![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > xpen | Structured version Visualization version GIF version |
Description: Equinumerosity law for Cartesian product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by NM, 24-Jul-2004.) (Proof shortened by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
xpen | ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relen 8891 | . . . . 5 ⊢ Rel ≈ | |
2 | 1 | brrelex1i 5689 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐶 ∈ V) |
3 | endom 8922 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | |
4 | xpdom1g 9016 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐴 ≼ 𝐵) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶)) | |
5 | 2, 3, 4 | syl2anr 598 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶)) |
6 | 1 | brrelex2i 5690 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V) |
7 | endom 8922 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐶 ≼ 𝐷) | |
8 | xpdom2g 9015 | . . . 4 ⊢ ((𝐵 ∈ V ∧ 𝐶 ≼ 𝐷) → (𝐵 × 𝐶) ≼ (𝐵 × 𝐷)) | |
9 | 6, 7, 8 | syl2an 597 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐵 × 𝐶) ≼ (𝐵 × 𝐷)) |
10 | domtr 8950 | . . 3 ⊢ (((𝐴 × 𝐶) ≼ (𝐵 × 𝐶) ∧ (𝐵 × 𝐶) ≼ (𝐵 × 𝐷)) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐷)) | |
11 | 5, 9, 10 | syl2anc 585 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐷)) |
12 | 1 | brrelex2i 5690 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐷 ∈ V) |
13 | ensym 8946 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
14 | endom 8922 | . . . . 5 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ≼ 𝐴) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≼ 𝐴) |
16 | xpdom1g 9016 | . . . 4 ⊢ ((𝐷 ∈ V ∧ 𝐵 ≼ 𝐴) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐷)) | |
17 | 12, 15, 16 | syl2anr 598 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐷)) |
18 | 1 | brrelex1i 5689 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V) |
19 | ensym 8946 | . . . . 5 ⊢ (𝐶 ≈ 𝐷 → 𝐷 ≈ 𝐶) | |
20 | endom 8922 | . . . . 5 ⊢ (𝐷 ≈ 𝐶 → 𝐷 ≼ 𝐶) | |
21 | 19, 20 | syl 17 | . . . 4 ⊢ (𝐶 ≈ 𝐷 → 𝐷 ≼ 𝐶) |
22 | xpdom2g 9015 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐷 ≼ 𝐶) → (𝐴 × 𝐷) ≼ (𝐴 × 𝐶)) | |
23 | 18, 21, 22 | syl2an 597 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐷) ≼ (𝐴 × 𝐶)) |
24 | domtr 8950 | . . 3 ⊢ (((𝐵 × 𝐷) ≼ (𝐴 × 𝐷) ∧ (𝐴 × 𝐷) ≼ (𝐴 × 𝐶)) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐶)) | |
25 | 17, 23, 24 | syl2anc 585 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐵 × 𝐷) ≼ (𝐴 × 𝐶)) |
26 | sbth 9040 | . 2 ⊢ (((𝐴 × 𝐶) ≼ (𝐵 × 𝐷) ∧ (𝐵 × 𝐷) ≼ (𝐴 × 𝐶)) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷)) | |
27 | 11, 25, 26 | syl2anc 585 | 1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 Vcvv 3444 class class class wbr 5106 × cxp 5632 ≈ cen 8883 ≼ cdom 8884 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-1st 7922 df-2nd 7923 df-er 8651 df-en 8887 df-dom 8888 |
This theorem is referenced by: map2xp 9094 unxpdom2 9201 sucxpdom 9202 xpnum 9892 infxpenlem 9954 infxpidm2 9958 xpdjuen 10120 mapdjuen 10121 pwdjuen 10122 djuxpdom 10126 ackbij1lem5 10165 canthp1lem1 10593 xpnnen 16098 qnnen 16100 rexpen 16115 met2ndci 23894 re2ndc 24180 dyadmbl 24980 opnmblALT 24983 mbfimaopnlem 25035 mblfinlem1 36161 |
Copyright terms: Public domain | W3C validator |