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Mirrors > Home > ILE Home > Th. List > addext | GIF version |
Description: Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5572. For us, it is proved a different way. (Contributed by Jim Kingdon, 15-Feb-2020.) |
Ref | Expression |
---|---|
addext | ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) # (𝐶 + 𝐷) → (𝐴 # 𝐶 ∨ 𝐵 # 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 496 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → 𝐴 ∈ ℂ) | |
2 | simplr 497 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → 𝐵 ∈ ℂ) | |
3 | 1, 2 | addcld 7252 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (𝐴 + 𝐵) ∈ ℂ) |
4 | simprl 498 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → 𝐶 ∈ ℂ) | |
5 | simprr 499 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → 𝐷 ∈ ℂ) | |
6 | 4, 5 | addcld 7252 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (𝐶 + 𝐷) ∈ ℂ) |
7 | 4, 2 | addcld 7252 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (𝐶 + 𝐵) ∈ ℂ) |
8 | apcotr 7826 | . . 3 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ (𝐶 + 𝐷) ∈ ℂ ∧ (𝐶 + 𝐵) ∈ ℂ) → ((𝐴 + 𝐵) # (𝐶 + 𝐷) → ((𝐴 + 𝐵) # (𝐶 + 𝐵) ∨ (𝐶 + 𝐷) # (𝐶 + 𝐵)))) | |
9 | 3, 6, 7, 8 | syl3anc 1170 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) # (𝐶 + 𝐷) → ((𝐴 + 𝐵) # (𝐶 + 𝐵) ∨ (𝐶 + 𝐷) # (𝐶 + 𝐵)))) |
10 | apadd1 7827 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 # 𝐶 ↔ (𝐴 + 𝐵) # (𝐶 + 𝐵))) | |
11 | 1, 4, 2, 10 | syl3anc 1170 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (𝐴 # 𝐶 ↔ (𝐴 + 𝐵) # (𝐶 + 𝐵))) |
12 | apadd2 7828 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 # 𝐷 ↔ (𝐶 + 𝐵) # (𝐶 + 𝐷))) | |
13 | 2, 5, 4, 12 | syl3anc 1170 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (𝐵 # 𝐷 ↔ (𝐶 + 𝐵) # (𝐶 + 𝐷))) |
14 | apsym 7825 | . . . . 5 ⊢ (((𝐶 + 𝐵) ∈ ℂ ∧ (𝐶 + 𝐷) ∈ ℂ) → ((𝐶 + 𝐵) # (𝐶 + 𝐷) ↔ (𝐶 + 𝐷) # (𝐶 + 𝐵))) | |
15 | 7, 6, 14 | syl2anc 403 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐶 + 𝐵) # (𝐶 + 𝐷) ↔ (𝐶 + 𝐷) # (𝐶 + 𝐵))) |
16 | 13, 15 | bitrd 186 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (𝐵 # 𝐷 ↔ (𝐶 + 𝐷) # (𝐶 + 𝐵))) |
17 | 11, 16 | orbi12d 740 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 # 𝐶 ∨ 𝐵 # 𝐷) ↔ ((𝐴 + 𝐵) # (𝐶 + 𝐵) ∨ (𝐶 + 𝐷) # (𝐶 + 𝐵)))) |
18 | 9, 17 | sylibrd 167 | 1 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) # (𝐶 + 𝐷) → (𝐴 # 𝐶 ∨ 𝐵 # 𝐷))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 662 ∈ wcel 1434 class class class wbr 3805 (class class class)co 5563 ℂcc 7093 + caddc 7098 # cap 7800 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3916 ax-pow 3968 ax-pr 3992 ax-un 4216 ax-setind 4308 ax-cnex 7181 ax-resscn 7182 ax-1cn 7183 ax-1re 7184 ax-icn 7185 ax-addcl 7186 ax-addrcl 7187 ax-mulcl 7188 ax-mulrcl 7189 ax-addcom 7190 ax-mulcom 7191 ax-addass 7192 ax-mulass 7193 ax-distr 7194 ax-i2m1 7195 ax-0lt1 7196 ax-1rid 7197 ax-0id 7198 ax-rnegex 7199 ax-precex 7200 ax-cnre 7201 ax-pre-ltirr 7202 ax-pre-ltwlin 7203 ax-pre-lttrn 7204 ax-pre-apti 7205 ax-pre-ltadd 7206 ax-pre-mulgt0 7207 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2612 df-sbc 2825 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-br 3806 df-opab 3860 df-id 4076 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-iota 4917 df-fun 4954 df-fv 4960 df-riota 5519 df-ov 5566 df-oprab 5567 df-mpt2 5568 df-pnf 7269 df-mnf 7270 df-ltxr 7272 df-sub 7400 df-neg 7401 df-reap 7794 df-ap 7801 |
This theorem is referenced by: mulext1 7831 abs00ap 10149 absext 10150 |
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