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Mirrors > Home > ILE Home > Th. List > mulext | GIF version |
Description: Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5599. For us, it is proved a different way. (Contributed by Jim Kingdon, 23-Feb-2020.) |
Ref | Expression |
---|---|
mulext | ⊢ (((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 | mulcld 7410 | . . 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 | mulcld 7410 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (𝐶 · 𝐷) ∈ ℂ) |
7 | 4, 2 | mulcld 7410 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (𝐶 · 𝐵) ∈ ℂ) |
8 | apcotr 7983 | . . 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 | mulext1 7988 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) # (𝐶 · 𝐵) → 𝐴 # 𝐶)) | |
11 | 1, 4, 2, 10 | syl3anc 1170 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 · 𝐵) # (𝐶 · 𝐵) → 𝐴 # 𝐶)) |
12 | mulext2 7989 | . . . . 5 ⊢ ((𝐷 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐶 · 𝐷) # (𝐶 · 𝐵) → 𝐷 # 𝐵)) | |
13 | 5, 2, 4, 12 | syl3anc 1170 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐶 · 𝐷) # (𝐶 · 𝐵) → 𝐷 # 𝐵)) |
14 | apsym 7982 | . . . . 5 ⊢ ((𝐷 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐷 # 𝐵 ↔ 𝐵 # 𝐷)) | |
15 | 5, 2, 14 | syl2anc 403 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (𝐷 # 𝐵 ↔ 𝐵 # 𝐷)) |
16 | 13, 15 | sylibd 147 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐶 · 𝐷) # (𝐶 · 𝐵) → 𝐵 # 𝐷)) |
17 | 11, 16 | orim12d 733 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (((𝐴 · 𝐵) # (𝐶 · 𝐵) ∨ (𝐶 · 𝐷) # (𝐶 · 𝐵)) → (𝐴 # 𝐶 ∨ 𝐵 # 𝐷))) |
18 | 9, 17 | syld 44 | 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 3811 (class class class)co 5590 ℂcc 7250 · cmul 7257 # cap 7957 |
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 3922 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-cnex 7338 ax-resscn 7339 ax-1cn 7340 ax-1re 7341 ax-icn 7342 ax-addcl 7343 ax-addrcl 7344 ax-mulcl 7345 ax-mulrcl 7346 ax-addcom 7347 ax-mulcom 7348 ax-addass 7349 ax-mulass 7350 ax-distr 7351 ax-i2m1 7352 ax-0lt1 7353 ax-1rid 7354 ax-0id 7355 ax-rnegex 7356 ax-precex 7357 ax-cnre 7358 ax-pre-ltirr 7359 ax-pre-ltwlin 7360 ax-pre-lttrn 7361 ax-pre-apti 7362 ax-pre-ltadd 7363 ax-pre-mulgt0 7364 ax-pre-mulext 7365 |
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 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-id 4083 df-xp 4406 df-rel 4407 df-cnv 4408 df-co 4409 df-dm 4410 df-iota 4933 df-fun 4970 df-fv 4976 df-riota 5546 df-ov 5593 df-oprab 5594 df-mpt2 5595 df-pnf 7426 df-mnf 7427 df-ltxr 7429 df-sub 7557 df-neg 7558 df-reap 7951 df-ap 7958 |
This theorem is referenced by: mulap0r 7991 lt2msq 8240 absext 10322 |
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