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Theorem leiso 14431

Description: Two ways to write a strictly increasing function on the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)

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

**Proof of Theorem leiso**
Step	Hyp	Ref	Expression
1		df-le 11221	. . . . . . 7 ⊢ ≤ = ((ℝ^* × ℝ^*) ∖ ^◡ < )
2	1	ineq1i 4182	. . . . . 6 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) = (((ℝ^* × ℝ^*) ∖ ^◡ < ) ∩ (𝐴 × 𝐴))
3		indif1 4248	. . . . . 6 ⊢ (((ℝ^* × ℝ^) ∖ ^◡ < ) ∩ (𝐴 × 𝐴)) = (((ℝ^ × ℝ^*) ∩ (𝐴 × 𝐴)) ∖ ^◡ < )
4	2, 3	eqtri 2753	. . . . 5 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) = (((ℝ^* × ℝ^*) ∩ (𝐴 × 𝐴)) ∖ ^◡ < )
5		xpss12 5656	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐴 ⊆ ℝ^) → (𝐴 × 𝐴) ⊆ (ℝ^ × ℝ^*))
6	5	anidms 566	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → (𝐴 × 𝐴) ⊆ (ℝ^* × ℝ^*))
7		sseqin2 4189	. . . . . . 7 ⊢ ((𝐴 × 𝐴) ⊆ (ℝ^* × ℝ^) ↔ ((ℝ^ × ℝ^*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
8	6, 7	sylib 218	. . . . . 6 ⊢ (𝐴 ⊆ ℝ^* → ((ℝ^* × ℝ^*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
9	8	difeq1d 4091	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → (((ℝ^* × ℝ^*) ∩ (𝐴 × 𝐴)) ∖ ^◡ < ) = ((𝐴 × 𝐴) ∖ ^◡ < ))
10	4, 9	eqtr2id 2778	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → ((𝐴 × 𝐴) ∖ ^◡ < ) = ( ≤ ∩ (𝐴 × 𝐴)))
11		isoeq2 7296	. . . 4 ⊢ (((𝐴 × 𝐴) ∖ ^◡ < ) = ( ≤ ∩ (𝐴 × 𝐴)) → (𝐹 Isom ((𝐴 × 𝐴) ∖ ^◡ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵)))
12	10, 11	syl 17	. . 3 ⊢ (𝐴 ⊆ ℝ^* → (𝐹 Isom ((𝐴 × 𝐴) ∖ ^◡ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵)))
13	1	ineq1i 4182	. . . . . 6 ⊢ ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ^* × ℝ^*) ∖ ^◡ < ) ∩ (𝐵 × 𝐵))
14		indif1 4248	. . . . . 6 ⊢ (((ℝ^* × ℝ^) ∖ ^◡ < ) ∩ (𝐵 × 𝐵)) = (((ℝ^ × ℝ^*) ∩ (𝐵 × 𝐵)) ∖ ^◡ < )
15	13, 14	eqtri 2753	. . . . 5 ⊢ ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ^* × ℝ^*) ∩ (𝐵 × 𝐵)) ∖ ^◡ < )
16		xpss12 5656	. . . . . . . 8 ⊢ ((𝐵 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) → (𝐵 × 𝐵) ⊆ (ℝ^ × ℝ^*))
17	16	anidms 566	. . . . . . 7 ⊢ (𝐵 ⊆ ℝ^* → (𝐵 × 𝐵) ⊆ (ℝ^* × ℝ^*))
18		sseqin2 4189	. . . . . . 7 ⊢ ((𝐵 × 𝐵) ⊆ (ℝ^* × ℝ^) ↔ ((ℝ^ × ℝ^*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
19	17, 18	sylib 218	. . . . . 6 ⊢ (𝐵 ⊆ ℝ^* → ((ℝ^* × ℝ^*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
20	19	difeq1d 4091	. . . . 5 ⊢ (𝐵 ⊆ ℝ^* → (((ℝ^* × ℝ^*) ∩ (𝐵 × 𝐵)) ∖ ^◡ < ) = ((𝐵 × 𝐵) ∖ ^◡ < ))
21	15, 20	eqtr2id 2778	. . . 4 ⊢ (𝐵 ⊆ ℝ^* → ((𝐵 × 𝐵) ∖ ^◡ < ) = ( ≤ ∩ (𝐵 × 𝐵)))
22		isoeq3 7297	. . . 4 ⊢ (((𝐵 × 𝐵) ∖ ^◡ < ) = ( ≤ ∩ (𝐵 × 𝐵)) → (𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
23	21, 22	syl 17	. . 3 ⊢ (𝐵 ⊆ ℝ^* → (𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
24	12, 23	sylan9bb 509	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom ((𝐴 × 𝐴) ∖ ^◡ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
25		isocnv2 7309	. . 3 ⊢ (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ^◡ < , ^◡ < (𝐴, 𝐵))
26		eqid 2730	. . . 4 ⊢ ((𝐴 × 𝐴) ∖ ^◡ < ) = ((𝐴 × 𝐴) ∖ ^◡ < )
27		eqid 2730	. . . 4 ⊢ ((𝐵 × 𝐵) ∖ ^◡ < ) = ((𝐵 × 𝐵) ∖ ^◡ < )
28	26, 27	isocnv3 7310	. . 3 ⊢ (𝐹 Isom ^◡ < , ^◡ < (𝐴, 𝐵) ↔ 𝐹 Isom ((𝐴 × 𝐴) ∖ ^◡ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵))
29	25, 28	bitri 275	. 2 ⊢ (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ((𝐴 × 𝐴) ∖ ^◡ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵))
30		isores1 7312	. . 3 ⊢ (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ≤ (𝐴, 𝐵))
31		isores2 7311	. . 3 ⊢ (𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
32	30, 31	bitri 275	. 2 ⊢ (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
33	24, 29, 32	3bitr4g 314	1 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∖ cdif 3914 ∩ cin 3916 ⊆ wss 3917 × cxp 5639 ^◡ccnv 5640 Isom wiso 6515 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-le 11221

This theorem is referenced by: leisorel 14432 icopnfhmeo 24848 iccpnfhmeo 24850 xrhmeo 24851

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