝐵 × 𝐵)))
36		isoeq3 7320	. . . 4 ⊢ (((𝐵 × 𝐵) ∖ ^◡ < ) = ( ≤ ∩ (𝐵 × 𝐵)) → (𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
37	35, 36	syl 18	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
38	4, 25, 37	3bitrd 308	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom < , ^◡ < (𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
39		isocnv2 7332	. . 3 ⊢ (𝐹 Isom ^◡ ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ^◡^◡ ≤ , ^◡ ≤ (𝐴, 𝐵))
40		isores2 7334	. . . 4 ⊢ (𝐹 Isom ^◡ ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ^◡ ≤ , ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
41		isores1 7335	. . . 4 ⊢ (𝐹 Isom ^◡ ≤ , ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
42	40, 41	bitri 278	. . 3 ⊢ (𝐹 Isom ^◡ ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
43		lerel 11275	. . . . 5 ⊢ Rel ≤
44		dfrel2 6190	. . . . 5 ⊢ (Rel ≤ ↔ ^◡^◡ ≤ = ≤ )
45	43, 44	mpbi 233	. . . 4 ⊢ ^◡^◡ ≤ = ≤
46		isoeq2 7319	. . . 4 ⊢ (^◡^◡ ≤ = ≤ → (𝐹 Isom ^◡^◡ ≤ , ^◡ ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ^◡ ≤ (𝐴, 𝐵)))
47	45, 46	ax-mp 5	. . 3 ⊢ (𝐹 Isom ^◡^◡ ≤ , ^◡ ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ^◡ ≤ (𝐴, 𝐵))
48	39, 42, 47	3bitr3ri 305	. 2 ⊢ (𝐹 Isom ≤ , ^◡ ≤ (𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
49	38, 48	bitr4di 292	1 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom < , ^◡ < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ^◡ ≤ (𝐴, 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∖ cdif 3910 ∩ cin 3912 ⊆ wss 3913 × cxp 5662 ^◡ccnv 5663 Rel wrel 5669 Isom wiso 6540 ℝ^*cxr 11244 < clt 11245 ≤ cle 11246

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5273 ax-pr 5407

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-ne 2965 df-ral 3086 df-rex 3096 df-rab 3424 df-v 3465 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-br 5114 df-opab 5178 df-id 5559 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-iota 6495 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-xr 11249 df-ltxr 11250 df-le 11251

This theorem is referenced by: xrge0iifhmeo 34273

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