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Theorem gtiso 31614
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
StepHypRef Expression
1 eqid 2736 . . . . 5 ((𝐴 × 𝐴) ∖ < ) = ((𝐴 × 𝐴) ∖ < )
2 eqid 2736 . . . . 5 ((𝐵 × 𝐵) ∖ < ) = ((𝐵 × 𝐵) ∖ < )
31, 2isocnv3 7277 . . . 4 (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵))
43a1i 11 . . 3 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵)))
5 df-le 11195 . . . . . . . . . 10 ≤ = ((ℝ* × ℝ*) ∖ < )
65cnveqi 5830 . . . . . . . . 9 ≤ = ((ℝ* × ℝ*) ∖ < )
7 cnvdif 6096 . . . . . . . . 9 ((ℝ* × ℝ*) ∖ < ) = ((ℝ* × ℝ*) ∖ < )
8 cnvxp 6109 . . . . . . . . . 10 (ℝ* × ℝ*) = (ℝ* × ℝ*)
9 ltrel 11217 . . . . . . . . . . 11 Rel <
10 dfrel2 6141 . . . . . . . . . . 11 (Rel < ↔ < = < )
119, 10mpbi 229 . . . . . . . . . 10 < = <
128, 11difeq12i 4080 . . . . . . . . 9 ((ℝ* × ℝ*) ∖ < ) = ((ℝ* × ℝ*) ∖ < )
136, 7, 123eqtri 2768 . . . . . . . 8 ≤ = ((ℝ* × ℝ*) ∖ < )
1413ineq1i 4168 . . . . . . 7 ( ≤ ∩ (𝐴 × 𝐴)) = (((ℝ* × ℝ*) ∖ < ) ∩ (𝐴 × 𝐴))
15 indif1 4231 . . . . . . 7 (((ℝ* × ℝ*) ∖ < ) ∩ (𝐴 × 𝐴)) = (((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) ∖ < )
1614, 15eqtri 2764 . . . . . 6 ( ≤ ∩ (𝐴 × 𝐴)) = (((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) ∖ < )
17 xpss12 5648 . . . . . . . . 9 ((𝐴 ⊆ ℝ*𝐴 ⊆ ℝ*) → (𝐴 × 𝐴) ⊆ (ℝ* × ℝ*))
1817anidms 567 . . . . . . . 8 (𝐴 ⊆ ℝ* → (𝐴 × 𝐴) ⊆ (ℝ* × ℝ*))
19 sseqin2 4175 . . . . . . . 8 ((𝐴 × 𝐴) ⊆ (ℝ* × ℝ*) ↔ ((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
2018, 19sylib 217 . . . . . . 7 (𝐴 ⊆ ℝ* → ((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
2120difeq1d 4081 . . . . . 6 (𝐴 ⊆ ℝ* → (((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) ∖ < ) = ((𝐴 × 𝐴) ∖ < ))
2216, 21eqtr2id 2789 . . . . 5 (𝐴 ⊆ ℝ* → ((𝐴 × 𝐴) ∖ < ) = ( ≤ ∩ (𝐴 × 𝐴)))
2322adantr 481 . . . 4 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → ((𝐴 × 𝐴) ∖ < ) = ( ≤ ∩ (𝐴 × 𝐴)))
24 isoeq2 7263 . . . 4 (((𝐴 × 𝐴) ∖ < ) = ( ≤ ∩ (𝐴 × 𝐴)) → (𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵)))
2523, 24syl 17 . . 3 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵)))
265ineq1i 4168 . . . . . . 7 ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ* × ℝ*) ∖ < ) ∩ (𝐵 × 𝐵))
27 indif1 4231 . . . . . . 7 (((ℝ* × ℝ*) ∖ < ) ∩ (𝐵 × 𝐵)) = (((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) ∖ < )
2826, 27eqtri 2764 . . . . . 6 ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) ∖ < )
29 xpss12 5648 . . . . . . . . 9 ((𝐵 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐵 × 𝐵) ⊆ (ℝ* × ℝ*))
3029anidms 567 . . . . . . . 8 (𝐵 ⊆ ℝ* → (𝐵 × 𝐵) ⊆ (ℝ* × ℝ*))
31 sseqin2 4175 . . . . . . . 8 ((𝐵 × 𝐵) ⊆ (ℝ* × ℝ*) ↔ ((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
3230, 31sylib 217 . . . . . . 7 (𝐵 ⊆ ℝ* → ((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
3332difeq1d 4081 . . . . . 6 (𝐵 ⊆ ℝ* → (((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) ∖ < ) = ((𝐵 × 𝐵) ∖ < ))
3428, 33eqtr2id 2789 . . . . 5 (𝐵 ⊆ ℝ* → ((𝐵 × 𝐵) ∖ < ) = ( ≤ ∩ (𝐵 × 𝐵)))
3534adantl 482 . . . 4 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → ((𝐵 × 𝐵) ∖ < ) = ( ≤ ∩ (𝐵 × 𝐵)))
36 isoeq3 7264 . . . 4 (((𝐵 × 𝐵) ∖ < ) = ( ≤ ∩ (𝐵 × 𝐵)) → (𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
3735, 36syl 17 . . 3 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
384, 25, 373bitrd 304 . 2 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
39 isocnv2 7276 . . 3 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵))
40 isores2 7278 . . . 4 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
41 isores1 7279 . . . 4 (𝐹 Isom ≤ , ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
4240, 41bitri 274 . . 3 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
43 lerel 11219 . . . . 5 Rel ≤
44 dfrel2 6141 . . . . 5 (Rel ≤ ↔ ≤ = ≤ )
4543, 44mpbi 229 . . . 4 ≤ = ≤
46 isoeq2 7263 . . . 4 ( ≤ = ≤ → (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
4745, 46ax-mp 5 . . 3 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵))
4839, 42, 473bitr3ri 301 . 2 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
4938, 48bitr4di 288 1 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  cdif 3907  cin 3909  wss 3910   × cxp 5631  ccnv 5632  Rel wrel 5638   Isom wiso 6497  *cxr 11188   < clt 11189  cle 11190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-xr 11193  df-ltxr 11194  df-le 11195
This theorem is referenced by:  xrge0iifhmeo  32517
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